Inverse scattering transform for a nonlocal derivative nonlinear Schrödinger equation

Theoretical and Mathematical Physics - We give a detailed discussion of a nonlocal derivative nonlinear Schrödinger (NL-DNLS) equation with zero boundary conditions at infinity in terms of the...     

Inverse theory of Schrödinger matrices

In this note we discuss the inverse spectral theory for Schrödinger matrices, in particular a conjecture of Gesztesy-Simon [1] on the number of distinct iso-spectral Schrödinger matrices. We consider 3 × 3 matrices and obtain counter examples to their conjecture.     

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.     

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.     

An inverse scattering problem for the Schrödinger equation with a potential having a periodic asymptotics

A theorem is proved regarding the expansion in the eigenfunctions of the one-dimensional Schrödinger equationL = –d ^z/dx ²+q(x)(–<x<)with a potential q(x), satisfying the condition

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.     

On a class of nonlinear inhomogeneous Schrödinger equation

In this paper, we study the inhomogeneous Schrödinger equation $$i\varphi_{t}=-\triangle\varphi -|x|^{b}|\varphi|^{p-1}\varphi,\quad x\in \mathbb{R}^{N}.$$ By using variational methods and a refined interpolation inequality, we establish some simple but sharp conditions on the solutions which exist globally or blow up in a finite time. An interesting result is that we obtain the existence of global solution for arbitrarily large data.     

The Cauchy problem of a focusing energy-critical nonlinear Schrödinger equation

In this paper, we combine variational methods and harmonic analysis to discuss the Cauchy problem of a focusing nonlinear Schrödinger equation. We study the global well-posedness, finite time blowup and asymptotic behavior of this problem. By Hamiltonian property, we establish two types of invariant evolution flows. Then from one flow and the stability of classical energy-critical nonlinear Schrödinger equation, we find that the solution exists globally and scattering occurs. Finally, we get a precise blowup criterion of this problem for positive energy initial data via the other flow.     

Existence of a ground state and scattering for a nonlinear Schrödinger equation with critical growth

We study the energy-critical focusing nonlinear Schrödinger equation with an energy-subcritical perturbation. We show the existence of a ground state in the four or higher dimensions. Moreover, we give a sufficient and necessary condition for a solution to scatter, in the spirit of Kenig and Merle (Invent Math 166:645–675, 2006).     

Existence of solutions for a nonperiodic semilinear Schrödinger equation

In this paper we consider a class of semilinear Schrödinger equation which terms are asymptotically periodic at infinity. Under a weaker superquadratic condition on the nonlinearity, the existence of a ground state solution is established. The main tools employed here to overcome the new difficulties are the concentration-compactness principle and the Local Mountain Pass Theorem.     

Inverse problem for Schrödinger equations with Yang–Mills potentials in a slab

In this work we consider the inverse boundary value problem for Schrödinger equations with Yang–Mills potentials in the domain of infinite slab type. We prove that the potentials can be determined uniquely up to a gauge equivalent class assuming that only partial measurements are known on the boundary hyperplanes.     

Mixed problem for a cubic Schrödinger evolution equation with a cubic dissipative term

The question of the global solvability of a mixed problem for a Schrödinger evolution equation with a cubic dissipative term is considered.     

Global well-posedness and scattering for a class of nonlinear Dunkl–Schrödinger equations

In this paper, we introduce a class of nonlinear Schrödinger equations associated with the Dunkl operators. In this regard, we study local and global well-posedness, and the scattering theory associated with these equations.