Self-Localized Solutions of the Kundu-Eckhaus Equation in Nonlinear Waveguides

In this paper we numerically analyze the 1D self-localized solutions of the Kundu-Eckhaus equation (KEE) in nonlinear waveguides using the spectral renormalization method (SRM) and compare our findings with those solutions of the nonlinear Schrodinger equation (NLSE). We show that single, dual and N-soliton solutions exist for the case with zero optical potentials, i.e. V=0. We also show that these soliton solutions do not exist, at least for a range of parameters, for the photorefractive lattices with optical potentials in the form of V=Io cos^2(x) for cubic nonlinearity. However, self-stable solutions of the KEE with saturable nonlinearity do exist for some range of parameters. We compare our findings for the KEE with those of the NLSE and discuss our results.Comment: Typos are corrected, 8 figures are adde

Rogue waves and other solutions of single and coupled Ablowitz–Ladik and nonlinear Schrödinger equations

We provide a simple technique for finding the correspondence between the solutions of Ablowitz–Ladik and nonlinear Schrodinger 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. We consider several examples. We provide tables, with selected solutions, which allow us to easily match the pairs of solutions. We show that our technique can be extended to the case of coupled Ablowitz–Ladik and nonlinear Schrodinger (i.e. Manakov) equations. We provide some new solutions

Brief communication: Modulation instability of internal waves in a smoothly stratified shallow fluid with a constant buoyancy frequency

Unexpectedly large displacements in the interior of the oceans are studied through the dynamics of packets of internal waves, where the evolution of these displacements is governed by the nonlinear Schrödinger equation. In cases with a constant buoyancy frequency, analytical treatment can be performed. While modulation instability in surface wave packets only arises for sufficiently deep water, “rogue” internal waves may occur in shallow water and intermediate depth regimes. A dependence on the stratification parameter and the choice of internal modes can be demonstrated explicitly. The spontaneous generation of rogue waves is tested here via numerical simulation.</p

Initial-Boundary Value Problem for Stimulated Raman Scattering Model: Solvability of Whitham Type System of Equations Arising in Long-Time Asymptotic Analysis

An initial-boundary value problem for a model of stimulated Raman scattering was considered in [Moskovchenko E.A., Kotlyarov V.P., J. Phys. A: Math. Theor. 43 (2010), 055205, 31 pages]. The authors showed that in the long-time range $t\to+\infty$ the $x>0$, $t>0$ quarter plane is divided into 3 regions with qualitatively different asymptotic behavior of the solution: a region of a finite amplitude plane wave, a modulated elliptic wave region and a vanishing dispersive wave region. The asymptotics in the modulated elliptic region was studied under an implicit assumption of the solvability of the corresponding Whitham type equations. Here we establish the existence of these parameters, and thus justify the results by Moskovchenko and Kotlyarov

Quasi-Gramian Solution of a Noncommutative Extension of the Higher-Order Nonlinear Schr\"odinger Equation

The nonlinear Schr{\"o}odinger (NLS) equation, which incorporates higher-order dispersive terms, is widely employed in the theoretical analysis of various physical phenomena. In this study, we explore the non-commutative extension of the higher-order NLS equation (HNLS). We treat real or complex-valued functions, such as g1 = g1(x, t) and g2 = g2(x, t), as non-commutative, and employ the Lax pair associated with the evolution equation as in the commutation case. We derive the quasi-Gramian solution of the system by employing a binary Darboux transformation (DT). Moreover, the solution can be used to study the stability of plane waves and to understand the generation of periodic patterns in the context of modulational instability.Comment: 20 pages, 32 figure