Positive solutions for a weakly coupled nonlinear Schrödinger system

Existence of a nontrivial solution is established, via variational methods, for a system of weakly coupled nonlinear Schrödinger equations. The main goal is to obtain a positive solution, of minimal action if possible, with all vector components not identically zero. Generalizations for nonautonomous systems are considered.     

Multiple solutions of a coupled nonlinear Schrödinger system

We will consider the relation between the number of positive standing waves solutions for a class of coupled nonlinear Schrödinger system in RN and the topology of the set of minimum points of potential V(x). The main characteristics of the system are that its functional is strongly indefinite at zero and there is a lack of compactness in RN. Combining the dual variational method with the Nehari technique and using the Concentration-Compactness Lemma, we obtain the existence of multiple solutions associated to the set of global minimum points of the potential V(x) for ? sufficiently small. In addition, our result gives a partial answer to a problem raised by Sirakov about existence of solutions of the perturbed system.     

Global existence of small classical solutions to nonlinear Schrödinger equations

We study the global Cauchy problem for nonlinear Schrödinger equations with cubic interactions of derivative type in space dimension n?3$n?3$. The global existence of small classical solutions is proved in the case where every real part of the first derivatives of the interaction with respect to first derivatives of wavefunction is derived by a potential function of quadratic interaction. The proof depends on the energy estimate involving the quadratic potential and on the endpoint Strichartz estimates.     

Quasi-periodic solutions in a nonlinear Schrödinger equation

In this paper, one-dimensional (1D) nonlinear Schrödinger equation

iut−uxx+mu+⁴|u|u=0     

Multiple solutions for a class of nonlinear Schrödinger equations

In this paper, we find new conditions to ensure the existence of infinitely many homoclinic type solutions for the Schrödinger equation

     

On the existence of standing wave solutions for a class of quasilinear Schrödinger systems

In this paper minimax methods in a suitable Orlicz space are employed to establish the existence of standing wave solutions for a class of quasilinear Schrödinger systems involving subcritical nonlinearities. The systems considered here can model an interaction phenomena in plasma physics.     

Exact solutions of a two-dimensional nonlinear Schrödinger equation

In the present study, we converted the resulting nonlinear equation for the evolution of weakly nonlinear hydrodynamic disturbances on a static cosmological background with self-focusing in a two-dimensional nonlinear Schrödinger (NLS) equation. Applying the function transformation method, the NLS equation was transformed to an ordinary differential equation, which depended only on one function ξ and can be solved. The general solution of the latter equation in ζ leads to a general solution of NLS equation. A new set of exact solutions for the two-dimensional NLS equation is obtained.     

On the existence of signed and sign-changing solutions for a class of superlinear Schrödinger equations

This paper deals with a semilinear Schrödinger equation whose nonlinear term involves a positive parameter λ and a real function f(u) which satisfies a superlinear growth condition just in a neighborhood of zero. By proving an a priori estimate (for a suitable class of solutions) we are able to avoid further restrictions on the behavior of f(u) at infinity in order to prove, for λ sufficiently large, the existence of one-sign and sign-changing solutions. Minimax methods are employed to establish this result.     

Infinitely many small solutions for a modified nonlinear Schrödinger equations

In the present paper, we study the Modified Nonlinear Schrödinger Equations (MNSE). Without any growth condition on the nonlinear term, we obtain the existence of infinitely many small solutions for MNSE by a dual approach.     

On the Cauchy problem for a coupled system of third-order nonlinear Schrödinger equations

We investigate some well-posedness issues for the initial value problem (IVP) associated with the system

     

On the nonlinear Schrödinger limit of the magnetic Zakharov system

In this paper, we study the limit behavior of a smooth solution for the magnetic type Zakharov system. As the parameters tend to infinity, we prove that solutions of the magnetic Zakharov system converge to the solution of the nonlinear Schrödinger equation. Moreover, the detailed convergence rate is also investigated.     

Global existence and compact attractors for the discrete nonlinear Schrödinger equation

We study the asymptotic behavior of solutions of discrete nonlinear Schrödinger-type (DNLS) equations. For a conservative system, we consider the global in time solvability and the question of existence of standing wave solutions. Similarities and differences with the continuous counterpart (NLS-partial differential equation) are pointed out. For a dissipative system we prove existence of a global attractor and its stability under finite-dimensional approximations. Similar questions are treated in a weighted phase space. Finally, we propose possible extensions for various types of DNLS equations.     

Asymptotics of odd solutions for cubic nonlinear Schrödinger equations

We consider the Cauchy problem for a cubic nonlinear Schrödinger equation in the case of an odd initial data from H²∩H0,2. We prove the global existence in time of solutions to the Cauchy problem and construct the modified asymptotics for large values of time.     

New solutions for nonlinear Schrödinger equations with critical nonlinearity

We consider the following nonlinear Schrödinger equations in Rn

     

Direct search for exact solutions to the nonlinear Schrödinger equation

A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schrödinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansätze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear Schrödinger equation. Various examples of exact solutions with constant, trigonometric function type, exponential function type and rational function amplitude are given upon careful analysis. A bifurcation phenomenon in the nonlinear Schrödinger equation is clearly exhibited during the solution process.     

Regular and self-similar solutions of nonlinear Schrödinger equations

We study the Cauchy problem for the nonlinear Schrödinger equations with nonlinear term |u|^ou. For some admissible α we show the existence of global solutions and we calculate the regularity of those solutions. Also we give some necessary conditions and some sufficient conditions on initial data for the existence of self-similar solutions.     

Global existence for some radial, low regularity nonlinear Schrödinger equations

We prove the nonlinear Schrödinger equation has a local solution for any energy - subcritical nonlinearity when u₀ is the characteristic function of a ball in Rn. Additionally, we establish the existence of a global solution for n?3 when and α?2. Finally, we establish the existence of a global solution when the initial function is radial, the nonlinear Schrödinger equation has an energy subcritical nonlinearity, and the initial function lies in Hρ+?(Rn)∩H1/2+?(Rn)∩H1/2+?,1(Rn).