An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity
Abstract
:1. Introduction
2. Nonlinear Schrödinger Equation
2.1. Ground States
2.1.1. The Subcritical Case: $p\in (2,6)$
 (1)
 if $p<4$, then there exists a ground state for every $\mu >0$;
 (2)
 if $p\ge 4$, then:
 (i)
 whenever$${\mu}^{\frac{p2}{6p}}\mathcal{K}>{N}^{\frac{4}{6p}}{c}_{p},$$$${c}_{p}:={\left[{\left(\frac{p(p4)}{16}\right)}^{\frac{2}{p2}}+\frac{p}{8}{\left(\frac{p(p4)}{16}\right)}^{\frac{4p}{p2}}\right]}^{\frac{p2}{6p}},$$
 (ii)
 whenever$${\mu}^{\frac{p2}{6p}}\mathcal{K}<{\left(\frac{p}{2}\right)}^{\frac{2}{6p}}\frac{\mathcal{C}{(\mathcal{G},p)}^{\frac{4p}{6p}}}{\mathcal{C}{(\mathcal{G},\infty )}^{p}},$$
2.1.2. The Critical Case: $p=6$
 a graph $\mathcal{G}$ is said to admit a cyclecovering if and only if every edge of $\mathcal{G}$ belongs to a cycle, namely either a loop (i.e., a closed path of consecutive bounded edges) or an unbounded path joining the endpoints of two distinct halflines (which are then identified as a single vertex at infinity);
 a graph $\mathcal{G}$ is said to possesses a terminal edge if and only if it contains an edge ending with a vertex of degree one.
 (i)
 if $\mathcal{G}$ has at least a terminal edge (as, for instance, in Figure 6a), then$${\mu}_{\mathcal{K}}={\mu}_{{\mathbb{R}}^{+}},\phantom{\rule{2.em}{0ex}}{\mathcal{E}}_{\mathcal{G}}(\mu ,\mathcal{K})=\infty \phantom{\rule{1.em}{0ex}}for\phantom{\rule{4.pt}{0ex}}all\phantom{\rule{4.pt}{0ex}}\mu >{\mu}_{\mathcal{K}},$$
 (ii)
 if $\mathcal{G}$ admits a cyclecovering (as, for instance, in Figure 6b), then$${\mu}_{\mathcal{K}}={\mu}_{\mathbb{R}}$$
 (iii)
 if $\mathcal{G}$ has only one halfline and no terminal edges (as, for instance, in Figure 6c), then$${\mu}_{{\mathbb{R}}^{+}}<{\mu}_{\mathcal{K}}<\sqrt{3}$$
 (iv)
 if $\mathcal{G}$ has no terminal edges, does not admit a cyclecovering, but presents at least two halflines (as, for instance, in Figure 6d), then$${\mu}_{{\mathbb{R}}^{+}}<{\mu}_{\mathcal{K}}\le {\mu}_{\mathbb{R}}$$
 (1)
 the sequence ${\mathcal{G}}_{\u03f5}^{1}$ can be constructed by considering a graph whose compact core does not admit a cyclecovering (see, e.g., Figure 7a) and letting the length of one of its cutedges, the edges whose removal disconnects the graph (e.g., $\widehat{e}$ in Figure 7a), go to infinity;
 (2)
 the sequence ${\mathcal{G}}_{\u03f5}^{2}$ can be constructed by considering a graph as in Figure 7b and letting the length of the compact core go to infinity keeping at the same time the total diameter of the compact core bounded (namely, thickening the compact core);
 (3)
 the sequences ${\mathcal{G}}_{\u03f5}^{3},{\mathcal{G}}_{\u03f5}^{4}$ can be constructed by considering a signpost graph (see, e.g., Figure 7c) and letting the length of its cutedge $\tilde{e}$ go to infinity and to zero, respectively.
2.2. Bound States
2.2.1. Existence Results
 detecting the energy levels at which the PalaisSmale condition is satisfied, namely detect the values $c\in \mathbb{R}$ such that any sequence $({u}_{n})\in {\mathcal{H}}_{\mu}(\mathcal{G})$ satisfying
 (i)
 $E({u}_{n},\mathcal{K},p)\to c$
 (ii)
 $\parallel d{E}_{{\mathcal{H}}_{\mu}(\mathcal{G})}({u}_{n},\mathcal{K},p){\parallel}_{{\mathbb{T}}_{{u}_{n}}^{\prime}{\mathcal{H}}_{\mu}(\mathcal{G})}\to 0$
(with ${\mathbb{T}}_{{u}_{n}}^{\prime}{\mathcal{H}}_{\mu}(\mathcal{G})$ denoting the topological dual of the tangent to the manifold ${\mathcal{H}}_{\mu}(\mathcal{G})$ at ${u}_{n}$) admits a subsequence converging in ${\mathcal{H}}_{\mu}(\mathcal{G})$;  constructing suitable minmax levels.
2.2.2. Nonexistence Results
 (i)
 if the graph $\mathcal{G}$ satisfies$${\mu}^{\frac{p2}{6p}}\mathcal{K}<\frac{\mathcal{C}{(\mathcal{G},p)}^{\frac{4p}{6p}}}{\mathcal{C}{(\mathcal{G},\infty )}^{p}},$$
 (ii)
 if $\mathcal{G}$ is a tree (i.e., no loops) with at most one pendant (see, e.g., Figure 8), then there is no bound state of mass μ with $\lambda \ge 0$, for every $\mu >0$.
3. Nonlinear Dirac Equation
3.1. Remarks on the Dirac Operator on Graphs
3.2. Bound States
4. Nonrelativistic Limit
5. Conclusion: A Brief Summary
 (i)
 the mentioned results only concerne the “free” selfadjoint extensions of the Laplacian and the Dirac operator introduced in Section 1 and Section 3.1, respectively;
 (ii)
 in order to find bound states, in the NLS case one fixes the mass μ and studies constrained critical points of the energy functional (thus providing no information on the frequencies λ that arise naturally as Lagrange multipliers), while in the NLD case one fixes the frequency ω and discusses the connected action functional (thus losing any information on the mass of the resulting critical points);
 (iii)
 the nonrelativistic limit must be considered (as above) a limit for a sequence of relativistic parameters ${c}_{n}\to \infty $ and a suitably “tuned” sequence of frequencies ${\omega}_{n}$;
 (iv)
 since (clearly) a ground state is a bound state too, the fourth column of the Table 1 must be meant to refer to those bound states which are not ground states;
 (v)
 the constants ${\mu}_{1},{\mu}_{2}$ are defined by: ${\mu}_{1}:={N}^{\frac{4}{p2}}{c}_{p}^{\frac{6p}{p2}}{\mathcal{K}}^{\frac{p6}{p2}}$ (with $N,\phantom{\rule{0.166667em}{0ex}}{c}_{p}$ introduced in Theorem 1–item (i)) and ${\mu}_{2}:={(p/2)}^{\frac{2}{p2}}\mathcal{C}{(\mathcal{G},p)}^{\frac{4p}{p2}}{\mathcal{K}}^{\frac{p6}{p2}}\mathcal{C}{(\mathcal{G},\infty )}^{\frac{p(p6)}{p2}}$ (with $\mathcal{C}(\mathcal{G},p),\phantom{\rule{0.166667em}{0ex}}\mathcal{C}(\mathcal{G},\infty )$ introduced in (10)–(11));
 (vi)
Author Contributions
Funding
Conflicts of Interest
References
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Exponents  Ground  Bound  Connection NLSNLD  

NLSE  $p\in (2,4)$ 
 (see box below)  (see box below) 
$p\in [4,6)$ 


 
$p=6$ 


 
$p>6$ 
 (see box above) 
 
NLDE  $p\in (2,6)$ 



$p=6$ 
 (see box above) 
 
$p>6$ 
 (see box above) 

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Borrelli, W.; Carlone, R.; Tentarelli, L. An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity. Symmetry 2019, 11, 169. https://doi.org/10.3390/sym11020169
Borrelli W, Carlone R, Tentarelli L. An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity. Symmetry. 2019; 11(2):169. https://doi.org/10.3390/sym11020169
Chicago/Turabian StyleBorrelli, William, Raffaele Carlone, and Lorenzo Tentarelli. 2019. "An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity" Symmetry 11, no. 2: 169. https://doi.org/10.3390/sym11020169