Steady Solitary and Periodic Waves in a Nonlinear Nonintegrable Lattice

Igor Andrianov,Andrey Bochkarev,Aleksandr Zemlyanukhin,Vladimir Erofeev

doi:10.3390/sym12101608

Abstract

In this paper, stationary solitary and periodic waves of a nonlinear nonintegrable lattice are numerically constructed using a two-stage approach. First, as a result of continualization, a nonintegrable generalized Boussinesq—Ostrovsky equation is obtained, for which the solitary-wave and periodic solutions are numerically found by the Petviashvili method. In the second stage, discrete analogs of the obtained solutions are used as initial conditions in the numerical simulation of the original lattice. It is shown that the initial perturbations constructed in this way propagate along the lattice without changing their shape.

Highlights

Nonlinear discrete mathematical models often arise in problems of the dynamics of deformable systems, in physics and biology
They appear either as a result of discretization of physically significant integrable partial differential equations (PDEs), such as the Korteweg–de Vries, the sine—Gordon, and the nonlinear Schrödinger equations, or when modeling systems that are discrete in nature
The problem is most often reduced to the study of a nonintegrable nonlinear differential-difference equation (DDE), the canonical example of which is the Fermi–Pasta–Ulam–Tsingou (FPUT) lattice [4]

Summary

Introduction

Nonlinear discrete mathematical models often arise in problems of the dynamics of deformable systems, in physics and biology. The problem is most often reduced to the study of a nonintegrable nonlinear differential-difference equation (DDE), the canonical example of which is the Fermi–Pasta–Ulam–Tsingou (FPUT) lattice [4]. Many nonintegrable evolutionary and quasi-hyperbolic equations have exact solitary-wave and periodic solutions [5]. In [8], a discrete version of the linearizable Eckhaus equation is introduced, which is a coupled system of two difference evolutionary equations This system is reduced to a single nonlinear first-order difference equation that does not have exact solutions and exhibits chaotic dynamics. The purpose of this study is to construct stable periodic and solitary-wave solutions of a nonintegrable DDE based on a combined approach. In the two sections, based on the Petviashvili method [28,29], steady solitary wave and periodic solutions of a full nonlinear PDE are numerically constructed. Using a discrete analog of the obtained solutions, a numerical simulation of the original nonlinear lattice is carried out, and the stable propagation of the disturbances is demonstrated

Structure and Equation of Motion of the Lattice

Transition from a Lattice Equation to a Continuous Equation

Checking the Correctness of the Transition to a Continuous Equation

Findings

Discussion and Conclusions