Publication Date:
2014-03-06

Description:
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 dn 2 ( x , m ) ± m cn ( x , m ) dn ( x , m ) , 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.

Print ISSN:
0022-2488

Electronic ISSN:
1089-7658

Topics:
Mathematics
,
Physics

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