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:

On a complex fundamental solution of the Schrödinger equation

A second-order Schrödinger differential operator of parabolic type is considered, for which an explicit form of a fundamental solution is derived. Such operators arise in many problems of physics, and the fundamental solution plays the role of the Feynman propagation function.     

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).     

Numerical approximation of solution for the coupled nonlinear Schrödinger equations

In this article, a compact finite difference scheme for the coupled nonlinear Schrödinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and convergence in maximum norm with order O(τ² + h⁴) are also proved by the discrete energy method. Finally, numerical results are provided to verify the theoretical analysis.     

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 =...     

Multiple positive solutions for a quasilinear system of Schrödinger equations

We consider the quasilinear system

On the solution of a generalized Schrödinger-type equation

Lu(t)+(u,F)g(t)=f(t), tS. , ( F, g). .

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The authors wish to thank Professor Yu. A. Rozanov for his help and discussions.     

On the well-posedness for stochastic fourth-order Schrödinger equations

The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations,

$iu_t + \Delta ^2 u + \varepsilon \Delta u + \lambda |u|^{p - 1} u = \dot \xi ,(t,x) \in \mathbb{R}^ + \times \mathbb{R}^n ,n \geqslant 1,\varepsilon \in \{ - 1,0, + 1\} ,$

, is investigated in this paper. The local well-posedness in the energy space H ²(? ⁿ ) are proved for \(p > \tfrac{{n + 4}}{{n + 2}}\), and p ≤ 2^# ? 1 if n ≥ 5. Global existence is also derived for either defocusing or focusing L ²-subcritical nonlinearities.

     

On decay of the Schrödinger resolvent

We strengthen the known Agmon-Jensen-Kato decay of the resolvent for a special case of the Schrödinger equation in arbitrary dimension n ≥ 1. The decay is of crucial importance in applications to linear and nonlinear hyperbolic PDEs.     

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).     

On a bifurcation value related to quasi-linear Schrödinger equations

By virtue of numerical arguments we study a bifurcation phenomenon occurring for a class of minimization problems associated with the so-called quasi-linear Schrödinger equation, object of various investigations in the last two decades.     

On the stability of self-similar solutions of 1D cubic Schrödinger equations

In this paper we will study the stability properties of self-similar solutions of $1$ D cubic NLS equations with time-dependent coefficients of the form 0.1 $$\begin{aligned} \displaystyle { iu_t+u_{xx}+\frac{u}{2} \left(|u|^2-\frac{A}{t}\right)=0, \quad A\in \mathbb{R }. } \end{aligned}$$ The study of the stability of these self-similar solutions is related, through the Hasimoto transformation, to the stability of some singular vortex dynamics in the setting of the Localized Induction Equation (LIE), an equation modeling the self-induced motion of vortex filaments in ideal fluids and superfluids. We follow the approach used by Banica and Vega that is based on the so-called pseudo-conformal transformation, which reduces the problem to the construction of modified wave operators for solutions of the equation $$\begin{aligned} iv_t+ v_{xx} +\frac{v}{2t}(|v|^2-A)=0. \end{aligned}$$ As a by-product of our results we prove that Eq. (0.1) is well-posed in appropriate function spaces when the initial datum is given by $u(0,x)= z_0 \mathrm p.v \frac{1}{x}$ for some values of $z_0\in \mathbb{C }\setminus \{ 0\}$ , and $A$ is adequately chosen. This is in deep contrast with the case when the initial datum is the Dirac-delta distribution.     

On the new critical exponent for the nonlinear Schrödinger equations

We consider the Cauchy problem for the pth order nonlinear Schrödinger equation in one space dimension $$\left\{\begin{array}{ll}iu_{t} + \frac{1}{2} u_{xx} = |u|^{p}, x \in {\bf R}, \, t > 0, \\ \qquad u(0, x) = u_{0} (x), \; x \in {\bf R},\end{array}\right.$$ where \({p > p_{s} = \frac{3 + \sqrt{17}}{2}}\) . We reveal that p = 4 is a new critical exponent with respect to the large time asymptotic behavior of solutions. We prove that if p _s < p < 4, then the large time asymptotics of solutions essentially differs from that for the linear case, whereas it has a quasilinear character for the case of p > 4.