Standing waves for a coupled system of nonlinear Schrödinger equations

Annali di Matematica Pura ed Applicata (1923 -) - We study the following system of nonlinear Schrödinger equations: $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\Delta u +a(x) u =...     

Global existence for a system of weakly coupled nonlinear Schrödinger equations

This paper discusses the weakly coupled nonlinear Schrödinger equations in the supercritical case. With the best constant of the Gagliardo-Nirenberg inequality, we derive a sufficient condition for the global existence of solutions; this condition is expressed in terms of stationary solutions (nonlinear ground state).     

Bifurcation in a multicomponent system of nonlinear Schrödinger equations

We consider the system ${-\Delta{u}_{j} + a(x)u_{j} = \mu_{j}u^{3}_{j} + \beta \sum_{k \neq j} u^{2}_{k}u_{j}}$ , u _j > 0, j = 1, . . . , n, on a possibly unbounded domain ${\Omega \subset \mathbb{R}^{N}, N \leq 3}$ , with Dirichlet boundary conditions. The system appears in nonlinear optics and in the analysis of mixtures of Bose–Einstein condensates. We consider the self-focussing (attractive self-interaction) case ${\mu_{1}, \ldots, \mu_{n} > 0}$ and take ${\beta \in \mathbb{R}}$ as bifurcation parameter. There exists a branch of positive solutions with uj/uk being constant for all ${j, k \in \{1, \ldots, n\}}$ . The main results are concerned with the bifurcation of solutions from this branch. Using a hidden symmetry we are able to prove global bifurcation even when the linearization has even-dimensional kernel (which is always the case when n > 1 is odd).     

Existence of solutions for a system of coupled nonlinear stationary bi-harmonic Schrödinger equations

We obtain existence and multiplicity results for the solutions of a class of coupled semilinear bi-harmonic Schrödinger equations. Actually, using the classical Mountain Pass Theorem and minimization techniques, we prove the existence of critical points of the associated functional constrained on the Nehari manifold.Furthermore, we show that using the so-called fibering method and the Lusternik–Schnirel’man theory there exist infinitely many solutions, actually a countable family of critical points, for such a semilinear bi-harmonic Schrödinger system under study in this work.     

A note on existence of antisymmetric solutions for a class of nonlinear Schr?dinger equations

We consider the nonlinear Schr?dinger equation

On a class of nonlinear Schrödinger equations

This paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic equation:

An Lp-theory for almost sure local well-posedness of the nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equations (NLS) on ${\mathbb{R}}^d$ with random and rough initial data. By working in the framework of $L^p({\mathbb{R}}^d)$ spaces, $p>2$, we prove almost sure local well-posedness for rougher initial data than those considered in the existing literature. The main ingredient of the proof is the dispersive estimate.     

Final state problem for a system of nonlinear Schrödinger equations with three wave interaction

We consider the asymptotic behavior of a solution to a system of quadratic nonlinear Schrödinger equations with three wave interaction in two dimensions. We construct a particular solution which has a mass transition phenomenon among three components periodically in time. This is based on the analysis for a system of ordinary differential equations which approximates the solution of the system of nonlinear Schrödinger equations.     

Existence of breathers for discrete nonlinear Schrödinger equations

In this paper, we obtain a new sufficient condition on the existence of breathers for the discrete nonlinear Schrödinger equations by using critical point theory in combination with periodic approximations. The classical Ambrosetti–Rabinowitz superlinear condition is improved.     

A note on existence of antisymmetric solutions for a class of nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation

$$-\triangle u + V(x)u= f(u)\quad {\rm in}\quad \mathbb{R}^N.$$

We assume that V is invariant under an orthogonal involution and show the existence of a particular type of sign changing solution. The basic tool employed here is the Concentration–Compactness Principle.

     

Loss of regularity for supercritical nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation with defocusing, smooth, nonlinearity. Below the critical Sobolev regularity, it is known that the Cauchy problem is ill-posed. We show that this is even worse, namely that there is a loss of regularity, in the spirit of the result due to G. Lebeau in the case of the wave equation. As a consequence, the Cauchy problem for energy-supercritical equations is not well-posed in the sense of Hadamard. We reduce the problem to a supercritical WKB analysis. For super-cubic, smooth nonlinearity, this analysis is new, and relies on the introduction of a modulated energy functional à la Brenier.     

Multiple positive solutions for a quasilinear system of Schrödinger equations

We consider the quasilinear system

Localized nodal solutions of higher topological type for semiclassical nonlinear Schrödinger equations

We investigate the existence of localized sign-changing solutions for the semiclassical nonlinear Schrödinger equation

$$\begin{aligned} -\epsilon ^2\Delta v+V(x)v=|v|^{p-2}v,\ v\in H^1(\mathbb {R}^N) \end{aligned}$$

where \(N\ge 2,\) \(2<p<2^*\), \(\epsilon >0\) is a small parameter, and V is assumed to be bounded and bounded away from zero. When V has a local minimum point P, as \(\epsilon \rightarrow 0\), we construct an infinite sequence of localized sign-changing solutions clustered at P and these solutions are of higher topological type in the sense that they are obtained from a minimax characterization of higher dimensional symmetric linking structure via the symmetric mountain pass theorem. It has been an open question whether the sign-changing solutions of higher topological type can be localized and our result gives an affirmative answer. The existing results in the literature have been subject to some geometrical or topological constraints that limit the number of localized sign-changing solutions. At a local minimum point of V, Bartsch et al. (Math Ann 338:147–185, 2007) proved the existence of N pairs of localized sign-changing solutions and D’Aprile and Pistoia (Ann Inst Hénri Poincare Anal Non Linéaire 26:1423–1451, 2009) constructed 9 pairs of localized sign-changing solutions for \(N\ge 3\). Our result gives an unbounded sequence of such solutions. Our method combines the Byeon and Wang’s penalization approach and minimax method via a variant of the classical symmetric mountain pass theorem, and is rather robust without using any non-degeneracy conditions.

     

A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations

Numerical Algorithms - In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and a...