We provide a simple technique for finding the correspondence between the solutions of AblowitzLadik and nonlinear Schrödinger equations. Even though they belong to different classes, in that one is continuous and one is discrete, there are matching solutions. This fact allows us to discern common features and obtain solutions of the continuous equation from solutions of the discrete equation. W...

In 1968 V.E. Zakharov derived the Nonlinear Schrödinger equation for the 2D water wave problem in the absence of surface tension, i.e., for the evolution of gravity driven surface water waves, in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet. In this paper we give a rigorous proof that the wave packets in the two-dimensional water...

A nonlinear Schrödinger equation with external potential −(t + b)−1 is considered and its explicit solutions are constructed.

In this paper, we are concerned with the semilinear Schrödinger equation (1.1) −∆Au− V (x)u = |u| ∗−2u , x ∈ R , where −∆A = (−i∇+A)2, u : R → C, N ≥ 3, 2∗ = 2N N−2 denotes the critical Sobolev exponent, A = (A1, A2, ..., AN ) : R N → R is the vector (or magnetic) potential, the coefficient V is the scalar (or electric) potential and may be signchanging. The nonlinear Schrödinger equation arise...

The perturbation theory based on the Riemann-Hilbert problem is constructed for the nonlinear Schrödinger equation. The second-order equations for the spectral data describing the soliton-wave interaction are obtained. The theory is applied to the parametrically driven, damped nonlinear Schrödinger equation. The parametrically driven NLS soliton is shown to become unstable, when the driving str...

We present a mass-preserving scheme for the stochastic nonlinear Schrödinger equation with multiplicative noise of Stratonovich type. It is a splitting scheme and we present an explicit formula for solving the sub-step related to the nonlinear part. The scheme is unconditionally stable in the L2 norm. For the linear stochastic Schrödinger equation, we prove that the scheme has a strong converge...

We rigorously study the long time dynamics of solitary wave solutions of the nonlinear Schrödinger equation in time-dependent external potentials. To set the stage, we first establish the well-posedness of the Cauchy problem for a generalized nonautonomous nonlinear Schrödinger equation. We then show that when the external potential varies slowly in space compared to the size of the soliton, th...

Optical dark solitons described by the generalized nonlinear Schrödinger equation are discussed, and the criterion of soliton instability is presented. This analytical criterion is confirmed numerically for an exactly solvable model of nonlinear saturation.

We find new periodic solutions of the Kadomtsev-Petviashvili (KP) equation, the nonlinear Schrödinger (NLS) equation, the λφ4 model, the sine-Gordon equation and the Boussinesq equation by making appropriate linear superpositions of known periodic solutions. This unusual procedure for generating solutions of nonlinear differential equations is successful as a consequence of some powerful, recen...