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1988•Journal of Mathematical PhysicsRequires access

Gauge transformation and the higher order Korteweg–de Vries equation

Yu-kun Zheng, W.L. Chan

Open publisher page 13 citations

Abstract

To the family of higher order Korteweg–de Vries (KdV) equations a new family of higher order mKdV equations depending on a parameter η is constructed. They are connected by an η2-dependent Miura transformation. A Bäcklund transformation is also established. Furthermore, a new gauge transformation is associated to this Bäcklund transformation. It enables one to derive auto-Bäcklund transformations for the families of higher order η2-mKdV and KdV equations, respectively. For illustration, four generations of explicit solutions of the second order KdV equation are presented.

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What this paper is about

To the family of higher order Korteweg–de Vries (KdV) equations a new family of higher order mKdV equations depending on a parameter η is constructed. They are connected by an η2-dependent Miura transformation. A Bäcklund transformation is also established. Furthermore, a new gauge transformation is associated to this Bäcklund transformation. It enables one to derive auto-Bäcklund transformations for the families of higher order η2-mKdV and KdV equations, respectively. For illustration, four generations of explicit solutions of the second order KdV equation are presented.

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Available abstract

To the family of higher order Korteweg–de Vries (KdV) equations a new family of higher order mKdV equations depending on a parameter η is constructed. They are connected by an η2-dependent Miura transformation. A Bäcklund transformation is also established. Furthermore, a new gauge transformation is associated to this Bäcklund transformation. It enables one to derive auto-Bäcklund transformations for the families of higher order η2-mKdV and KdV equations, respectively. For illustration, four generations of explicit solutions of the second order KdV equation are presented.

Key concepts: Korteweg–de Vries equation, Transformation (genetics), Mathematics, Order (exchange), Gauge (firearms), Mathematical physics, Gauge theory, Dispersionless equation

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