Numerical Calculation of Conservation Laws in Solitonic Interactions

Abstract

The aim of this paper was the numerical computing of the first conservation law present in the interactions between solitons generated from numerical solutions of the Korteweg de Vries (KdV) equation. As follows from the theory of partial differential equations in which appears solitons, like some solutions of the KdV, many interesting behaviors come associated with them, in particular the conservation laws derived from it. Only the first three ones have direct interpretation or are associated with physical quantities, such as conservation of energy, momentum or mass density. The remaining are conservation laws in the strict sense of physics but do not have a clearly associated with magnitudes that usually appear in physical processes. The KdV equation constitutes one of the called “integrable systems” and in this way the conservations laws derived from it, have special interest for applications on the study of solitons’s propagation. These non-linear traveling waves that can be found from tsunamis in the oceans, transmission of information by transoceanic fiber optic communications, transmitted information through neural microtubules in living beings have a relevant technological and scientific interest. So the knowledge of conservation laws in the interaction between solitons becomes very interesting from both point of view technological and for the study of certain process in living beings. To obtain the explicit form of the conservation laws derived from the KdV equation, mainly due to the large number of terms that arise in the mathematical expression of them, there were developed a computational code implemented in a symbolic calculus platform that facilitated their algebraic handling. The conservation laws thus obtained were numerically evaluated during the computing integration of the KdV equation, based on an spectral type scheme. Their solutions represent the interaction of two solitons. Among the obtained results it can be pointed out that if the generation of solitons by solving the KdV equation has a high degree of approximation, the integrals of motion does not require highly sophisticated numerical algorithms to be satisfied and also make a numerical validation of the integration schemes of the nonlinear equation under study.