New classes of solutions in the coupled PT symmetric nonlocal nonlinear Schrödinger equations with four wave mixing

Abstract

We investigate generalized nonlocal coupled nonlinear Schorödinger equation containing Self-Phase Modulation, Cross-Phase Modulation and Four Wave Mixing involving nonlocal interaction. By means of Darboux transformation we obtained a family of exact breathers and solitons including the Peregrine soliton, Kuznetsov-Ma breather, Akhmediev breather along with all kinds of soliton-soliton and breather-soltion interactions. We analyze and emphasize the impact of the four-wave mixing on the nature and interaction of the solutions. We found that the presence of Four Wave Mixing converts a two-soliton solution into an Akhmediev breather. In particular, the inclusion of Four Wave Mixing results in the generation of a new solutions which is spatially and temporally periodic called "Soliton (Breather) lattice".

Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation

“…Also this equation has unusual properties of the exact soliton and breather solutions, particularly, solitons could blow up in finite time and the NNLS equation supports both dark and anti-dark soliton solutions simultaneously (see e.g. [2,6,46,28,40,45,47] and references therein; see also [48], where the general soliton solutions for the coupled Schrödinger equations (1.3) are found). Apart from deriving exact solutions of the NNLS equation, it is important, in the both mathematical and physical perspective, to consider initial value problems with general initial data.…”

Peregrine Solitons of the Higher-Order, Inhomogeneous, Coupled, Discrete, and Nonlocal Nonlinear Schrödinger Equations

“…7.12 The Generalized PT-Symmetric Nonlocal Coupled NLSE With Nonlocal SPM, XPM, and FWM of the Following Form [273] i…”

Inverse scattering transforms and soliton solutions of nonlocal reverse‐space nonlinear Schrödinger hierarchies

“…Three types of nonlocal nonlinear Schrödinger (NLS) equation arises while taking group reductions. 1 The corresponding inverse scattering transforms have been recently established for the scalar case [2][3][4][5][6] and the multicomponent case, 7,8 and soliton solutions have been constructed from the Riemann-Hilbert problems whose jump is the identity, 8,9 through Darboux transformations, [10][11][12] and by the Hirota bilinear method. 13 Some other multicomponent generalizations 1,14,15 and nonlocal integrable equations 16 were also presented.…”