Linear Superposition in Nonlinear Equations

Abstract

Even though the KdV and modified KdV equations are nonlinear, we show that suitable linear combinations of known periodic solutions involving Jacobi elliptic functions yield a large class of additional solutions. This procedure works by virtue of some remarkable new identities satisfied by the elliptic functions.

7 pages, 1 figure

References in corpus (2)

• Cyclic Identities Involving Jacobi Elliptic Functions
• Some Exact Results for Mid-Band and Zero Band-Gap States of Associated Lame Potentials

Cited by in corpus (13)

• Representations and classification of traveling wave solutions to Sinh-G{ö}rdon equation
• Cyclic Identities Involving Jacobi Elliptic Functions
• Cyclic Identities Involving Jacobi Elliptic Functions. II
• Superposition in nonlinear wave and evolution equations
• Periodic Solutions of Nonlinear Equations Obtained by Linear Superposition
• Existence of superposition solutions for pulse propagation in nonlinear resonant media
• Linear superposition in the general heavenly equation
• Superposition principle and composite solutions to coupled nonlinear Schrödinger equations
• A Generalization of Landen's Quadratic Transformation Formulas for Jacobi Elliptic Functions
• Generalized Landen Transformation Formulas for Jacobi Elliptic Functions
• New superposed elliptic solutions for a class of nonlinear evolution equations
• Nonlinear extension of the J-matrix method of scattering: A toy model
• Cyclic Identities Involving Ratios of Jacobi Theta Functions