Asymptotics of Regular and Irregular Solutions in Chains of Coupled van der Pol Equations

Abstract

Chains of coupled van der Pol equations are considered. The main assumption that motivates the use of special asymptotic methods is that the number of elements in the chain is sufficiently large. This allows moving from a discrete system of equations to the use of a continuity argument and obtaining an integro-differential boundary value problem as the initial model. In the study of the behaviour of all its solutions in a neighbourhood of the equilibrium state, infinite-dimensional critical cases arise in the problem of the stability of solutions. The main results include the construction of special families of quasi-normal forms, namely non-linear boundary value problems of either Schrödinger or Ginzburg–Landau type. Their solutions make it possible to determine the main terms of the asymptotic expansion of both regular and irregular solutions to the original system. The main goal is the study of chains with diffusion- and advective-type couplings, as well as fully connected chains.

1. Formulation of the Problem

The classical van der Pol equation

$$\ddot{u}+a\dot{u}+u=f(u,\dot{u}),f(u,\dot{u})=\dot{u}{u}^2$$

arises in many applied problems. Here, we consider a system of N coupled van der Pol equations

$${\ddot{u}}_j+a{\dot{u}}_j+u=f(u_j,{\dot{u}}_j)+d\sum _{j=1}^N{a}_{ij}{u}_i,j=1,\dots N.$$

(1)

Systems of this type have been considered in the works of many authors (see, for example, [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]). We will assume that the coupling coefficients of each j element are the same for all j, i.e., $a_{ij}=a_{i-j}$, and that the chain is circular, i.e, the $j\pm N$ element coincides with the j element.

Let $x_j=2\pi j{N}^{-1}(j=1,\dots N)$ be uniformly distributed points on some circle with angular coordinate $x_j$. Function $u_j\left(t\right)$ is the value of $u(t,x)$ at $x=x_j$.

In this paper, we will consider several of the most common types of coupling, namely the following.

1.

Diffusion chains. For such chains, we assume that

$$a_1=a_{-1}=1,a_0=2,a_j=0\mathrm{at}j=2,\dots N-2,$$

i.e., the second term on the right hand side of (1) has the form

$$d[u(t,x_{j+1})-2u(t,x_j)+u(t,x_{j-1})].$$

(2)

Sometimes, in the case of $d<0$, (2) is called anti-diffusion.

2.

One-way coupled chains. They are determined by the relations

$$a_1=1,a_0=-1,a_j=0\mathrm{at}j=2,\dots N-1,$$

that means that in (1) we have

$$d[u(t,x_{j+1})-u(t,x_{j-1})].$$

(3)

Sometimes the coupling (3) is called advective. Another variant of the advective coupling is often used, when

$$a_1=1,a_{-1}=1.$$

In this case, we have

$$d[u(t,x_{j+1})-u(t,x_{j-1})].$$

3.

Chains with two-way or semi-diffusion coupling. Here,

$$a_1=a_{-1}=1,a_0=a_2=\dots =a_{N-2}=0$$

and the analogue of (2) and (3) has the form

$$d[u(t,x_{j+1})+u(t,x_{j-1})].$$

(4)

4.

Fully coupled chains. In this case, we assume that

$$d\sum _{i=1}^N{a}_{ij}{u}_i=d\frac{1}{N}\sum _{i=1}^N{a}_{i}u(t,x_i).$$

The next assumption is central. We will assume that the number of N elements in (1) is sufficiently large, which means that the parameter $\epsilon =2\pi N^{-1}$ satisfies the relation

$$0<\epsilon \ll 1.$$

This condition allows us to naturally pass from the fixed variables $u(t,x_j)(j=1,\dots ,N)$ to the continuous variable $x\in [0,2\pi ]$ of function $u(t,x)$. Then, system (1) can be written in a more general form as

$$\frac{{\partial }^{2}u}{\partial t^2}+a\frac{\partial u}{\partial t}+u=f\left(u,\frac{\partial u}{\partial t}\right)+d\underset{-\infty }{\overset{+\infty }{\int }}F(s,\epsilon )u(t,x+s)ds,$$

(5)

with periodic boundary conditions

$$u(t,x+2\pi )\equiv u(t,x).$$

(6)

Note that the last term on the right-hand side of (5) can also be written as an integral

$$d\underset{0}{\overset{2\pi }{\int }}\tilde{F}(s,\epsilon )u(t,x+s)ds.$$

We will describe function $F(s,\epsilon )$ in more detail. For the values $k=0$ and $k=\pm 1$, we set

$$F_k\left(s\right)=\frac{1}{2\epsilon \sigma \sqrt{\pi }}exp\left(-{\left(\epsilon \sigma \right)}^{-2}{(s+\epsilon k)}^2\right),\sigma >0\left(\underset{-\infty }{\overset{+\infty }{\int }}{F}_k\left(s\right)ds=1\right).$$

Note that for a fixed function $u(t,x)$, for $\sigma \to 0$, we have

$$\underset{\sigma \to 0}{lim}\underset{-\infty }{\overset{+\infty }{\int }}{F}_k\left(s\right)u(t,x+s)ds=u(t,x+\epsilon k).$$

Function

$$F(s,\epsilon )=F_1\left(s\right)-2F_0\left(s\right)+F_{-1}\left(s\right)$$

(7)

generalises the diffusion coupling, since it satisfies

$$\underset{\sigma \to 0}{lim}\underset{-\infty }{\overset{+\infty }{\int }}F(s,\epsilon )u(t,x+s)ds=u(t,x+\epsilon )-2u(t,x)+u(t,x-\epsilon ).$$

Note that the coupling of diffusion type (3) is the simplest difference approximation of the diffusion operator ${\partial }^{2}u/\partial x^2$.

For semi-diffusion coupling, it is natural to assume that

$$F(s,\epsilon )=F_1\left(s\right)+F_{-1}\left(s\right).$$

The expression (3) is the simplest difference approximation of the advection (transfer) operator $\partial u/\partial x$. At the same time, we will also consider a coupling “similar” to (4):

$$u(t,x+\epsilon )-u(t,x-\epsilon ),$$

(8)

since it also approximates the advection operator to the same extent. In relation to the function $F(s,\epsilon )$, for (3), the following expression arises:

$$F(s,\epsilon )=F_1\left(s\right)-F_0\left(s\right),$$

whereas for (8)

$$F(s,\epsilon )=F_1\left(s\right)-F_{-1}\left(s\right).$$

Note that the following relations

$$\underset{-\infty }{\overset{+\infty }{\int }}{F}_k\left(s\right)exp\left(ims\right)ds=cos\left(\epsilon m\right)exp(-{\sigma }^2{\left(\epsilon m\right)}^2),\left(\underset{-\infty }{\overset{+\infty }{\int }}{F}_k\left(s\right)ds=1\right)$$

indicate the convenience of representing $F(s,\epsilon )$ in terms of functions of the form $F_k\left(s\right)$. Functions $F(s,\epsilon )$ of this type have been used in [16]. Similar expressions for $F\left(s\right)$, where $F\left(s\right)=Const·exp[-{\sigma }^2|s\left|\right]$, are also given in [17]; however, they are less convenient.

For fully coupled chains, for $N\gg 1$, the representation

$$d\frac{1}{N}\sum _{j=1}^N{a}_{i}u(t,x_i)=d\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}\phi \left(s\right)u(t,x+s)ds$$

(9)

arises naturally. Note that the case when

$$\phi \left(s\right)\equiv 1$$

seems to be the most important.

In this paper, we will study the behaviour of all the solutions to (5) and (6) with sufficiently small initial conditions (in the norm) under the constraints (7)–(9). More precisely, we will study the asymptotics as $\epsilon \to 0$, for all $t\in (t_0,\infty ),x\in [0,2\pi ]$ of functions that are sufficiently small and satisfy the boundary value problems (5) and (6) with a high degree of accuracy in $\epsilon$ (uniformly in $t,x$).

We fix the space of initial conditions

$$u(0,x)\in C_{[0,2\pi ]},\frac{\partial u}{\partial t}{|}_{t=0}\in C_{[0,2\pi ]}$$

to be the phase space of the boundary value problem (5) and (6). In addition to the model in (5) and (6), we consider another model by replacing $u(t,x+s)$ with $\partial u(t,x+s)/\partial t$ in the integral of the right-hand side of (5), namely

$$\frac{{\partial }^{2}u}{\partial t^2}+a\frac{\partial u}{\partial t}+u=f\left(u,\frac{\partial u}{\partial t}\right)+d\underset{-\infty }{\overset{+\infty }{\int }}F(s,\epsilon )\frac{\partial }{\partial t}u(t,x+s)ds.$$

We shall comment on the similarities and differences between the solutions of such models.

When studying the local behaviour of solutions, the main focus is on the study of the properties of boundary value problems linearised at zero with $2\pi$-periodic boundary conditions (6)

$$\frac{{\partial }^{2}u}{\partial t^2}+a\frac{\partial u}{\partial t}+u=d\underset{-\infty }{\overset{+\infty }{\int }}F(s,\epsilon )u(t,x+s)ds$$

(10)

and

$$\frac{{\partial }^{2}u}{\partial t^2}+a\frac{\partial u}{\partial t}+u=d\underset{-\infty }{\overset{+\infty }{\int }}F(s,\epsilon )\frac{\partial u(t,x+s)}{\partial t}ds.$$

In turn, the behaviour of solutions to these boundary value problems is determined by the location of the roots of the characteristic equation

$${\lambda }^2+a\lambda +1=d\underset{-\infty }{\overset{+\infty }{\int }}F(s,\epsilon )exp\left(iks\right)ds,$$

(11)

where $k=0,\pm 1,\pm 2,\dots$.

We now formulate some statements of a general plan which are analogues of Lyapunov’s theorems on stability in the first approximation.

Proposition 1. 

Let all roots of the characteristic Equation (11) have negative real parts and be separated from zero at $\epsilon \to 0$. Then, the zero equilibrium of (5) and (6) is asymptotically stable, and all solutions from some sufficiently small ε-independent neighbourhood of the zero equilibrium state tend to zero as $t\to \infty$.

Proposition 2. 

Let the characteristic Equation (11) have a positive real part separated from zero at $\epsilon \to 0$. Then, the zero equilibrium in (5) and (6) is unstable, and no attractors of this boundary value problem can exist in a sufficiently small ε-independent neighbourhood of the zero equilibrium state.

Thus, only such critical cases need to be considered when the characteristic Equation (11) does not have roots with positive real parts separated from zero as $\epsilon \to 0$, but there are roots whose real parts tend to zero as $\epsilon \to 0$. Below, we study the local behaviour of solutions to the boundary value problem (5) and (6) in critical cases.

An important feature of these critical cases is the fact that an infinite number of roots of the corresponding characteristic equations tend to the imaginary axis as $\epsilon \to 0$. Thus, we can say that critical cases of infinite dimension are realised. Critical cases of this type have been considered in [18,19,20]. The methodology of these works will be essentially used here.

The main result is the construction of special non-linear forms, the so-called quasi-normal forms (QNF), namely boundary value problems of parabolic type. They do not contain small parameters and their non-local dynamics determines the local behaviour of solutions to the original boundary value problems.

Solutions of quasi-normal form determine the principal terms of the asymptotic expansions of the original boundary value problems. As a rule, they are classical Ginzburg–Landau equations or integro-differential equations. Numerical methods are well developed for such equations. For the original boundary value problems, the use of numerical methods is difficult, since they are singularly perturbed and the solutions contain components that rapidly oscillate in the space or time variable. Therefore, despite the apparent complexity, quasi-normal forms are simpler objects, ready for further study using standard methods. In addition, the above quasi-normal forms allow one to immediately draw a conclusion, for example, about multistability (for quasi-normal forms in which continuum parameters are present) and about high sensitivity to parameter changes. The conclusion that many of the presented quasi-normal forms are characterised by complex irregular dynamics follows from the well-known results of the analysis of this type of system (see, for example, [21]).

The parameter $\sigma$ in the formulas for $F(s,\epsilon )$ has a clear meaning. It defines a set of chain elements that significantly affect each specific element. This effect is weaker the farther the elements are from each other. For $\sigma =0$, additional critical cases arise when infinitely many roots of the characteristic equation, which correspond to harmonics with arbitrarily large numbers, have asymptotically small real parts. To describe the dynamic properties of the problem in this situation, the condition $\sigma \ll 1$ is considered. Note that the quasi-normal form in such cases acquire an additional spatial variable, which means that there is a tendency to complicate the oscillations.

Note that the condition that the real parts of the roots of the characteristic equation are nonpositive implies that

$$a\ge 0.$$

(12)

Below, when considering critical cases for fixed coefficients $a_0$ and $d_0$, these coefficients will vary, i.e., we will assume that

$$a=a_0+{\epsilon }^2{a}_1,d=d_0+{\epsilon }^2{d}_2.$$

(13)

This is a quite important remark, since for $a>0$, all solutions of the van der Pol equation tend to zero as $t\to \infty ,$ and, for $a_0=0$ and $a_1<0$, this equation has a stable limit cycle.

The solutions studied in this paper are conditionally divided into two types: regular and irregular. Regular solutions are solutions “well dependent” on the parameter $\epsilon$, i.e., solutions for which the following asymptotic representation holds:

$$u(t,x+\epsilon )=u(t,x)+\epsilon \frac{\partial }{\partial x}u(t,x)+\frac{1}{2}{\epsilon }^2\frac{{\partial }^{2}u}{\partial y^2}u(t,x)+\dots .$$

Irregular solutions have a more complex structure, which consists of a superposition of functions smoothly (regularly) depending on the parameter $\epsilon$ and functions smoothly depending on the parameter ${\epsilon }^{-1}$.

Note that for some types of couplings, only regular or only irregular solutions can exist or both can exist simultaneously.

Regarding the methodology, for finite-dimensional critical cases in the problem of the stability of solutions, as a rule, it is possible to substantiate statements about the existence of local invariant integral manifolds whose dimension is determined by the dimension of critical cases. The initial system on such manifolds is represented as special non-linear systems of ordinary differential equations, which are called normal forms. Their non-local solutions determine the local behaviour of all solutions to the original systems as $t\to \infty$.

In this paper, we study situations where the critical cases have infinite dimensions. It is difficult, and sometimes even impossible, to substantiate the existence of an invariant manifold if it does not coincide with the entire phase space. Nevertheless, it is possible to use the formalism of the method of normal forms, which is based on the use of the structure of “critical” solutions of linearised equations. Such constructions were successfully used, for example, in the works [22,23]. The fact that there is no quadratic non-linearity in the original van der Pol equation considerably simplifies the corresponding calculations, but is not fundamental. All constructions extend to non-linear second order equations with quadratic and cubic non-linearities of general form.

The main assumption that opens the way to the application of special asymptotic methods is that the number of elements N in the chain is sufficiently large. Thus, the small parameter $\epsilon \sim N^{-1}$ arises in a natural way.

The main result is the construction of the so-called quasi-normal forms and continuum families of quasi-normal forms, which are special non-linear distributed boundary value problems that do not contain a small parameter. The non-local structure of the solutions of these quasi-normal forms determines the principal terms of the asymptotic expansions of the original problem.

The paper is organised as follows. The main content deals with the study of chains with diffusion-type couplings. Section 2, Section 3 and Appendix A.1 are devoted to this. In Appendix A.2, the results are extended to equations in which, instead of the van der Pol non-linearity, there is a conservative non-linearity describing dislocations in a solid.

In Section 4, the dynamic properties of chains with advective coupling are studied, which is an equally interesting problem. In Section 5, we will discuss the asymptotics of solutions in fully connected chains. We note that, on one hand, the results of the above sections differ significantly from each other, and on the other hand, they serve as an important addition to the existing studies.

2. Asymptotic Behaviour of Solutions in Chains with Diffusion-Type Connections

The linearised zero boundary value problem for chains (5) and (6) with diffusion-type couplings has the form (10), (6), where $F(s,\epsilon )=F_1\left(s\right)-2F_0\left(s\right)+F_{-1}\left(s\right)$. Substituting $u=exp(ikx+\lambda t)$ into these equations, we arrive at the characteristic equation

$${\lambda }^2+a\lambda +1=-4d{sin}^2\left(\frac{z}{2}\right)exp(-{\delta }^2{z}^2),$$

(14)

where $z=\epsilon k,k=0,\pm 1,\pm 2,\dots$. In order for this equation to have no roots with positive real parts, it is necessary and sufficient that the following condition is satisfied

$$1+4d\underset{z}{max}\left({sin}^2\frac{z}{2}·exp(-{\delta }^2{z}^2)\right)\ge 0.$$

(15)

In what follows, this condition is assumed to be satisfied.

2.1. Asymptotic Behaviour of Regular Solutions

Given that

$$a_0>0$$

for some fixed integer k, all roots of (14) have negative real parts separated from the imaginary axis as $\epsilon \to 0$. Therefore, the critical case in the stability problem is realised only for

$$a_0=0.$$

Then, the roots ${\lambda }_k\left(\epsilon \right)$ and ${\overline{\lambda }}_k\left(\epsilon \right)$ are complex and

$${\lambda }_k\left(\epsilon \right)=i+\epsilon {\lambda }_{1k}+o\left({\epsilon }^2\right),{\lambda }_{1k}=-\frac{1}{2}{a}_1+\frac{id{k}^2}{2}.$$

Equation (10) has a set of solutions

$$u(t,x,\epsilon )=\sum _{k=-\infty }^{\infty }{\xi }_k\left(\tau \right)exp\left(it\right),$$

(16)

and ${\xi }_k\left(\tau \right)=exp\left[({\lambda }_{1k}+o\left({\epsilon }^2\right))\tau \right],$ where $\tau ={\epsilon }^{2}t$ is a slow temporal variable. The expression (16) can be written as $u(t,x,\epsilon )=\xi (\tau ,x),$ where the Fourier coefficients $\xi (\tau ,x)$ are equal to ${\xi }_k\left(\tau \right)$. Solutions to the non-linear boundary value problem (5) and (6) are then sought in the form of a formal series

$$u(t,x,\epsilon )=\epsilon \left(\xi (\tau ,x)exp\left(it\right)+\overline{cc}\right)+{\epsilon }^3{u}_3(t,\tau ,x)+\dots .$$

(17)

The expression $\overline{cc}$ in (17) and below denotes the complex conjugate of the previous term. Here, $\xi (\tau ,x)$ is an unknown function and $u_3(t,\tau ,x)$ is $2\pi$ periodic in t and x. We substitute (17) into (5) and collect the coefficients of the same powers of $\epsilon$. Then, for $u_3=u_{30}+{\overline{u}}_{30}$, we arrive at

$$\frac{{\partial }^2{u}_{30}}{\partial t^2}+u_{30}=\left[-i{a}_1\xi -2i\frac{\partial \xi }{\partial \tau }+d\frac{{\partial }^2\xi }{\partial x^2}-i\xi {\left|\xi \right|}^2\right]exp\left(it\right)+i{\xi }^{3}exp\left(3it\right).$$

This equation is solvable with respect to $\xi (\tau ,x)$ in the specified class of functions if and only if the equalities

$$2\frac{\partial \xi }{\partial \tau }=-id\frac{{\partial }^2\xi }{\partial x^2}-a_1\xi -\xi {\left|\xi \right|}^2,\xi (\tau ,x+2\pi )\equiv \xi (\tau ,x)$$

(18)

are satisfied.

The boundary value problem (18) plays the role of a normal form. Its solutions bounded at $\tau \to \infty ,x\in [0,2\pi ]$ determine, according to (17), the asymptotics of regular solutions to the original boundary value problem (5) and (6).

At $a_1<0$, the boundary value problem (18) has an infinite number of periodic solutions ${\rho }_{0}exp\left[ikx+id{k}^2\tau /2\right]$. Note also the formula for $y(\tau ,x)={\left|\xi (\tau ,x)\right|}^2$ is

$$\underset{0}{\overset{2\pi }{\int }}\left[\frac{\partial y}{\partial \tau }+a_{1}y+y^2\right]dx=0.$$

2.2. Asymptotic Behaviour of Irregular Solutions

We fix $\delta >0$ arbitrarily and define the function

$$\gamma \left(\delta \right)={\left[1+4d{sin}^2\left(\frac{\delta }{2}\right)exp(-{\delta }^2{\sigma }^2)\right]}^{1/2}.$$

Note that

$${\gamma }^{\prime }\left(\delta \right)=2d{\gamma }^{-1}\left(\delta \right)exp(-{\delta }^2{\sigma }^2)sin\left(\frac{\delta }{2}\right)·\left[cos\frac{\delta }{2}-2\delta {\sigma }^{2}sin\frac{\delta }{2}\right].$$

Additionally, we assume that ${\gamma }^{\prime }\left(\delta \right)\ne 0$, i.e.,

$$\delta \ne \pi n\mathrm{and}ctg\frac{\delta }{2}\ne 2\delta {\sigma }^2.$$

(19)

We denote the value that completes the expression $\delta {\epsilon }^{-1}$ to an integer by $\theta =\theta (\delta ,\epsilon )\in [0,1)$. Consider the set of integers

$$K_{\epsilon }\left(\delta \right)=\{\delta {\epsilon }^{-1}+\theta +m,m=0,\pm 1,\pm 2,\dots \}.$$

Thus, $\theta \left(\epsilon \right)$ is a piecewise linear function, which, for $\epsilon \to 0$, runs over all values from zero to one infinitely many times. For the values $k\in K_{\epsilon }\left(\delta \right)$, we consider the asymptotics of the roots ${\lambda }_m^{+}\left(\epsilon \right)$ and ${\overline{\lambda }}_m^{+}\left(\epsilon \right)$ of the characteristic Equation (14). We obtain

$${\lambda }_m^{+}\left(\epsilon \right)=i\gamma \left(\delta \right)+i\epsilon (\theta +m){\gamma }^{\prime }\left(\delta \right)+{\epsilon }^2{\lambda }_{m2}^{+}\left(\epsilon \right),$$

$${\lambda }_m^{-}\left(\epsilon \right)=-i\gamma \left(\delta \right)+i\epsilon (\theta +m){\gamma }^{\prime }\left(\delta \right)+{\epsilon }^2{\lambda }_{m2}^{-}\left(\epsilon \right),$$

where $m=0,\pm 1,\pm 2,\dots$ and ${\lambda }_{m2}^{\pm }\left(\epsilon \right)$ are some functions bounded as $\epsilon \to 0$, whose explicit form is not required.

The linearised boundary value problem (10), (6) has solutions

$$exp\left(i\left(\frac{\delta }{\epsilon }+\theta +m\right)x+{\lambda }_m^{\pm }\left(\epsilon \right)t\right).$$

Hence, it follows that the solutions to the same boundary value problem are the set of solutions

$$\begin{array}{cc}\hfill u=& \sum _{k=-\infty }^{\infty }({\xi }_{m}exp\left(i(\delta {\epsilon }^{-1}+\theta +m)x+{\lambda }_m^{+}\left(\epsilon \right)t\right)+\overline{cc}+\hfill \\ \hfill & +{\eta }_{m}exp\left(i(\delta {\epsilon }^{-1}+\theta +m)x+{\lambda }_m^{-}\left(\epsilon \right)t\right)+\overline{cc}),\hfill \end{array}$$

where the quantities ${\xi }_m$ and ${\eta }_m$ are arbitrary. This expression can be rewritten in the form

$$\begin{array}{c}\hfill u_1=U_1(t,\tau ,x,\epsilon )=exp\left[i(\delta {\epsilon }^{-1}+\theta )x+i(\gamma \left(\delta \right)+\epsilon \theta {\gamma }^{\prime }\left(\delta \right))t\right](\xi (\tau ,x^{+})+\overline{cc})+\\ \hfill +exp\left[i(\delta {\epsilon }^{-1}+\theta )x-i(\gamma \left(\delta \right)+\epsilon \theta {\gamma }^{\prime }\left(\delta \right))t\right](\eta (\tau ,x^{-})+\overline{cc}),\end{array}$$

(20)

where $\tau ={\epsilon }^{2}t,$

$$x^{\pm }=x\pm \epsilon {\gamma }^{\prime }\left(\delta \right)t,$$

and the Fourier coefficients of the functions $\xi (\tau ,x)$ and $\eta (\tau ,x)$ satisfy the following

$${\xi }_m\left(\tau \right)={\xi }_{m}exp\left({\lambda }_{m2}^{+}\left(\epsilon \right)\tau \right),{\eta }_m\left(\tau \right)={\eta }_{m}exp\left({\lambda }_{m2}^{-}\left(\epsilon \right)\tau \right).$$

Note that due to condition (19), the values $x^{\pm }$ do not coincide with x. Based on the obtained representation of solutions of the linear equation, we shall look for solutions to the non-linear boundary value problem (5) and (6) in the form

$$u(t,x,\epsilon )=\epsilon u_1+{\epsilon }^2{u}_2+{\epsilon }^3{u}_3.$$

(21)

Functions $u_j$ are $2\pi {\left(\gamma \left(\delta \right)+\epsilon \theta {\gamma }^{\prime }\left(\delta \right)\right)}^{-1}$ periodic in t and $2\pi$ periodic in x. The expression for $u_1$ is the same as (20), with the only difference that now $\xi (\tau ,x)$ and $\eta (\tau ,x)$ are unknown complex amplitudes.

We substitute (21) into (5) and successively collect coefficients of the same powers of $\epsilon$. In the first step, we arrive at an equation for $u_1$, whose solution is presented in (20). In the second step, we obtain the same equation for $u_2:$

$$\frac{{\partial }^2{u}_2}{\partial t^2}+u_2=d\underset{-\infty }{\overset{\infty }{\int }}F(s,\epsilon )u_2(t,\tau ,x+s)ds.$$

We fix $u_2$ to be the function

$$u_2=E^{+}(t,x,\epsilon )f^{+}(t_1,\tau ,x)+\overline{cc}+E^{-}(t,x,\epsilon )f^{-}(t_1,\tau ,x)+\overline{cc},$$

(22)

where

$$E^{\pm }(t,\tau ,x)=exp\left(i(\delta {\epsilon }^{-1}+\theta )x\pm i(\gamma \left(\delta \right)+\epsilon \theta {\gamma }^{\prime }\left(\delta \right))t\right),t_1=\epsilon t,$$

and the expression $f(t_1,\tau ,x)$ is not defined at this step. It will be chosen in the next step.

The next step is central. We obtain an equation for $u_3$ of the form

$$\begin{array}{cc}\hfill & \frac{{\partial }^2{u}_3}{\partial t^2}+u_3-d\underset{-\infty }{\overset{\infty }{\int }}F(s,\epsilon )u_3(t,\tau ,x+s)ds=E^{+}(t,\tau ,x)B_0^{+}(t_1,x,\tau )+\hfill \\ \hfill & +E^{-}(t,\tau ,x)B_0^{-}(t_1,x,\tau )+E^{+}(t,\tau ,x)B^{+}(x^{+},\tau )+E^{-}(t,\tau ,x)B^{-}(x^{-},\tau )+\overline{cc}+\hfill \\ \hfill & +E^{3}B(t_1,x,x^{+},x^{-},\tau )+\overline{cc}.\hfill \end{array}$$

Here, we use the notation

$$\begin{array}{cc}\hfill B_0^{+}(t_1,x,\tau )& =-2i\gamma \left(\delta \right)\left[\left(\frac{\partial f^{+}}{\partial t_1}+\frac{{\gamma }^{\prime }\left(\delta \right)}{\gamma \left(\delta \right)}\frac{\partial f^{+}}{\partial x}\right)+2\xi \left({\left|\eta \right|}^2-{M\left(\right|\eta |}^2)\right)\right],\hfill \\ \hfill B_0^{-}(t_1,x,\tau )& =2i\gamma \left(\delta \right)\left[\left(\frac{\partial f^{-}}{\partial t_1}-\frac{{\gamma }^{\prime }\left(\delta \right)}{\gamma \left(\delta \right)}\frac{\partial f^{-}}{\partial x}\right)-2\eta \left({\left|\xi \right|}^2-{M\left(\right|\xi |}^2)\right)\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill B^{+}(x^{+},\tau )& =-i\gamma \left(\delta \right)\frac{\partial \xi }{\partial \tau }-i{\gamma }^{″}\left(\delta \right)\left(-\frac{{\partial }^2\xi }{\partial x^2}+2i\theta \frac{\partial \xi }{\partial x}+{\theta }^2\xi \right)-i\gamma \left(\delta \right)a_1\xi -\hfill \\ \hfill & -{\xi \left(\right|\xi |}^2+{2M\left(\right|\eta |}^2\left)\right),\hfill \\ \hfill B^{-}(x^{-},\tau )& =i\gamma \left(\delta \right)\frac{\partial \eta }{\partial \tau }+i{\gamma }^{″}\left(\delta \right)\left(-\frac{{\partial }^2\eta }{\partial x^2}+2i\theta \frac{\partial \eta }{\partial x}+{\theta }^2\eta \right)+i\gamma \left(\delta \right)a_1\eta -\hfill \\ \hfill & -{\eta \left(\right|\eta |}^2+{2M\left(\right|\xi |}^2\left)\right).\hfill \end{array}$$

In these formulas

$$M\left(\phi \left(x\right)\right)=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}\phi \left(x\right)dx.$$

The expression for $B(t_1,x,x^{+},x^{-},\tau )$ is not given, since it will not be used to find $\xi$ and $\eta$, only when deriving a formula to determine $u(t,x,\epsilon )$ in (21). Note that $B_0^{\pm }$ and $B^{\pm }$ have different arguments. Therefore, to solve Equation (22) in the specified class of functions, it is necessary to require that the equalities

$$B_0^{\pm }\equiv B^{\pm }\equiv 0$$

are satisfied. We consider the equalities $B_0^{\pm }=0$ as equations for the unknown functions $f^{\pm }(t_1,x)$. Its solutions can be written explicitly. One can verify that

$$\begin{array}{c}\hfill f^{+}(t_1,x,\tau )=-{\left(2{\gamma }^{\prime }\left(\delta \right)\right)}^{-1}\xi (\tau ,x^{+})\underset{0}{\overset{x^{-}}{\int }}\left({\left|\eta (\tau ,x)\right|}^2-{M\left(\right|\eta (\tau ,x)|}^2)\right)dx,\\ \hfill f^{-}(t_1,x,\tau )={\left(2{\gamma }^{\prime }\left(\delta \right)\right)}^{-1}\eta (\tau ,x^{-})\underset{0}{\overset{x^{+}}{\int }}\left({\left|\xi (\tau ,x)\right|}^2-{M\left(\right|\xi (\tau ,x)|}^2)\right)dx.\end{array}$$

(23)

Then, from the equalities $B^{\pm }=0$, we obtain that $\xi$ and $\eta$ satisfy the following

$$2i\gamma \left({\delta }_0\right)\frac{\partial \xi }{\partial \tau }={\gamma }^{″}\left({\delta }_0\right)\left[\frac{{\partial }^2\xi }{\partial x^2}+2i\theta \frac{\partial \xi }{\partial x}-{\theta }^2\xi \right]-i\gamma \left({\delta }_0\right)a_1\xi -i\gamma \left({\delta }_0\right){\xi \left(\right|\xi |}^2+{2M\left(\right|\eta |}^2\left)\right),$$

(24)

$$-2i\gamma \left({\delta }_0\right)\frac{\partial \eta }{\partial \tau }={\gamma }^{″}\left({\delta }_0\right)\left[\frac{{\partial }^2\eta }{\partial x^2}-2i\theta \frac{\partial \eta }{\partial x}-{\theta }^2\eta \right]+i\gamma \left({\delta }_0\right)a_1\eta +i\gamma \left({\delta }_0\right){\eta \left(\right|\eta |}^2+{2M\left(\right|\xi |}^2\left)\right),$$

(25)

$$\xi (\tau ,x+2\pi )\equiv \xi (\tau ,x),\eta (\tau ,x+2\pi )\equiv \eta (\tau ,x).$$

(26)

The main result is that the boundary value problem (24)–(26) plays the role of a normal form for (5) and (6), when considering solutions with modes from the set $K_{\epsilon }\left(\delta \right)\cap K_{\epsilon }(-\delta )$. In order to formulate the corresponding result more precisely, we introduce some notation. We arbitrarily fix ${\theta }_0\in [0,1)$. By ${\epsilon }_n={\epsilon }_n(\delta ,{\theta }_0)$, we denote a sequence ${\epsilon }_n\to 0(n\to \infty ),$ where the value of ${\theta }_0({\epsilon }_n,\delta )$ does not change.

Theorem 1. 

We fix arbitrarily ${\theta }_0\in [0,1)$. Let $\xi (\tau ,x),\eta (\tau ,x)$ be a solution of the boundary value problem (24)–(26). Then, for sufficiently small values of $\epsilon ={\epsilon }_n(\delta ,{\theta }_0)$, the function (21) for $\xi =\xi (\tau ,x^{+}),\eta =\eta (\tau ,x^{-})$ satisfies the boundary value problem (5) and (6) up to $o\left({\epsilon }_n^3\right)$.

We note again that, in this section, we have studied the asymptotic behaviour of solutions whose modes belong to the set $K\left(\delta \right)$. It is natural to call the value $\delta {\epsilon }^{-1}+\theta$ the base mode for such solutions.

2.3. Quasinormal Form in the Case ${\gamma }^{\prime }\left(\delta \right)=0$

Let the condition

$${\gamma }^{\prime }\left({\delta }_0\right)=0$$

(27)

be satisfied for some value $\delta ={\delta }_0$. For example, ${\delta }_0$ could be ${\delta }_0=2\pi n_0,$ where $n_0>0$ and is an integer. The expression $2\pi n_0{\epsilon }^{-1}$ is an integer because $\epsilon =2\pi N^{-1}$, which means $\theta (2\pi n_0,\epsilon )=0$.

Under condition (27), the constructions are substantially simplified. This follows from the fact that $x^{+}=x^{-}=x$. In the asymptotic representation (21), we have $u_2=0,$ and hence $f^{\pm }=0$. Here, we restrict ourselves to presenting the final boundary value problem for the amplitudes $\xi (\tau ,x)$ and $\eta (\tau ,x)$:

$$\begin{array}{c}\hfill 2i\gamma \left({\delta }_0\right)\frac{\partial \xi }{\partial \tau }={\gamma }^{″}\left({\delta }_0\right)\left[\frac{{\partial }^2\xi }{\partial x^2}+2i\theta \frac{\partial \xi }{\partial x}-{\theta }^2\xi \right]-i\gamma \left({\delta }_0\right)a_1\xi -i\gamma \left({\delta }_0\right){\xi \left(\right|\xi |}^2+{2\left|\eta \right|}^2),\end{array}$$

(28)

$$\begin{array}{c}\hfill -2i\gamma \left({\delta }_0\right)\frac{\partial \eta }{\partial \tau }={\gamma }^{″}\left({\delta }_0\right)\left[\frac{{\partial }^2\eta }{\partial x^2}-2i\theta \frac{\partial \eta }{\partial x}-{\theta }^2\eta \right]+i\gamma \left({\delta }_0\right)a_1\eta +i\gamma \left({\delta }_0\right){\eta \left(2\right|\xi |}^2+{\left|\eta \right|}^2),\end{array}$$

(29)

$$\xi (\tau ,x+2\pi )\equiv \xi (\tau ,x),\eta (\tau ,x+2\pi )\equiv \eta (\tau ,x).$$

(30)

The final result is the assertion of Theorem 1. It should be noted that in the case of ${\delta }_0=2\pi n_0$ in (28)–(30), and hence in Theorem 1, the value $\theta$ is zero.

2.4. Quasinormal Form in the Case $\gamma \left(\delta \right)=0$

Let $\gamma \left({\delta }_0\right)=0$ be satisfied for some ${\delta }_0>0$. In this case, we conclude that

$$1+4d{sin}^2\frac{{\delta }_0}{2}·exp(-{\delta }^2{\sigma }^2)=0.$$

(31)

Hence, taking into account condition (15), we conclude that ${\gamma }^{\prime }\left({\delta }_0\right)=0,{\gamma }^{″}\left({\delta }_0\right)<0$ and ${\delta }_0$ is the first positive root of the equation

$$4ctg\frac{{\delta }_0}{2}={\sigma }^2{\delta }_0.$$

Finding ${\delta }_0,$ from (31) we find the value of the coefficient d. For coefficient a in (5), it is convenient to take the equality $a=\epsilon a_1$.

In the case of (31), the situation is even more simplified. In the analogue of the asymptotic representation (21), we have

$$u(t,x,\epsilon )={\epsilon }^{1/2}\left(\xi (t_1,x)exp\left(i({\delta }_0{\epsilon }^{-1}+\theta )x\right)+\overline{cc}\right)+{\epsilon }^{3/2}{u}_3(t_1,x),t_1=\epsilon t.$$

(32)

Substituting (32) into (5), after some calculations we obtain the quasi-normal form

$$\begin{array}{cc}\hfill \frac{{\partial }^2\xi }{\partial t_1^2}+a_1\frac{\partial \xi }{\partial t_1}=& -{\gamma }^{″}\left({\delta }_0\right)\left(\frac{{\partial }^2\xi }{\partial x^2}-i\theta \frac{\partial \xi }{\partial x}-{\theta }^2\xi \right)-2{\left|\xi \right|}^2\frac{\partial \xi }{\partial t_1}-{\xi }^2\frac{\partial \overline{\xi }}{\partial t_1},\hfill \\ \hfill & \xi (\tau ,x+2\pi )\equiv \xi (\tau ,x)\hfill \end{array}$$

(33)

as the main result. Note that the linear and non-linear components in (33) differ significantly from the above quasi-normal forms.

In Appendix A.1, we consider the question of constructing a CNF for finding the amplitudes of multi-frequency solutions.

3. Equations with a Small Parameter $\mathbf{\sigma }$

In this section, we assume that the parameter $\sigma$ appearing in the definition of $F(s,\epsilon )$ is sufficiently small:

$$\sigma =\epsilon {\sigma }_1.$$

(34)

Therefore, in the case under consideration, the function $F_0(s,\epsilon )$ is essentially “closer” to the $\delta$-function.

In addition, here we study the influence of variations in the value of N on the asymptotic behaviour of the solutions. We fix an arbitrary integer value c, and let the number of elements in (1) be $N+c$.

3.1. Asymptotic Behaviour of Solutions with One Base Mode

We set

$$\mu =\frac{2\pi }{N+c}.$$

On the right-hand side of Equation (5), the parameter $\epsilon$ is replaced by $\mu$; that is, instead of $F(s,\epsilon )$, we have $F(s,\mu )$:

$$\frac{{\partial }^{2}u}{\partial t^2}+\epsilon a_1\frac{\partial u}{\partial t}+u+f(u,\dot{u})=d\underset{-\infty }{\overset{+\infty }{\int }}F(s,\mu )u(t,x+s)ds.$$

(35)

Considering that $\epsilon =2\pi N^{-1},$ we obtain

$$\mu =\epsilon \left[1-\frac{c}{2\pi }\epsilon +\frac{c^2}{{\left(2\pi \right)}^2}{\epsilon }^2+\dots \right].$$

The boundary value problem (35), (6) linearised at zero has the characteristic equation

$${\lambda }^2+\epsilon a_1\lambda +1=-4d{sin}^2\left(\frac{z}{2}\right)exp(-{\epsilon }^2{\sigma }_1^2{z}^2),z=\mu k,k=0,\pm 1,\pm 2,\dots .$$

(36)

As in the previous section, we consider the question of constructing the asymptotes of solutions to (35), (6) based on modes from $K\left(\delta \right)=\{\delta {\epsilon }^{-1}+\theta +m,m=0,\pm 1,\pm 2,\dots ,\delta >0\}$. We consider the most important case when $\delta$ is not an integer multiple of $2pi:\delta \ne 2{\pi }_j$.

The roots ${\lambda }_m\left(\epsilon \right)(m=0,\pm 1,\pm 2,\dots )$ of the characteristic Equation (36) corresponding to modes from $K\left(\delta \right)$ have amplitude

$${\lambda }_m\left(\epsilon \right)=i\gamma \left(\delta \right)+\epsilon {\lambda }_{m1}+{\epsilon }^2{\lambda }_{m2}+\dots ,\gamma \left(\delta \right)={\left(1+4{sin}^2\frac{\delta }{2}\right)}^{1/2},$$

where

$${\lambda }_{m1}=id{\left(2\gamma \left(\delta \right)\right)}^{-1}sin\left(\delta \right)(m+\theta -\delta c{\left(2\pi \right)}^{-1}),$$

$${\lambda }_{m2}=-\frac{1}{2}{a}_1-i{A}_0-i{A}_1(m+\theta -\delta c{\left(2\pi \right)}^{-1})-i{A}_2{(m+\theta -\delta c{\left(2\pi \right)}^{-1})}^2.$$

The coefficients $A_0,A_1$ and $A_2$ satisfy

$$\begin{array}{cc}\hfill A_0& =2d{\sigma }^2{\delta }^2{\left(sin\frac{\delta }{2}\right)}^2·{\gamma }^{-1}\left(\delta \right),\hfill \\ \hfill A_1& =dcsin\delta ·{\left(4\pi \gamma \left(\delta \right)\right)}^{-1},\hfill \\ \hfill A_2& =d[{(sin\delta )}^2-2{\gamma }^2\left(\delta \right)cos\delta ]{\left(8\gamma \left(\delta \right)\right)}^{-3}.\hfill \end{array}$$

The solution of the linearised equation for (35) corresponding to the root ${\lambda }_m\left(\epsilon \right)$ can be written as

$$u_m(t,x,\epsilon )={\xi }_m(\tau ,x^{+})E^{+}+{\eta }_m(\tau ,x^{-})E^{-},$$

where $\tau ={\epsilon }^{2}t,x^{\pm }=x\pm 2\epsilon d,$

$$\begin{array}{c}\hfill E^{\pm }=exp\left[i(\delta {\epsilon }^{-1}+\theta -c\delta {\left(2\pi \right)}^{-1})x\pm i(\gamma \left(\delta \right)+\epsilon {\left(2\gamma \left(\delta \right)\right)}^{-1}·(\theta -\delta c{\left(2\pi \right)}^{-1})sin\delta )t\right],\\ \hfill {\xi }_m\left(\tau \right)={\xi }_{m}exp\left(({\lambda }_{m2}+O\left(\epsilon \right))\tau \right),{\eta }_m\left(\tau \right)={\eta }_{m}exp\left(({\overline{\lambda }}_{m2}+O\left(\epsilon \right))\tau \right),\end{array}$$

where ${\xi }_m$ and ${\eta }_m$ are arbitrary complex constants. As in the previous sections, we seek the asymptotes of the solutions to the non-linear boundary value problem (5) and (6) in the form

$$\begin{array}{cc}\hfill u=& {\epsilon }^{1/2}\left(\xi (\tau ,x)E^{+}+\overline{cc}+\eta (\tau ,x)E^{-}+\overline{cc}\right)+\epsilon \left(f^{+}{E}^{+}+\overline{cc}+f^{-}{E}^{-}+\overline{cc}\right)+\hfill \\ & +{\epsilon }^{3/2}{u}_3(t,\tau ,x)+\dots .\hfill \end{array}$$

(37)

The functions $f^{\pm }=f^{\pm }(t_1,\tau ,x)$ are defined by the same formulas as in (23), where $t_1=\epsilon t$ and

$$\gamma \left(\delta \right)={\left(1+4{sin}^2\frac{\delta }{2}\right)}^{1/2}.$$

We substitute (37) into (5). We collect the coefficients at ${\epsilon }^{3/2},$ and in order to determine the known amplitudes $\xi$ and $\eta$, we obtain the boundary value problem

$$\begin{array}{cc}\hfill \frac{\partial \xi }{\partial \tau }=& -\frac{1}{2}{a}_1\xi -i{A}_0\xi -i{A}_1\left(\theta -c\delta {\left(2\pi \right)}^{-1}-\frac{i\partial }{\partial x}\right)\xi -i{A}_2{\left(\theta -c\delta {\left(2\pi \right)}^{-1}-\frac{i\partial }{\partial x}\right)}^2\xi -\hfill \\ & -\frac{1}{2}\xi \left({\left|\xi \right|}^2+{2M\left(\right|\eta |}^2)\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \frac{\partial \eta }{\partial \tau }=& -\frac{1}{2}{a}_1\eta +i{A}_0\eta +i{A}_1\left(\theta -c\delta {\left(2\pi \right)}^{-1}-\frac{i\partial }{\partial x}\right)\eta +i{A}_2{\left(\theta -c\delta {\left(2\pi \right)}^{-1}-\frac{i\partial }{\partial x}\right)}^2\eta -\hfill \\ & -\frac{1}{2}\eta \left({\left|\eta \right|}^2+{2M\left(\right|\xi |}^2)\right),\hfill \end{array}$$

$$\xi (\tau ,x+2\pi )\equiv \xi (\tau ,x),\eta (\tau ,x+2\pi )\equiv \eta (\tau ,x).$$

The main result is that this boundary value problem plays the role of a normal form. For every fixed ${\theta }_0\in [0,1)$, its non-local solutions bounded as $\tau \to \infty ,x\in [0,2\pi ]$, allow us to construct asymptotes for sufficiently small functions ${\epsilon }_n={\epsilon }_n\left({\theta }_0\right)$ that satisfy the original boundary value problem (5) and (6) up to $o\left({\epsilon }^{3/2}\right)$.

Remark 1. 

The roles of the coefficients c and ${\sigma }_1$ are determined by the above formulas. Under the conditions of the next Section 3.2, these coefficients play a much more important role.

3.2. The Case of an Infinite Set of Basic Modes

Significantly new and interesting points arise when considering the asymptotics of solutions containing various infinite sets of basic modes. Here, we emphasise an important class of solutions whose base modes belong to the set

$$K_{\infty }\left(\delta \right)=\{\delta {\epsilon }^{-1}+\theta +2\pi n{\epsilon }^{-1}+m,m,n=0,\pm 1,\pm 2,\dots \}.$$

The roots ${\lambda }_{mn}\left(\epsilon \right)$ of the characteristic equation corresponding to these modes have the asymptotics

$${\lambda }_{mn}\left(\epsilon \right)=i\gamma \left(\delta \right)+\epsilon {\lambda }_{mn1}+{\epsilon }^2{\lambda }_{mn2},$$

where

$$\begin{array}{cc}\hfill {\lambda }_{mn1}& =ib\left(m-cn+\theta -\frac{\delta c}{2\pi }\right),b={\left(\gamma \left(\delta \right)\right)}^{-1}sin\delta ,\hfill \\ \hfill {\lambda }_{mn2}& =-\frac{1}{2}{a}_1+B_1\left({(m-cn)}^2+2(m-cn)\left(\theta -\frac{\delta c}{2\pi }\right)+{\left(\theta -\frac{\delta c}{2\pi }\right)}^2\right)+B_2{(\delta +2\pi n)}^2,\hfill \\ \hfill B_1& =-i{\left(2\gamma \left(\delta \right)\right)}^{-1}\left({\left(\gamma \left(\delta \right)\right)}^{-2}{sin}^2\delta -cos\delta \right),B_2=-i2{\left(\gamma \left(\delta \right)\right)}^{-1}{\sigma }_1^2{sin}^2\frac{\delta }{2}.\hfill \end{array}$$

The solutions of the linear boundary value problem $u_{mn}(t,x,\epsilon )$ corresponding to ${\lambda }_{mn}\left(\epsilon \right)$ can be written as

$$\begin{array}{cc}\hfill & u_{mn}(t,x,\epsilon )={\xi }_{mn}\left(\tau \right)exp\left[i(\delta {\epsilon }^{-1}+\theta )x+im{x}^{+}+2\pi in{y}^{+}+i\left(\gamma \left(\delta \right)+\epsilon b\left(\theta -\frac{\delta c}{2\pi }\right)\right)t\right],\hfill \\ \hfill & x^{\pm }=x\pm \epsilon bt,y^{\pm }=y\mp \epsilon cbt,y=2\pi {\epsilon }^{-1}x.\hfill \end{array}$$

Therefore, in order to construct the asymptotes of the solutions to the boundary value problem, we use the expressions

$$u(t,x,\epsilon )=U(t,x,\epsilon )+O\left({\epsilon }^2\right)$$

(38)

and

$$\begin{array}{cc}\hfill U(t,x,\epsilon )=& {\epsilon }^{1/2}\left(\xi (\tau ,x^{+},y^{+})E_c^{+}+\overline{cc}+\eta (\tau ,x^{-},y^{-})E_c^{-}+\overline{cc}\right)+\hfill \\ \hfill & +\epsilon \left(f_c^{+}(t_1,x,y)E_c^{+}+\overline{cc}+f_c^{-}(t_1,x,y)E_c^{-}+\overline{cc}\right)+\hfill \\ \hfill & +{\epsilon }^{3/2}\left(H_1^{+}(\tau ,x,y)E_c^{+}+\overline{cc}+H_1^{-}(\tau ,x,y)E_c^{-}+\overline{cc}\right)+\hfill \\ \hfill & +H_3^{+}(\tau ,x,y){\left(E_c^{+}\right)}^3+\overline{cc}+H_3^{-}(\tau ,x,y){\left(E_c^{-}\right)}^3+\overline{cc}\hfill \end{array}$$

(39)

where $t_1=\epsilon t$ and $E_c^{\pm }$ is given by

$$E_c^{\pm }=exp\left[i\left(\delta {\epsilon }^{-1}+\theta -\frac{\delta c}{2\pi }\right)x\pm i\left(\gamma \left(\delta \right)+\epsilon b\left(\theta -\frac{\delta c}{2\pi }\right)\right)t\right].$$

Substituting (38) and (39) into (5), after simple calculations, we obtain equations for $H_1^{\pm }$ and $H_3^{\pm }$. The functions $H_3^{\pm }$ are determined from these equations, and the solvability conditions for $H_1^{\pm }$ allow us to determine $f_c^{\pm }$ and obtain equations for $\xi (\tau ,x,y)$ and $\eta (\tau ,x,y)$. The difficulty lies in the fact that the equations for determining $H_1^{\pm }$ include functions whose spatial arguments $x^{+},y^{+}$ and $x^{-},y^{-}$ are different; for some of these are $x^{+}$ and $y^{+}$, while for others $x^{-}$ and $y^{-}$. In non-linearity, these arguments are present in different factors. The purpose of the constructions being carried out is to obtain systems of boundary value problems for determining the unknown functions $\xi$ and $\eta$ with the same arguments.

In order to have arbitrariness in the choice of functions $f^{\pm }$, we consider the question of the solvability in the class of $2\pi$-periodic functions in x and y of the equation

$$\frac{\partial g(x,y)}{\partial x}-c\frac{\partial g(x,y)}{\partial y}=p(x,y),$$

(40)

where $p(x,y)$ is $2\pi$ periodic in x and y. To do this, we introduce some notation. By $D,J$ and $J_0$, we denote [24] operators defined on continuously differentiable functions $v(x,y)$ of two variables x and y, acting according to the rules

$$\begin{array}{ccc}\hfill Dv(x,y)& =& \frac{\partial v}{\partial x}-c\frac{\partial v}{\partial y},\hfill \\ \hfill Jv(x,y)& =& \underset{0}{\overset{x}{\int }}v(s,cx+y-cs)ds,\hfill \\ \hfill J_{0}v(x,y)& =& \frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}v(s,cx+y-cs)ds.\hfill \end{array}$$

Under the condition

$$J_{0}p(x,y)=0$$

there exists a periodic solution of Equation (40) in both variables, which is given by

$$g(x,y)=Jp(x,y).$$

Note that an arbitrary $2\pi$-periodic function $p(x,y)$ in x and y can be represented as

$$p=J_{0}p-(1-J_0)p$$

and the following is satisfied

$$J_0(p-J_{0}p)=0.$$

Below, we shall need the following relations, which follow from these definitions:

$$Dv(cx+y)=0,D(J-J_0)v(x,y)=DJv(x,y)=v(x,y).$$

The cubic non-linearity $R=(i\gamma \xi E_c^{+}+\overline{cc}+i\gamma \eta E_c^{-}+\overline{cc})·{(\xi E_c^{+}+\overline{cc}+\eta E_c^{-}+\overline{cc})}^2$ for $E_c^{\pm }$ implies the following

$$R=-{i\gamma \left(\delta \right)\xi \left(\right|\xi |}^2+{2\left|\eta \right|}^2{)-i\gamma \left(\delta \right)\eta (\left|\eta \right|}^2+{2\left|\xi \right|}^2).$$

It is convenient to represent the above expression as a sum of four terms

$$\begin{array}{c}\hfill R_1=-{i\gamma \left(\delta \right)\xi \left(\right|\xi |}^2+2J_0{\left|\eta \right|}^2),R_2=-{i\gamma \left(\delta \right)\xi \left(2\right|\eta |}^2-2J_0{\left|\eta \right|}^2),\\ \hfill R_3=-{i\gamma \left(\delta \right)\eta \left(\right|\eta |}^2+2J_0{\left|\xi \right|}^2),R_4=-{i\gamma \left(\delta \right)\eta \left(2\right|\xi |}^2-2J_0{\left|\xi \right|}^2).\end{array}$$

The functions $R_1$ and $R_3$ depend only on $x^{+},y^{+}$ and $x^{-},y^{-}$, respectively, while both of the functions $R_2$ and $R_4$ depend on all arguments. We manage the arbitrariness in the choice of $f^{\pm }$ in such a way as to “remove” the terms $R_2$ and $R_4$ in the equation for $H_1^{\pm }$. From here, we arrive at:

$$\begin{array}{c}\hfill 2i\gamma \left(\delta \right)\frac{\partial f^{+}}{\partial t_1}=2i(sin\delta )D{f}^{+}-i\gamma \left(\delta \right)\xi \left({2\left|\eta \right|}^2-2J_0{\left|\eta \right|}^2\right),\end{array}$$

(41)

$$\begin{array}{c}\hfill -2i\gamma \left(\delta \right)\frac{\partial f^{-}}{\partial t_1}=2i(sin\delta )D{f}^{-}-i\gamma \left(\delta \right)\eta \left({2\left|\xi \right|}^2-2J_0{\left|\xi \right|}^2\right).\end{array}$$

(42)

In (41), we set $f^{+}(t_1,x,y)=\xi (\tau ,x^{+},y^{+})·g(t_1,x,y)$. Then, we obtain

$$2i\gamma \left(\delta \right)\frac{\partial g^{+}}{\partial t_1}-2i(sin\delta )·D{g}^{+}=-2i\gamma \left(\delta \right)\left(|\eta (\tau ,x^{-},y^{-}){|}^2-J_0{\left|\eta (\tau ,x^{-},y^{-})\right|}^2\right).$$

Then, for $x=x^{-}-b{t}_1,y=y^{-}+cb{t}_1$, we arrive at the equation for $g^{+}=g^{+}(x,y)$:

$$bD{g}^{+}={\left|\eta (\tau ,x,y)\right|}^2-J_0{\left(\right|\eta (\tau ,x,y)|}^2).$$

Hence, we conclude that

$$g^{+}(x,y)=b^{-1}\left(J\left(\right|\eta (\tau ,x,y){|}^2-J_0\left|\eta (\tau ,x,y){|}^2\right)\right).$$

Analogously, we find that $f^{-}(t_1,x,y)=\eta (\tau ,x^{-},y^{-})·g(t_1,x,y)$ and

$$g^{-}(x,y)=-b^{-1}\left(J\left(\right|\xi (\tau ,x,y){|}^2-J_0\left|\xi (\tau ,x,y){|}^2\right)\right).$$

Let us formulate the main result. Consider the boundary value problem

$$\begin{array}{cc}\hfill \frac{\partial \xi }{\partial \tau }=& {\left(\delta -i\frac{\partial }{\partial y}\right)}^2\xi +i{B}_1\left[-D^2-2i\left(\theta -\frac{c\delta }{2\pi }\right)D+{\left(\theta -\frac{c\delta }{2\pi }\right)}^2\right]\xi +\hfill \\ \hfill & +i{B}_2\left(\theta -\frac{c\delta }{2\pi }-iD\right)\xi -\frac{1}{2}{a}_1\xi -\frac{1}{2}\xi \left({\left|\xi \right|}^2+2J_0{\left(\right|\eta |}^2)\right),\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill \frac{\partial \eta }{\partial \tau }=& {\left(\delta -i\frac{\partial }{\partial y}\right)}^2\eta -i{B}_1\left[-D^2-2i\left(\theta -\frac{c\delta }{2\pi }\right)D+{\left(\theta -\frac{c\delta }{2\pi }\right)}^2\right]\eta -\hfill \\ \hfill & -i{B}_2\left(\theta -\frac{c\delta }{2\pi }-iD\right)\eta -\frac{1}{2}{a}_1\eta -\frac{1}{2}\eta \left({\left|\eta \right|}^2+2J_0{\left(\right|\xi |}^2)\right),\hfill \end{array}$$

(44)

$$\xi (\tau ,x+2\pi ,y)\equiv \xi (\tau ,x,y)\equiv \xi (\tau ,x,y+2\pi ),$$

(45)

$$\eta (\tau ,x+2\pi ,y)\equiv \eta (\tau ,x,y)\equiv \eta (\tau ,x,y+2\pi ).$$

(46)

Recall that

$$Df=\frac{\partial f}{\partial x}-c\frac{\partial f}{\partial y}.$$

By ${\epsilon }_k\left({\theta }_0\right)(k=k_0,k_0+1,\dots )$, we denote a sequence such that ${\epsilon }_k\left({\theta }_0\right)\to 0$ for $k\to \infty$ and $\theta \left({\epsilon }_k\left({\theta }_0\right)\right)={\theta }_0$.

Theorem 2. 

Let (34) be satisfied. We fix arbitrarily $\delta \ne 2\pi k(k=0,1,\dots ),{\theta }_0\in [0,1)$ and an integer c. In addition, let $\mu =N+c$ and let $\left(\xi (\tau ,x,y),\eta (\tau ,x,y)\right)$ for their derivatives with respect to τ and let their second order derivatives with respect to x and y be bounded functions as $\tau \to \infty ,x\in [0,2\pi ],y\in [0,2\pi ]$. Moreover, let $\left(\xi (\tau ,x,y),\eta (\tau ,x,y)\right)$ be a solution to the boundary value problem (43)–(46) for $\theta ={\theta }_0$. Then, the function $U(t,x,{\epsilon }_k)$, for $\tau ={\epsilon }_k\left({\theta }_0\right)t$, $x^{\pm }=x\pm {\epsilon }_{k}bt,$ $y^{\pm }=y\mp {\epsilon }_{k}cbt,$ and $y=2\pi {\epsilon }_k^{-1}x$ satisfies the boundary value problem (5) and (6) up to $o\left({\epsilon }_k^3\right)$.

3.3. Examples

We consider two cases. In the first of them, we assume that $K_{\infty }=\{2\pi n{\epsilon }^{-1}+m;m,n=0,\pm 1,\pm 2,\dots \}$ and $\sigma =\epsilon {\sigma }_1,\delta ={\epsilon }^{2}t,y=2\pi {\epsilon }^{-1}x,a={\epsilon }^2{a}_1,d=d_0+{\epsilon }^2{d}_1$ and $4d_0>-1$. Then, the final quasi-normal form is

$$\begin{array}{c}\hfill \frac{\partial \xi }{\partial \tau }=-\frac{a_1}{2}\xi -i\frac{d_0}{8}{D}^2\xi -i\frac{d_0{\sigma }_1^2}{2}\frac{{\partial }^2\xi }{\partial y^2}-\frac{1}{2}\xi {\left|\xi \right|}^2,\\ \hfill \xi (\tau ,x+2\pi ,y)\equiv \xi (\tau ,x,y)\equiv \xi (\tau ,x,y+2\pi ).\end{array}$$

The function $\xi (\tau ,x,y)$ determines the asymptotes of the solutions of (5) and (6) according to the formula

$$u(t,x,\epsilon )={\epsilon }^{1/2}\left(\xi (\tau ,x,y)exp\left(it\right)+\overline{cc}\right)+O\left({\epsilon }^{3/2}\right).$$

It is interesting to compare this result with the situation considered in Appendix A.1.2 (see Formula (A8)).

In the second case, we assume that

$$d_0=-\frac{1}{4}.$$

It is convenient to assume that the value of N is even. Otherwise, it suffices to replace the integer c by $c+1$. Then, the expression $\pi {\epsilon }^{-1}$ is an integer, and hence $\theta =0$.

Consider the basic set of modes $K_{\infty }=\{\pi (2n+1){\epsilon }^{-1}+m;m,n=0,\pm 1,\pm 2,\dots \}$. Here, we assume that $\sigma =\epsilon {\sigma }_1,a=\epsilon a_1,t_1=\epsilon t$ and $d=d_0+{\epsilon }^2{d}_1$. As a result, we arrive at the real quasi-normal form

$$\begin{array}{c}\hfill \frac{{\partial }^2\xi }{\partial t_1^2}+a_1\frac{\partial \xi }{\partial t_1}+d_1\xi =4^{-2}{D}^2\xi +{\sigma }_1^2\frac{{\partial }^2\xi }{\partial y^2}-{\xi }^2\frac{\partial \xi }{\partial t_1},\\ \hfill \xi (\tau ,x+2\pi ,y)\equiv \xi (\tau ,x,y)\equiv -\xi (\tau ,x,y+\pi )\end{array}$$

where $u(t,x,\epsilon )={\epsilon }^{1/2}\xi (\tau ,x,y)+O\left({\epsilon }^{3/2}\right)$. The obtained result differs significantly from the case considered in Section 2.4.

In Appendix A.2, the results obtained are applied to the problem of dislocations in a solid.

4. Advective Coupling

Here, we consider the equation

$$\frac{{\partial }^{2}u}{\partial t^2}+a\frac{\partial u}{\partial t}+u=f\left(u,\frac{\partial u}{\partial t}\right)+d\underset{-\infty }{\overset{\infty }{\int }}(F_{\epsilon }\left(s\right)-F_{-\epsilon }\left(s\right))u(t,x+s)ds$$

(47)

with boundary conditions (6).

The equation linearised at zero has the form

$$\frac{{\partial }^{2}u}{\partial t^2}+a\frac{\partial u}{\partial t}+u=d\underset{-\infty }{\overset{\infty }{\int }}(F_{\epsilon }\left(s\right)-F_{-\epsilon }\left(s\right))u(t,x+s)ds.$$

(48)

We investigate the roots of the characteristic equation for (48)

$${\lambda }^2+a\lambda +1=2idg\left(z\right)sinz,$$

(49)

where $g\left(z\right)=exp\left(-{\sigma }^2{z}^2\right),z=\epsilon k,k=0,\pm 1,\pm 2,\dots$. Let the following nondegeneracy condition be satisfied

$$a>0.$$

(50)

We consider separately two cases depending on the value of parameter $\delta$. First, in Section 4.1, we will focus on “average” values of this parameter, i.e., the value $\delta >0$ is supposed to be somehow fixed. In Section 4.2, we will consider the case of sufficiently small values of $\delta$.

4.1. The Case of “Average” Values of $\sigma$

For $d=0$, all roots of (49) have negative real parts. We denote the smallest (if it exists) value of the parameter d by $d_0^{+}>0$, for which there exists a value $z_0^{+}$ such that, for $d=d_0^{+}$ and $z=z_0^{+}$, Equation (49) has a purely imaginary root $\lambda =i\omega (\omega >0)$. We denote the largest (if it exists) value of the parameter d by $d_0^{-}<0$, for which there exists $z_0^{-}$ such that, for $d=d_0^{-}$ and $z=z_0^{-}$, Equation (49) has a purely imaginary root $\lambda =i\omega (\omega >0)$. Let $z_0$ be the first positive root of equation $tgz={\left(2\sigma z\right)}^{-1}$. We set $g_0=\underset{z}{max}g\left(z\right)sinz$. Then, $g_0=g\left(z_0\right)sin{z}_0$.

Lemma 1. 

We fix arbitrarily $a_0>0$. We have:

$$\omega =1,d_0^{+}=\frac{1}{2}a{g}_0^{-1},d_0^{-}=-\frac{1}{2}a{g}_0^{-1},z_0^{+}=z_0,z_0^{-}=-z_0.$$

Lemma 2. 

Assume that the following relations hold:

$$d_0^{-}<d<d_0^{+}.$$

Then, all roots of (49) have negative real parts and are separated from zero as $\epsilon \to 0$.

Lemma 3. 

Let one of the followings relations,

$$d_0>d_0^{+}\mathrm{or}{d}_0<d_0^{-},$$

hold. Then, Equation (49) has a positive real part separated from zero as $\epsilon \to 0$.

The proofs of these Lemmas are simple, and thus we omit them.

From Lemmas 1–3, it follows that, for $d=d_0^{\pm }$, in the boundary value problem (48), (6), the critical cases are realised in the stability problem. In (47) we set

$$a=a_0+{\epsilon }^2{a}_1,d=d_0^{+}+{\epsilon }^2{d}_1.$$

(51)

Let us find the asymptotes as $\epsilon \to 0$ of all the roots ${\lambda }_m^{\pm }\left(\epsilon \right)$ and ${\overline{\lambda }}_m^{\pm }\left(\epsilon \right)(m=0,\pm 1,\pm 2,\dots )$ of Equations (49) whose real parts tend to zero. We first introduce some more notation. Let $\Theta =\Theta (\epsilon ,z)\in [0,1)$ be the value that completes the expression $\left|z\right|{\epsilon }^{-1}$. By $g^0$, we denote the expression

$${\left.g^0=\frac{d^2}{d{z}^2}(g\left(z\right)sinz)\right|}_{z=z_0}.$$

Note that $g^0<0$. Consider the set of integer values $k=z_0{\epsilon }^{-1}+\Theta +m(m=0,\pm 1,\pm 2,\dots )$. Then, substituting them in (49), we have the equality $z=z_0+\epsilon (\Theta +m)$.

Lemma 4. 

The assympotic equalities

$${\lambda }_m^{+}\left(\epsilon \right)=i+{\epsilon }^2(L_0{(\Theta +m)}^2+L_1)+O\left({\epsilon }^4\right),$$

(52)

where

$$\begin{array}{cc}\hfill L_0& ={(a_0+4)}^{-1}(i{a}_0-2)2d_0{g}^0,\hfill \\ \hfill L_1& ={(a_0+4)}^{-1}(i{a}_0-2)(2d_0{g}_0-a_1),\hfill \end{array}$$

hold.

Note that formula (52) does not change when the value $d_0^{+}$ in (51) is replaced by $d_0^{-}$.

Equation (48), for $m=0,\pm 1,\pm 2,\dots$, has the solution

$$u_m(t,x,\epsilon )=exp\left[i\left(z_0^{\pm }{\epsilon }^{-1}+\Theta +m\right)x+(i+{\epsilon }^2(L_0{(\Theta +m)}^2+L_1)+O\left({\epsilon }^4\right))t\right].$$

At the next stage, we introduce the non-linear boundary value problem

$$\frac{\partial \xi }{\partial \tau }=L_0\frac{{\partial }^2\xi }{\partial x^2}+2i{L}_0\Theta \frac{\partial \xi }{\partial x}+(L_1-L_0{\Theta }^2)\xi +\sigma \xi {\left|\xi \right|}^2,$$

(53)

$$\xi (\tau ,x+2\pi )\equiv \xi (\tau ,x),$$

(54)

where $\sigma =-{(a_0^2+4)}^{-1}(2+i{a}_0)$. Moreover, we define function $U_0(\tau ,x)$ by

$$U_0(\tau ,x)={\xi }^{3}iAexp\left(i\left(3\left(z_0{\epsilon }^{-1}+\Theta \right)\right)x+3it\right)-{\overline{\xi }}^{3}i\overline{A}exp\left(-i\left(3\left(z_0{\epsilon }^{-1}+\Theta \right)x\right)-3it\right),$$

where $A=-{(a_0^2+4)}^{-1}(2+i{a}_0)$. Below, by ${\epsilon }_n={\epsilon }_n\left({\Theta }_0\right)$ we denote a sequence ${\epsilon }_n\to 0,$ on which $A=-{(a_0^2+4)}^{-1}(2+i{a}_0)$. We formulate the main result of this section.

Theorem 3. 

Let conditions (51) be satisfied. We arbitrarily fix ${\Theta }_0\in [0,1)$. Let $\xi (\tau ,x)$ be a bounded, for $\tau \to \infty ,x\in [0,2\pi ]$, solution to the boundary problem (53) and (54) for $\Theta ={\Theta }_0$. Then, the function

$$\begin{array}{cc}\hfill u_0(t,x,\epsilon )=& \epsilon (\xi (\tau ,x)exp(i(z_0{\epsilon }^{-1}+\Theta )x+it)+\hfill \\ \hfill & +\overline{\xi }(\tau ,x)exp(-i(z_0{\epsilon }^{-1}+\Theta )x-it))+{\epsilon }^3{U}_0(\tau ,x),\hfill \end{array}$$

for $\epsilon ={\epsilon }_n\left({\Theta }_0\right)$, satisfies the boundary value problem (47), (6) up to $o\left({\epsilon }_n^3\left({\Theta }_0\right)\right)$.

4.2. Advective Connection for Small Values of the Parameter $\sigma$

Here, we assume that the parameter $\sigma$ appearing in $g\left(z\right)$ is sufficiently small. For some ${\sigma }_1$, the condition

$$\sigma =\epsilon {\sigma }_1$$

(55)

is satisfied. In this case, we have $\omega =1,d_0^{+}=a_0/2,d_0^{-}=-a_0/2$. The main difference is that for each z, the equality $g\left(z\right)=1-{\epsilon }^2{\sigma }_1^2{z}^2+O\left({\epsilon }^4\right)$ holds. Thus, the values of $z_0^{\pm }$ are not uniquely determined: $z_0^{+}=2\pi n+\pi /2,z_0^{-}=2\pi n-\pi /2(n=0,\pm 1,\pm 2,\dots )$.

Let us make one simplifying assumption. Let the number of elements of the considered chain N be a multiple of four. Then, the values of $z_n^{\pm }{\epsilon }^{-1}$ are integers for all $n,$ and therefore $\Theta (\epsilon ,z_n^{\pm })=0$.

Asymptotic formulas similar to (52) for the roots of ${\lambda }_{m,n}^{+}\left(\epsilon \right),{\overline{\lambda }}_{m,n}^{+}\left(\epsilon \right)$, $(m,n=0,\pm 1,\pm 2$, $\dots )$, whose real parts tend to zero as $\epsilon \to 0,$ have the form

$${\lambda }_{mn}^{+}\left(\epsilon \right)=i+{\epsilon }^2{(a_0^2+4)}^{-1}(2+i{a}_0)\left[d_1-a_1-d_0\left(\frac{1}{2}{m}^2+{\sigma }^2{\left(2\pi n+\frac{\pi }{2}\right)}^{2}2\right)\right]+O\left({\epsilon }^4\right).$$

These roots correspond to solutions of the linear boundary value problem

$$u_{m,n}(t,x)=exp\left[i\left(\left(2\pi n+\frac{\pi }{2}\right){\epsilon }^{-1}+m\right)x+(i+O\left({\epsilon }^2\right))t\right].$$

Therefore, we seek formal solutions of (47) in the form

$$\begin{array}{cc}\hfill u(t,x,\epsilon )=& \epsilon \left(\xi (\tau ,x,y)exp(i\pi {\left(2\epsilon \right)}^{-1}x+it)+\overline{\xi }(\tau ,x,y)exp(-i\pi {\left(2\epsilon \right)}^{-1}x-it)\right)+\hfill \\ \hfill & +{\epsilon }^{3}U(t,\tau ,x,y),\tau ={\epsilon }^{2}t,y=n{\epsilon }^{-1}x,\hfill \end{array}$$

(56)

where $U(t,\tau ,x,y)$ is $2\pi$ periodic with respect to t and x and 1-periodic with respect to y.

Substitute (56) into (47). After standard operations, we obtain an equation for $U(t,\tau ,x,y)$. From the condition of its solvability in the indicated class of functions, we arrive at a boundary value problem for determining the amplitude $\xi (\tau ,x,y):$

$$\begin{array}{cc}\hfill \frac{\partial \xi }{\partial \tau }=& {(a_0^2+4)}^{-1}(2+i{a}_0)[-d_0\left(\frac{1}{2}\frac{{\partial }^2\xi }{\partial x^2}+{\sigma }^2\frac{{\partial }^2\xi }{\partial y^2}+i{\sigma }^2\frac{\pi }{2}\frac{\partial \xi }{\partial y}\right)+\hfill \\ \hfill & +\left(d_1-a_1-d_0{\sigma }^2\frac{{\pi }^2}{4}\right)\xi -\xi {\left|\xi \right|}^2],\hfill \end{array}$$

(57)

$$\xi (\tau ,x+2\pi ,y)\equiv \xi (\tau ,x,y)\equiv \xi (\tau ,x,y+1).$$

(58)

Thus, we justify the following result.

Theorem 4. 

Let (50), (51) and (55) be satisfied and ${\xi }_0(\tau ,x,y)$ be bounded as $\tau \to \infty ,x\in [0,2\pi ],y\in [0,1]$, by the solution of the boundary value problem (57) and (58). Then, function (56) for $\xi ={\xi }_0(\tau ,x,y)$ satisfies the boundary value problem (47), (6) up to $o\left({\epsilon }^3\right)$.

Note that Equations (53) and (57) are equations of Ginzburg–Landau type. It is known (see, for example, [21]) that their solutions can have complex structures, including irregular. In this respect, Equation (57) is much more complicated than (53) because it contains two spatial variables.

5. Fully Coupled Chains of van der Pol Equations

It suffices to consider the most important example of such chains of the form

$$\ddot{u_j}+{\epsilon }^2{a}_1\dot{u_j}+u_j-\dot{u}{u}^2=d\frac{1}{N}\sum _{k+1}^n{u}_k,j=1,\dots ,N.$$

For sufficiently large values of N, we pass to the boundary value problem

$$\frac{{\partial }^{2}u}{\partial t^2}+{\epsilon }^2{a}_1\frac{\partial u}{\partial t}+u-u^2\frac{\partial u}{\partial t}=d\underset{0}{\overset{1}{\int }}u(t,s)ds,$$

(59)

$$u(t,x+1)\equiv u(t,x).$$

(60)

The behaviour of the solutions of this boundary value problem differs significantly from the cases considered above. Here, we briefly dwell on the consideration of two cases depending on the value of the coefficient d.

5.1. The Case of Small Values of the Coefficient d

We assume that for some fixed value d, the condition

$$d=\epsilon d_1.$$

(61)

For $\epsilon =0$, the linearised boundary value problem

$$\frac{{\partial }^{2}u}{\partial t^2}+u=0,u(t,x+1)\equiv u(t,x)$$

has the same characteristic equation,

$${\lambda }^2+1=0,$$

(62)

for all modes $exp\left(2\pi ikx\right),k=0,\pm 1,\pm 2,\dots$. Therefore, under condition (61), we consider the critical case of an infinite set of pairs of purely imaginary roots with resonances $1:1:\dots$. We seek solutions of the boundary value problem (59) and (60) based on solutions of (62) in the form

$$u(t,x,\epsilon )=U(t,x,\epsilon )+O\left({\epsilon }^4\right),$$

(63)

$$\begin{array}{cc}U(t,x,\epsilon )=& \epsilon \left(\xi (\tau ,x)exp\left(it\right)+\overline{cc}\right)+{\epsilon }^3(u_{31}(\tau ,x)exp\left(it\right)+\overline{cc}+\\ & +u_{33}(\tau ,x)exp\left(3it\right)+\overline{cc}),\tau ={\epsilon }^{2}t.\end{array}$$

(64)

Substituting (63) and (64) into (59) and performing standard operations, we find an expression for $u_{33},$ and from the condition that the equation is solvable with respect to $u_{31}$, we obtain a boundary value problem for determining the unknown amplitude $\xi (\tau ,x)$. As a result, we arrive at the relations

$$\frac{\partial \xi }{\partial \tau }=-\frac{1}{2}{a}_1\xi -\frac{id}{2}\underset{0}{\overset{1}{\int }}\xi (\tau ,s)ds-\frac{1}{2}\xi {\left|\xi \right|}^2,$$

(65)

$$\xi (\tau ,x+1)\equiv \xi (\tau ,x).$$

(66)

$$u_{31}(\tau ,x)=0,u_{33}(\tau ,x)=-\frac{i}{8}{\xi }^3(\tau ,x).$$

The next statement says that this boundary value problem plays the role of a quasi-normal form for (59) and (60).

Theorem 5. 

Let (61) be satisfied and the boundary value problem (65) and (66) have a bounded solution $\xi (\tau ,x)$, for $\tau \to \infty ,x\in [0,1]$. Then, function $U(t,\tau ,x)$ satisfies the original boundary value problem (59) and (60) up to $O\left({\epsilon }^4\right)$.

5.2. Quasi-Normal Form for “Average” Values of the Parameter d

A boundary value problem linearised at zero for (59) and (60),

$$\frac{{\partial }^{2}u}{\partial t^2}+{\epsilon }^2{a}_1\frac{\partial u}{\partial t}+u=d\underset{0}{\overset{1}{\int }}u(t,s)ds,u(t,x+1)\equiv u(t,x),$$

(67)

has the characteristic equation

$${\lambda }_k^2+{\epsilon }^2{a}_1{\lambda }_k+1=\left\{\begin{array}{cc}d,\hfill & \mathrm{if}k=0,\hfill \\ 0\hfill & \mathrm{if}k\ne 0.\hfill \end{array}\right.$$

Therefore, there is a pair of roots ${\lambda }_0\left(\epsilon \right)$ and ${\overline{\lambda }}_0\left(\epsilon \right),$ where ${\lambda }_0\left(\epsilon \right)=i{\omega }_0+O\left(\epsilon \right)$ and ${\omega }_0={(1-d)}^{1/2}$, and infinitely many identical pairs of roots ${\lambda }_k\left(\epsilon \right)$ and ${\overline{\lambda }}_k\left(\epsilon \right)$ and ${\lambda }_k\left(\epsilon \right)=i+O\left(\epsilon \right)$$(k=\pm 1,\pm 2,\dots )$. These roots correspond to solutions of the linear boundary value problem (67)

$$u_k(t,x,\epsilon )=exp(2\pi ikx+{\lambda }_k\left(\epsilon \right)t).$$

Following the above method, we look for solutions to the non-linear boundary value problem (59) and (60) in the form $u(t,x,\epsilon )=U(t,x,\epsilon )+O\left({\epsilon }^4\right),$ where

$$U(t,x,\epsilon )=\epsilon \left(\xi (\tau ,x)exp\left(it\right)+\overline{cc}+\eta \left(\tau \right)exp\left(i{\omega }_{0}t\right)+\overline{cc}\right)+{\epsilon }^3{u}_3(t,\tau ,x).$$

(68)

It is important to keep in mind that the Fourier coefficient of function $\xi (\tau ,x)$ for a zero-mode harmonic is zero, so

$$M\left(\xi (\tau ,x)\right)\equiv \underset{0}{\overset{1}{\int }}\xi (\tau ,x)dx=0.$$

We substitute (68) into (59). After standard operations, we arrive at a system of equations for determining the unknown amplitudes $\xi (\tau ,x)$ and $\eta \left(\tau \right)$:

$$\frac{\partial \xi }{\partial \tau }=-\frac{1}{2}{a}_1\xi -\frac{1}{2}\left[{\xi \left|\xi \right|}^2-{M\left(\xi \right|\xi |}^2{)+2\xi |\eta |}^2\right],$$

(69)

$$\xi (\tau ,x+1)\equiv \xi (\tau ,x).$$

(70)

$$\frac{\partial \eta }{\partial \tau }=-\frac{1}{2}{a}_1\eta -\frac{1}{2}\eta \left[{\left|\eta \right|}^2+{2M\left(\right|\xi |}^2)\right].$$

(71)

Let us formulate the final result.

Theorem 6. 

Let $d<1$ and the boundary value problem (69)–(71) have a bounded solution $\xi (\tau ,x),\eta \left(\tau \right)$ as $\tau \to \infty ,x\in [0,1]$. Then, function $U(t,\tau ,x,\epsilon )$ satisfies the boundary value problem (59) and (60) up to $O\left({\epsilon }^4\right)$.

In Appendix A.3, the results obtained are applied to the problem of vibrations of pedestrian bridges.

6. Conclusions

Non-linear integro-differential boundary value problems that arise in the study of various chains of coupled van der Pol equations were investigated. Critical cases are singled out in the problem of the stability of the equilibrium state. An important conclusion is that these critical cases have infinite dimensions. A special technique was developed for studying the local behaviour of solutions in critical cases based on the construction of quasi-normal forms for finding the amplitude of solutions to the original boundary value problem. The above quasi-normal forms can be conditionally divided into three groups. The first group includes continuum families of equations depending on some parameters of Schrödinger type, in which the linear part is the same as in the Schrödinger equation. The number of parameters included in such quasi-normal forms is determined by the number of basic—asymptotically large—modes of the studied classes of solutions. Accordingly, the critical cases in the problem of stability are almost naturally called continual here. For an infinitely large number of such modes, an equation of the Schrödinger type arises with two spatial variables. Such quasi-normal forms are characteristic of chains with diffusion-type couplings. They are discussed in Section 2 and Section 3 and Appendix A.1.

The second type of quasi-normal forms describes solutions that include one basic mode. It is determined from the condition of the presence of a “point” critical case in the problem under study. Such quasi-normal forms are Ginzburg–Landau-type boundary value problems. Here, we can talk about complex dynamics which are characteristic of the Ginzburg–Landau equation with one or two spatial variables. Hence, it follows that in the original problem, the structure of the solutions can be complex. Quasi-normal forms of the second group are presented in Section 4.

Quasi-normal forms of the third group are typical for problems describing fully connected chains of equations. These quasi-normal forms are special non-linear integro-differential equations. Section 5 shows that they can have interesting families of solutions which are discontinuous in the spatial variable.

Note that in each of the above problems, we study the asymptotes of both regular (i.e., solutions that smoothly depend on a small parameter) and irregular solutions that have regular components, as well as solutions that smoothly depend on some large parameter. In the latter case, this leads to the appearance of solutions rapidly oscillating in the spatial variable.

In this work, with the help of solutions to quasi-normal forms, functions are constructed that satisfy the original boundary value problem with high accuracy. Even in regular cases, we are not talking about the asymptotes of exact solutions. The same conclusion also applies to the works of other authors (see, for example, [25]). From the point of view of problems of mathematical physics, the obtained conclusions are sufficient. In some cases, more accurate results can be obtained. For example, in the case when a quasi-normal form has a rough solution periodic in $\tau$ and some nondegeneracy-type conditions are satisfied, then it is possible to substantiate the existence of an exact solution (torus) for the original boundary value problem of the same stability and with the same asymptotes as the cycle in the quasi-normal form.

The proposed methods can also be used to study chains with other types of couplings with more general non-linearities, as well as with Neumann- or Dirichlet-type boundary conditions. In this connection, we note the problem of dislocations in a solid given as an application.

Note that under the condition ${\gamma }^{\prime }\left(\delta \right)\ne 0$, where $\gamma \left(\delta \right)$ is the oscillation frequency of solutions with base mode $\delta {\epsilon }^{-1}$, the solutions contain different spatial variables $x^{\pm }$ and $y^{\pm }$. In order to obtain a quasi-normal form with the same spatial variables and spatial derivatives, certain efforts had to be made. Auxiliary functions $f^{\pm }$ were introduced and special partial differential equations were solved to determine these functions.

It is worth noting that, under the condition $\gamma \left(\delta \right)=0$, i.e., in the non-oscillatory case, the corresponding quasi-normal forms differ significantly from the quasi-normal forms in the case of $\gamma \left(\delta \right)\ne 0$ (see Formula (33) and the formulas in Section 3.3). Note that oscillations in chains can also occur when the van der Pol equation itself has only a stable stationary solution.

It may be of interest to study chains of coupled van der Pol equations in which the integral term is replaced by

$$d\underset{-\infty }{\overset{\infty }{\int }}F(s,\epsilon )\frac{\partial u(t,x+s)}{\partial t}ds.$$

The proposed methods also extend to the study of solutions with an arbitrary number of basic modes, including those in the presence of resonance relations. In particular, the role of resonance relations is illustrated by considering problems with an infinite number of basic modes.

We dwell separately on the role of the parameters $\sigma ,\theta$ and c. The parameter $\sigma$ characterises the depth of the connection between the elements of the chain. For a large $\sigma$, this relationship is significant only between neighbouring elements, while for a small $\sigma$, the influence of relatively distant elements increases. If for a relatively large $\sigma$, the quasi normal form contains derivatives with respect to only one spatial variable, then for a small $\sigma$, differential operators contain derivatives with respect to two spatial variables. Obviously, the complexity of the solutions in the latter case increases.

Many of the above quasi-normal forms contain the parameter $\theta$, which varies from 0 to 1 depending on the number of elements, N, in the chain. For different $\theta$, the properties of the solutions of quasi-normal forms can change significantly [26]. Therefore, as N increases, an infinite process of direct and inverse bifurcations in the quasi-normal form is possible, and hence it is possible in the original system.

The integer parameter c also shows the changes in the properties of solutions when the number of elements in the chain changes from N to $N+c$. It is worth noting that this parameter is the coefficient of the second spatial derivative in problems with an infinite number of basic modes of solutions.