In this work we want to describe a discrete version of the well–known Adomian decomposition method (ADM) applied to the discrete nonlinear Schrödinger equation (NLS). The ADM was introduced by Adomian [3], [11] in the early 1980s to solve nonlinear ordinary and partial differential equations. The discrete nonlinear Schrödinger equation is omnipresent [8] in applied sciences, e.g. describing the...

We interpret the lens transformation (a variant of the pseudoconformal transformation) as a pseudoconformal compactification of spacetime, which converts the nonlinear Schrödinger equation (NLS) without potential with a nonlinear Schrödinger equation with attractive harmonic potential. We then discuss how several existing results about NLS can be placed in this compactified setting, thus offeri...

We implement the dressing method for a novel integrable generalization of the nonlinear Schrödinger equation. As an application, explicit formulas for the N-soliton solutions are derived. Moreover, as a by-product of the analysis, we find a simplification of the formulas for the N-solitons of the derivative nonlinear Schrödinger equation given by Huang and Chen. AMS Subject Classification (2000...

We prove that the non-radial sign-changing solutions to the nonlinear Schrödinger equation ∆u− u + |u|p−1u = 0 in R , u ∈ H(R ) constructed by Musso, Pacard, and Wei [19] are non-degenerate. This provides the first example of a non-degenerate sign-changing solution to the above nonlinear Schrödinger equation with finite energy.

The nonlinear Schrödinger-Helmholtz (SH) equation in N space dimensions with 2σ nonlinear power was proposed as a regularization of the classical nonlinear Schrödinger (NLS) equation. It was shown that the SH equation has a larger regime (1 ≤ σ < 4 N ) of global existence and uniqueness of solutions compared to that of the classical NLS (0 < σ < 2 N ). In the limiting case where the Schrödinger...

We interpret the lens transformation (a variant of the pseudoconformal transformation) as a pseudoconformal compactification of spacetime, which converts the nonlinear Schrödinger equation (NLS) without potential with a nonlinear Schrödinger equation with attractive harmonic potential. We then discuss how several existing results about NLS can be placed in this compactified setting, thus offeri...

We consider a nonlinear Schrödinger equation in R3 with a bounded local potential. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data is small and is near some nonlinearexcited state. We give a sufficient condition on the initial data so that the solution to the nonlinear Schrödinger equation approac...

The nonlinear Schrödinger equation is of tremendous interest in both theory and applications. Various regimes of pulse propagation in optical fibers are modeled by some form of the nonlinear Schrödinger equation. In this paper we introduce sequential and parallel numerical methods for numerical simulations of the 1+ 2 dimensional nonlinear Schrödinger type equations. The parallel methods are im...

We make a detailed study on the dynamics of gain-assisted superluminal optical solitons in a three-state active-Raman-gain medium at room temperature. Using a method of multiple-scales we derive a high-order nonlinear Schrödinger equation with correction terms contributed from differential gain, nonlinear dispersion, delay in nonlinear refractive index, and third-order dispersion of the system....

The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg-de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report...