Functionals with values in the non‐archimedean field of laurent series and their applications to the equations of elasticity theory. I¹

Dedicated to Tatsiana Radyna

Abstract

Functionals with values in Non‐Archimedean field of Laurent series applied to the definition of generalized solution (in the form of soliton) of the Hopf equation. Calculation method for the profile of infinitely narrow soliton is proposed. Applying this method, calculations of profiles are reduced to the nonlinear system of algebraic equations in 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. Example and numerical test are considered.

Funkcionalai su reikšmemis ne‐archimediniuose Laurent'o seku laukuose pritaikyti apibrežti apibendrintaji Hop'o lygties sprendini solitono pavidalu. Pasiūlytas skaitinis algoritmas begalo siauro solitono profilio radimui. Taikant ši metoda, profilio radimas suvedamas j netiesines algebriniu lygčiu sistemos erdveje R^{n+ 1} ,n > 1, sprendima. Parodyta, kad kai kuriuos sprendinius galima surasti naudojant Niutono iteracini metoda. Pateikiami pavyzdžiai ir skaitiniai testai.

Key words:

• generalized functions
• distributions
• conservation law
• Hopf equation
• soliton
• shock waves

Notes

Dedicated to Tatsiana Radyna