Four-Wave Semidiscrete Nonlinear Integrable System with 𝒫𝒯-Symmetry

Abstract

The new type of third-order spectral operator suitable to generate new multifield semidiscrete nonlinear systems with two coupling parameters in the framework of zero-curvature equation is proposed. The evolution operator corresponding to the first integrable system in an infinite hierarchy is explicitly recovered and the general form of first integrable system is isolated. The generalized procedure for the direct recursive development of infinite hierarchy of local conservation laws is presented and several lowest local conservation laws and local conserved densities are found. The reduction to the real field amplitudes in general system with unfixed sampling functions is made and the symmetric parametrization of field amplitudes allowing to exclude the redundant field function and to resolve the problem of sampling fixation for the particular realization of reduced integrable system is considered. This parametrization gives rise to the four field nonlinear model which in certain intervals of adjustable coupling parameter could serve as a semidiscrete analogy to the beam-plasma interaction system.

Coupled Nonlinear Dynamics in the Three-Mode Integrable System on a Regular Chain

“…In contrast, there exists the generalized procedure [18,19,31,32] permitting to develop an infinite set of local conservation laws recursively without references to auxiliary spectral data. By definition, any local conservation law associated with some semidiscrete system given on an infinite quasi-one-dimensional lattice can be written in the forṁ…”

Prototype and reduced nonlinear integrable lattice systems with the modulated pulson behavior

“…In contrast, there exists the generalized direct procedure [29,24,27,28] permitting to develop an infinite set of local conservation laws recursively without references to auxiliary spectral data. By definition, any local conservation law associated with some semi-discrete system given on an infinite quasi-one-dimensional lattice can be written in the form…”

“…Among other semi-discrete nonlinear integrable systems the integrable systems generated by the various auxiliary linear problems of third [21,22,23,24,25,27,28] or more higher [29,30] order deserve a special attention as the systems comprising several subsystems of distinct physical origins.…”