Exact Solution of Modified Veselov–Novikov Equation and Some Applications in the Game Theory

Damir Kurmanbayev

doi:10.1155/2020/9740638

Abstract

A method of finding exact solutions of the modified Veselov–Novikov (mVN) equation is constructed by Moutard transformations, and a geometric interpretation of these transformations is obtained. An exact solution of the mVN equation is found on the example of a higher order Enneper surface, and given transformations are applied in the game theory via Kazakh proverbs in terms of trees.

Highlights

Consider the following problem with the initial data: ψ􏽥t Aψ􏽥, (1) ψ􏽥|t 0 ψ0, (2)where A (z3/zz3). e following theorems are principal results of this paper.Theorem 1
Physical interpretation of exact solutions of the modified Veselov–Novikov equation corresponding to inverted Enneper surfaces analogically could be represented by solitons for mixed coupled nonlinear Schrodinger equations [3]
Equations by Moutard transformations on the example of a first-order Enneper surface. eorem 1 generates all minimal surfaces and leads to the regular solutions of modified Veselov–Novikov (mVN) equations formulated in eorem 2 by given transformations

Summary

Introduction

Consider the following problem with the initial data (this is the minimal surface): ψ􏽥t Aψ􏽥,. If ψ0 is holomorphic function, the general solution of (1)-(2) is given by the following formula:. Physical interpretation of exact solutions of the modified Veselov–Novikov equation (given below) corresponding to inverted Enneper surfaces analogically could be represented by solitons (depending on different parameters) for mixed coupled nonlinear Schrodinger equations [3]. The strategy of this work is to find the solution of the following modified Veselov–Novikov (mVN) equation [5]: U􏽥 t. Taimanov [7] found blowing-up solutions of mVN equations by Moutard transformations on the example of a first-order Enneper surface. Eorem 1 generates all minimal surfaces (higher order Enneper surface, catenoid, and helicoid) and leads to the regular solutions of mVN equations formulated in eorem 2 by given transformations. E previous matrix is chosen so that the system (9) is equivalent to Manakov’s L, A, B triple [5,9], which is an operator representation of mVN equations. (3) e real-valued function U􏽥 and the function V􏽥 satisfy the mVN equations

Preliminaries

Solution of the mVN Equation

Main Results

Conclusions