Abstract Formulation of the Miura Transform

Yoritaka Iwata

doi:10.3390/math8050747

Abstract

Miura transform is known as the transformation between Korweg de-Vries equation and modified Korweg de-Vries equation. Its formal similarity to the Cole-Hopf transform has been noticed. This fact sheds light on the logarithmic type transformations as an origin of a certain kind of nonlinearity in the soliton equations. In this article, based on the logarithmic representation of operators in infinite-dimensional Banach spaces, a structure common to both Miura and Cole-Hopf transforms is discussed. In conclusion, the Miura transform is generalized as the transform in abstract Banach spaces, and it is applied to the higher order abstract evolution equations.

Highlights

IntroductionThe Korteweg-de-Vries equation (KdV equation, for short) and the modified Korweg de-Vries equation (mKdV equation, for short) are known as nonlinear equations holding the soliton solutions
The Korteweg-de-Vries equation (KdV equation, for short) and the modified Korweg de-Vries equation are known as nonlinear equations holding the soliton solutions.Let u and v be the solutions of the KdV equation and mKdV equation, respectively
It is applied to the higher order abstract evolution equations

Summary

Introduction

The Korteweg-de-Vries equation (KdV equation, for short) and the modified Korweg de-Vries equation (mKdV equation, for short) are known as nonlinear equations holding the soliton solutions. Let u and v be the solutions of the KdV equation and mKdV equation, respectively. Let functions u and v be the general solutions that satisfy [KdV] ∂t u − 6u∂ x u + ∂3x u = 0,. For a recent result associated with the well-posedness of the KdV equations, the existence and uniqueness of the solution of semilinear KdV equations in non-parabolic domain is obtained in [1]. The Miura transform is generalized as the transform in the abstract spaces. The essence of several nonlinear transforms are pined downed within the theory of abstract equations defined in a general Banach spaces. The structure of the general solutions of second order abstract evolution equations are presented in association with the Miura transform.

Nonlinear Transform Associated with the Riccati’s Equation

Miura Transform and Cole-Hopf Transform

Logarithmic Representation of Infinitesimal Generators

Generalization of Miura Transform

Second Order Abstract Evolution Equations

Conclusions