Periodic Solutions of Nonlinear Equations Obtained by Linear Superposition

Abstract

We show that a type of linear superposition principle works for several nonlinear differential equations. Using this approach, we find periodic solutions of the Kadomtsev-Petviashvili (KP) equation, the nonlinear Schrodinger (NLS) equation, the $\lambda {\varphi }^4$ model, the sine-Gordon equation and the Boussinesq equation by making appropriate linear superpositions of known periodic solutions. This unusual procedure for generating solutions is successful as a consequence of some powerful, recently discovered, cyclic identities satisfied by the Jacobi elliptic functions.

19 pages, 4 figures

References in corpus (3)

• Linear Superposition in Nonlinear Equations
• Cyclic Identities Involving Jacobi Elliptic Functions
• A Generalization of Landen's Quadratic Transformation Formulas for Jacobi Elliptic Functions

Cited by in corpus (11)

• Elliptic solutions to a generalized BBM equation
• Cyclic Identities Involving Jacobi Elliptic Functions. II
• Superposition in nonlinear wave and evolution equations
• Linear superposition in the general heavenly equation
• Mass-imbalance induced structures of binary atomic mixtures in box potentials
• Superposition principle and composite solutions to coupled nonlinear Schrödinger equations
• Quench Dynamics and Emergence of Phase Separation in Two-Component Atomic Bose Gases at Zero Temperature and above the BEC Critical Temperature
• 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
• Cyclic Identities Involving Ratios of Jacobi Theta Functions