Using commutation methods, the authors present a general formalism to construct Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) \(N\)-soliton solutions relative to arbitrary (m)KdV background solutions. As an illustration of these techniques, the authors combine them with algebro-geometric methods and Hirota's \(\tau\)-function approach to systematically derive the (m)KdV \(N\)-soliton solutions on quasi-periodic finite-gap backgrounds.

Readership

Graduate students, research mathematicians, and theoretical physicists interested in soliton mathematics.

Table of Contents

• Introduction
• Quasi-periodic finite-gap (m)KdV-solutions
• (m)KdV-soliton solutions on quasi-periodic finite-gap backgrounds
• I. The single commutation method
• (m)KdV-soliton solutions on quasi-periodic finite-gap backgrounds
• II. The double commutation method
• Appendix A: Single commutation methods
• Appendix B: Double commutation methods
• Appendix C: Lax pairs, \(\tau\)-functions and Bäcklund transformations
• Appendix D: (m)KdV-soliton solutions relative to general backgrounds
• Appendix E: Hyperelliptic curves and theta functions
• References