### Abstract

For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions cn(x, m) and dn(x, m) with modulus m, then it also admits solutions in terms of their sum as well as difference. We have checked this in the case of several nonlinear equations such as the nonlinear Schrödinger equation, MKdV, a mixed KdV-MKdV system, a mixed quadratic-cubic nonlinear Schrödinger equation, the Ablowitz-Ladik equation, the saturable nonlinear Schrödinger equation, λϕ^{4}, the discrete MKdV as well as for several coupled field equations. Further, for a large number of nonlinear equations, we show that whenever a nonlinear equation admits a periodic solution in terms of dn^{2}(x, m), it also admits solutions in terms of , even though cn(x, m)dn(x, m) is not a solution of these nonlinear equations. Finally, we also obtain superposed solutions of various forms for several coupled nonlinear equations.

Received 02 December 2013
Accepted 11 February 2014
Published online 05 March 2014

Acknowledgments:
This work was supported in part by the U.S. Department of Energy.

Article outline:

I. INTRODUCTION
II. AS EXACT SOLUTIONS OF CONTINUUM NONLINEAR EQUATIONS
A. NLS equation
B. MKdV equation
C. ϕ^{2}-ϕ^{4} model
D. ϕ^{2}-ϕ^{3}-ϕ^{4} case
E. Mixed KdV-MKdV system
F. Mixed quadratic-cubic NLS equation
III. COUPLED CONTINUUM FIELD THEORIES WITH SOLUTIONS IN BOTH FIELDS
A. Coupled ϕ^{4} field theories
B. Coupled NLS-MKdV model
C. Coupled NLS model
IV. DISCRETE NONLINEAR EQUATIONS
A. Ablowitz-Ladik model
B. Saturable DNLS equation
C. Discrete λϕ^{4}
D. Discrete cubic-quintic model
E. Discrete MKdV model
V. SUPERPOSED SOLUTIONS
A. KdV equation
B. Quadratic NLS equation
C. ϕ^{3} field theory
VI. COUPLED FIELD THEORIES WITH SOLUTIONS
A. Quadratic NLS-KdV coupled theory
B. NLS-MKdV coupled field theory
C. Coupled NLS models
VII. MIXED AND SOLUTIONS IN COUPLED FIELD THEORIES
A. Coupled NLS-KdV fields
B. KdV-MKdV coupled system
C. Quadratic NLS-MKdV coupled model
D. Quadratic NLS-NLS coupled model
VIII. SUMMARY AND CONCLUSIONS

Commenting has been disabled for this content