295 research outputs found
Superposition of Elliptic Functions as Solutions For a Large Number of Nonlinear Equations
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 , 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 Schr\"odinger equation, MKdV, a mixed
KdV-MKdV system, a mixed quadratic-cubic nonlinear Schr\"odinger equation, the
Ablowitz-Ladik equation, the saturable nonlinear Schr\"odinger equation,
, 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) \pm \sqrt{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.Comment: 40 pages, no figure
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