Abstract
This paper proposes a method for constructing partial differential equation (PDE) systems with chaotic solitons by using truncated normal forms of an ordinary differential equation (ODE). The construction is based mainly on the fact that the existence of a soliton in a PDE system is equal to that of a homoclinic orbit in a related ODE system, and that chaos of ?i’lnikov homoclinic type in the ODE imply that the soliton in the PDE changes its profile chaotically along propagation direction. It is guaranteed that the constructed systems can self-generate chaotic solitons without any external perturbation but with constrained wave velocities in a rigorously mathematical sense.
Highlights
Chaotic solitons have been a subject of many theoretical papers over the last decades of years
This paper proposes a method for constructing partial differential equation (PDE) systems with chaotic solitons by using truncated normal forms of an ordinary differential equation (ODE)
The central question addressed in this paper is: How can one construct a partial differential equation (PDE) system that self-generates a chaotic solitary-wave pulse that exists in a rigorous sense? There are few rigorous results that addresses this question
Summary
Chaotic solitons have been a subject of many theoretical papers over the last decades of years. The central question addressed in this paper is: How can one construct a partial differential equation (PDE) system that self-generates a chaotic solitary-wave pulse that exists in a rigorous sense? Rigorous results for the generation of chaotic solitons are obtained through perturbations to a known system, e.g., Schrödinger equation and Ginzburg-Landau equation [58], that can generate solitons This method is not regular since, in practice, it can be applied only to particular examples. It is not difficult to imagine that the corresponding soliton in the original PDE system should be a chaotic soliton since its profile impossibly behaves regularly in time Such a relation between some solutions of ODE and PDE systems is the basis of our constructing PDE systems that self-generate chaotic solitons. CHEN system is ensured by that of one homoclinic orbit of an ODE system whereas its chaotic behavior (i.e., the soliton changes its profile chaotically along propagation direction) results from the chaos property of the ODE system which is guaranteed by Ši’lnikov’s homoclinic theorem [11]
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