Functionals with values in the Non-Archimedean field of Laurent series and their applications to the equations of elasticity theory

TL;DR

This paper introduces a novel approach using Non-Archimedean Laurent series functionals to define generalized solutions for the Hopf and elasticity equations, enabling algebraic and numerical analysis of solitons and shock waves.

Contribution

It develops a new calculation method for soliton and shock wave profiles by reducing the problem to algebraic equations and applying Newton iteration, with practical examples and tests.

Findings

Profiles of solitons and shock waves can be computed via algebraic systems.

Newton iteration effectively finds solutions to these algebraic systems.

The method is applicable to multidimensional elasticity equations.

Abstract

Functionals with values in Non-Archimedean field of Laurent series applied to the definition of generalized solution (in the form of soliton and shock wave) of the Hopf equation and equations of elasticity theory. Calculation method for the profile of infinitely narrow soliton and shock wave is proposed. Applying this method, calculations of profiles are reduced to the nonlinear system of algebraic equations in $\mathbf{R}^{n+1}$, $n>1$. It is shown that there is a possibility to find out some of the solutions of this system using the Newton iteration method. Examples and numerical tests are considered.