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We study the Derivative Nonlinear Schr\"odinger equation for generic initial data in a weighted Sobolev space that can support bright solitons (but exclude spectral singularities). Drawing on previous well-posedness results, we give a full…
We review recent results on global wellposedness and long-time behavior of smooth solutions to the derivative nonlinear Schr\"{o}dinger (DNLS) equation. Using the integrable character of DNLS, we show how the inverse scattering tools and…
We study the Derivative Nonlinear Schr\"odinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but excluding spectral singularities). We prove global well-posedness and give a full…
We are interested in solutions of the nonlinear Klein-Gordon equation (NLKG) in $\mathbb{R}^{1+d}$, $d\ge1$, which behave as a soliton or a sum of solitons in large time. In the spirit of other articles focusing on the supercritical…
In a previous paper (Matsuno 2011 {\it J. Phys. A: Math. Theore.} {\bf 44} 495202), we have presented a determinantal expression of the bright $N$-soliton solution for a multi-component modified nonlinear Schr\"odinger (NLS) system with…
In this paper we consider the existence of multi-soliton structures for the nonlinear Klein-Gordon equation (NLKG) in R^{1+d}. We prove that, independently of the unstable character of (NLKG) solitons, it is possible to construct a…
We introduce a setting based on the one-dimensional (1D) nonlinear Schroedinger equation (NLSE) with the self-focusing (SF) cubic term modulated by a singular function of the coordinate, |x|^{-a}. It may be additionally combined with the…
We consider the Cauchy problem for the Gross-Pitaevskii (GP) equation. Using the DBAR generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of the GP in the…
This article presents a concise survey of basic discrete and semi-discrete nonlinear models which produce two- and three-dimensional (2D and 3D) solitons, and a summary of main theoretical and experimental results obtained for such…
We derive a class of discrete nonlinear Schr{\"o}dinger (DNLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic problem. It is demonstrated that the derived class of…
We analytically study the large-time asymptotics of the solution of the defocusing modified Korteweg-de Vries (mKdV) equation under a symmetric non-vanishing background, which supports the emergence of solitons. It is demonstrated that the…
General $N$-solitons in three recently-proposed nonlocal nonlinear Schr\"odinger equations are presented. These nonlocal equations include the reverse-space, reverse-time, and reverse-space-time nonlinear Schr\"odinger equations, which are…
This article provides a naturel sequel of previous works [6, 4] regarding the stability of travelling waves for a general one-dimensional Schr\"odinger equation (N LS) with non-zero condition at infinity. The aim of this article is twofold.…
We study the Derivative Nonlinear Schr\"odinger (DNLS). equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities corresponding to algebraic solitons). We show…
For both the cubic Nonlinear Schr\"odinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension we consider the set ${\bf M}_N$ of pure $N$-soliton states, and their associated multisoliton…
In this work, we carry out a rigorous analysis of a multi-soliton solution of the focusing nonlinear Schr\"{o}dinger equation as the number, $N$, of solitons grows to infinity. We discover configurations of $N$-soliton solutions which…
We develop a general classification of the infinite number of families of solitons and soliton complexes in the one-dimensional Gross-Pitaevskii/nonlinear Schrodinger equation with a nonlinear lattice pseudopotential, i.e., periodically…
The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3\kappa u_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as a model delineating the propagation of shallow water waves, stands as a completely integrable system…
We consider a system of generalized coupled Discrete Nonlinear Schr\"{o}dinger (DNLS) equations, derived as a tight-binding model from the Gross-Pitaevskii-type equations describing a zigzag chain of weakly coupled condensates of…
We consider soliton resolution for the Calogero--Moser derivative nonlinear Schr\"odinger equation (CM-DNLS). A rigorous PDE analysis of (CM-DNLS) was recently initiated by G\'erard and Lenzmann, who demonstrated its Lax pair structure.…