Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term

The orbital stability of standing waves of nonlinear Schrödinger equations with a general nonlinear term is investigated in this paper. We study the corresponding minimizing problem with L ²-constraint: $$E_\alpha = \inf\left\{\frac{1}{2}\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx - \int\limits_{\mathbb{R}^N} F(|u|) dx; u \in H^1(\mathbb{R}^N), \|u\|_{L^2(\mathbb{R}^N)}^2=\alpha\right\}.$$ We discuss when a minimizing sequence with respect to E _α is precompact. We prove that there exists α ₀ ≥ 0 such that there exists a global minimizer if α > α ₀ and there exists no global minimizer if α < α ₀. Moreover, some almost critical conditions which determine α ₀ = 0 or α ₀ > 0 are established, and the existence results with respect to ${E_{\alpha_0}}$ under some conditions are obtained.     

Higher dimensional vortex standing waves for nonlinear Schrödinger equations

We study standing wave solutions to nonlinear Schrödinger equations, on a manifold with a rotational symmetry, which transform in a natural fashion under the group of rotations. We call these vortex solutions. They are higher dimensional versions of vortex standing waves that have been studied on the Euclidean plane. We focus on two types of vortex solutions, which we call spherical vortices and axial vortices.     

Instability of standing waves for a class of inhomogeneous Schrödinger equations with harmonic potential

Ricerche di Matematica - This notes studies the inhomogeneous non-linear Schrödinger equations with a harmonic potential $$\begin{aligned} i\partial _tu +\Delta u-|x|^2u+|x|^{b}|u|^{p-1}u=0....     

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 =...     

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:

Existence of standing waves of nonlinear Schrödinger equations with potentials vanishing at infinity

For a singularly perturbed nonlinear elliptic equation ${\epsilon }^2\mathrm{\Delta }u?V(x)u+u^p=0$, $x\in {\mathbb{R}}^N$, we prove the existence of bump solutions concentrating around positive critical points of V when nonnegative V is not identically zero for $p\in (\frac{N}{N?2},\frac{N+2}{N?2})$ or nonnegative V satisfies ${\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}_{|x|\to \infty }V(x){|x|}^2\log |x|>0$ for $p=\frac{N}{N?2}$.     

Vector solitary waves for a class of nonlocal nonlinear Schrödinger system

In this paper, we study a class of one-dimensional nonlocal nonlinear Schrödinger system, which describes two-color optical beams propagating through a cell with nematic liquid crystals. The existence of local and global solutions is derived first upon applying the Strichartz's estimates, conservation laws, and fixed points theorem. Then, we prove the existence of positive normalized vector solitary wave solutions by using variational approach and the concentration-compactness technique.     

Radial symmetry of standing waves for nonlinear fractional Hardy–Schrödinger equation

In this paper, by applying the method of moving planes, we conclude the conclusions for the radial symmetry of standing waves for a nonlinear Schrödinger equation involving the fractional Laplacian and Hardy potential. First, we prove the radial symmetry of solution under the condition of decay near infinity. Based upon that, under the condition of no decay, by the Kelvin transform, we establish the results for the non-existence and radial symmetry of solution.     

Solitary waves for a class of quasilinear Schrödinger equations in dimension two

In this paper we prove the existence and concentration behavior of positive ground state solutions for quasilinear Schrödinger equations of the form ?ε ²Δu + V(z)u ? ε ² [Δ(u ²)]u = h(u) in the whole two-dimension space where ε is a small positive parameter and V is a continuous potential uniformly positive. The main feature of this paper is that the nonlinear term h(u) is allowed to enjoy the critical exponential growth with respect to the Trudinger–Moser inequality and also the presence of the second order nonhomogeneous term [Δ(u ²)]u which prevents us to work in a classical Sobolev space. Using a version of the Trudinger–Moser inequality, a penalization technique and mountain-pass arguments in a nonstandard Orlicz space we establish the existence of solutions that concentrate near a local minimum of V.     

Standing waves with a critical frequency for nonlinear Schrödinger equations,II

For elliptic equations of the form , where the potential V satisfies , we develop a new variational approach to construct localized bound state solutions concentrating at an isolated component of the local minimum of V where the minimum value of V can be positive or zero. These solutions give rise to standing wave solutions having a critical frequency for the corresponding nonlinear Schrödinger equations. Our method allows a fairly general class of nonlinearity f(u) including ones without any growth restrictions at large.Received: 5 July 2002, Accepted: 24 October 2002, Published online: 14 February 2003The research of the first author was supported by Grant No. 1999-2-102-003-5 from the Interdisciplinary Research Program of the KOSEF.     

Instability of standing waves for a weakly coupled non-linear Schrödinger system

This article discusses the weakly coupled non-linear Schrödinger equations. With the variational characterization of the ground state solutions, the potential well argument and the concavity method, we derive a sharp condition for blow-up and global existence to the solutions of the Cauchy problem. At the same time, we also prove the instability of standing waves.     

Shock waves in discrete nonlinear Schrödinger equations

It is shown that in nonlinear differential–difference equations, which in the continuum limit are reduced to the nonlinear Schrödinger equation, localized excitations, reminding shock waves in liquids and gasses can propagate. Such waves may have either the conventional profile of shock waves or a shape of dark pulses evolving against a background. At the initial stages of evolution the shock waves are described by the equation u_t+uu_x=0 and split out in a train of soliton-like pulses after the shock is developed.     

Standing waves for discrete nonlinear Schrödinger equations: sign-changing nonlinearities

We consider the discrete nonlinear Schrödinger equation with infinitely growing potential and sign-changing power nonlinearity. Making use of critical point theory, we prove an existence and multiplicity result for standing wave solutions.