Maximum decay rate for the nonlinear Schrödinger equation

In this paper, we consider global solutions for the following nonlinear Schrödinger equation in with and We show that no nontrivial solution can decay faster than the solutions of the free Schrödinger equation, provided that u(0) lies in the weighted Sobolev space in the energy space, namely or in according to the different cases.     

Modified scattering for the critical nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation

$i{u}_t+\mathrm{\Delta }u=\lambda |u{|}^{\frac{2}{N}}u$

in all dimensions $N\ge 1$, where $\lambda \in \mathbb{C}$ and $?\lambda \le 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to \infty$, like $t^{?\frac{N}{2}}$ if $?\lambda =0$ and like ${(t\log ?t)}^{?\frac{N}{2}}$ if $?\lambda <0$. Moreover, we give an asymptotic expansion of those solutions as $t\to \infty$. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.     

A parametric spline scheme for the nonlinear Schrödinger equation

We develop a numerical method based on parametric adaptive quintic spline functions for solving the nonlinear Schrödinger (NLS) equation. The truncation error is theoretically analyzed. Based on the von Neumann method and the linearization technique, stability analysis of the method is studied and the method is shown to be unconditionally stable. Two invariants of motion related to mass and momentum are calculated to determine the conservation properties of the problem. Finally, some numerical tests are presented to illustrate the method’s efficiency.     

Thick clusters for the radially symmetric nonlinear Schrödinger equation

This article is devoted to the study of radially symmetric solutions to the nonlinear Schrödinger equation

$\varepsilon^2 \Delta u - V(r)u + |u|^{p-1}u = 0\, {\rm in} B,\quad \frac{\partial u}{\partial n} = 0\, {\rm on}\,{\partial}B,$

where B is a ball in \({\mathbb{R}}^N\) , 1 <  p <  (N +  2)/(N ? 2), N ≥ 3 and the potential V is radially symmetric. We construct positive clustering solutions in an annulus having O(1/?) critical points, as well as sign changing solutions with O(1/?) zeroes concentrating near zero.

     

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix $\bar \partial $ problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.     

Normalized bound states for the nonlinear Schrödinger equation in bounded domains

Given \(\rho >0\), we study the elliptic problem

$$\begin{aligned} \text {find } (U,\lambda )\in H^1_0(\Omega )\times {\mathbb {R}}\text { such that } {\left\{ \begin{array}{ll} -\Delta U+\lambda U=|U|^{p-1}U\\ \int _{\Omega } U^2\, dx=\rho , \end{array}\right. } \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain and \(p>1\) is Sobolev-subcritical, searching for conditions (about \(\rho \), N and p) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when p is \(L^2\)-subcritical, i.e. \(1<p<1+4/N\), the problem admits solutions for every \(\rho >0\). In the \(L^2\)-critical and supercritical case, i.e. when \(1+4/N \le p < 2^*-1\), we show that, for any \(k\in {\mathbb {N}}\), the problem admits solutions having Morse index bounded above by k only if \(\rho \) is sufficiently small. Next we provide existence results for certain ranges of \(\rho \), which can be estimated in terms of the Dirichlet eigenvalues of \(-\Delta \) in \(H^1_0(\Omega )\), extending to changing sign solutions and to general domains some results obtained in Noris et al. in Anal. PDE 7:1807–1838, 2014 for positive solutions in the ball.

     

A semi-discrete scheme for the stochastic nonlinear Schrödinger equation

Summary. We study the convergence of a semi-discretized version of a numerical scheme for a stochastic nonlinear Schrödinger equation. The nonlinear term is a power law and the noise is multiplicative with a Stratonovich product. Our scheme is implicit in the deterministic part of the equation as is usual for conservative equations. We also use an implicit discretization of the noise which is better suited to Stratonovich products. We consider a subcritical nonlinearity so that the energy can be used to obtain an a priori estimate. However, in the semi discrete case, no Ito formula is available and we have to use a discrete form of this tool. Also, in the course of the proof we need to introduce a cut-off of the diffusion coefficient, which allows to treat the nonlinearity. Then, we prove convergence by a compactness argument. Due to the presence of noise and to the implicit discretization of the noise, this is rather complicated and technical. We finally obtain convergence of the discrete solutions in various topologies. Mathematics Subject Classification (2000):35Q55, 60H15, 65M06, 65M12     

Breathers in a damped-driven nonlinear Schrödinger equation

We construct time-periodic solitons of the parametrically driven damped nonlinear Schrödinger equation as solutions of a boundary value problem for the space-dependent Fourier coefficients.     

Soliton dynamics for the nonlinear Schrödinger equation with magnetic field

The semiclassical regime of a nonlinear focusing Schrödinger equation in presence of non-constant electric and magnetic potentials V, A is studied by taking as initial datum the ground state solution of an associated autonomous stationary equation. The concentration curve of the solutions is a parameterization of the solutions of the second order ordinary equation \({\ddot x=-\nabla V(x)-\dot x\times B(x)}\), where \({B=\nabla\times A}\) is the magnetic field of a given magnetic potential A.     

A note on the illposedness for anisotropic nonlinear Schrödinger equation

In this short note, we show the illposedness of anisotropic Schroedinger equation in L^2 if the growth of nonlinearity is larger than a threshold power pc which is also the critical power for blowup, as Fibich, Ilan and Schochet have pointed out recently. The illposedness in anisotropic Sobolev space Hk,d-d^2s,s where 0 〈 s 〈 sc, sc =d/2-k/4-2/p-1, and the illposedness in Sobolev space of negative order H^s, s 〈 0 are also proved.     

The inverse problem for a nonlinear Schrödinger equation

Using variational methods, the solution of the inverse problem of finding the refractive index of a nonlinear medium in a multidimensional Schrödinger equation is studied. The correctness of the statement of the problem under consideration is investigated, and a necessary condition that must be satisfied by the solution of this problem is found. Bibliography: 8 titles.     

Attractors for discrete nonlinear Schrödinger equation with delay

In this paper,we first prove the existence of the global attractor Aν ∈ C（[-ν,0],2）（ν 0） for a weak damping discrete nonlinear Schrdinger equation with delay.Then we consider an upper semi-continuity of Aν as ν → 0＋.     

Pfaffian solution for coupled discrete nonlinear Schrödinger equation

The Pfaffian solution for the coupled discrete nonlinear Schrödinger equation is studied by using the direct method of soliton theory. The bilinear form of the equation contains a new Pfaffian identity. The Pfaffian representation of Toeplitz determinant is also derived.     

Space-time scattering for the Schrödinger equation

Results are obtained on the scattering theory for the Schrödinger equation $i\partial _t u(t,x) = - \Delta _x u(t,x) + V(t,x)u(t,x) + F(u(t,x))$ in spacesL ^r (R;L ^q (R ^d )) for a certain range ofr, q, the so-called space-time scattering. In the linear case (i.e.F≡)) the relation with usual configuration space scattering is established.