Separability in 1+1 Dimensions in Classical Nonlinear Fields

TL;DR

This paper investigates the conditions under which nonlinear systems in 1+1 dimensions are capable of having separable solitary wave solutions, focusing on the properties of systems that allow initial conditions composed of multiple solitary waves.

Contribution

It introduces the concept of separable systems that permit initial conditions with multiple solitary waves and analyzes their properties and behaviors in such systems.

Findings

Separable systems have two distinct sets of solitons.

Collisions between different sets of solitons may not exhibit soliton-like behavior.

Systems with a periodic potential and zero as a vacuum point are separable.

Abstract

Solitary wave and soliton solutions of nonlinear equations are well known for physicists. A soliton is a solitary wave with some outstanding features which make it reasonable to be studied seriously in nonlinear systems. In fact most of the nonlinear systems which have solitary wave solutions, has no soliton solutions. To realize a solitary wave as a soliton, we must prepare some initial conditions to collide two or more solitary wave solutions. In fact it is not possible to prepare such initial conditions for any nonlinear system with solitary wave solutions. In this paper we study the conditions that a system should have, to prepare a combination of its single solitary wave solutions as an initial condition for collision. These systems accept a combination of separated single solitary waves as an initial condition, so we call them separable systems. We see a system with periodic potential that zero is one of its vacuum points is separable. We observed that separable systems have two distinct set of solitons, but in general, if we collide members of different sets, they may not behave like solitons.