T.R.O. Melvin, A.R. Champneys, and D.E. Pelinovsky

Discrete traveling solitons in the Salerno model

SIAM Journal of Applied Dynamical Systems 8, 689709 (2009)

We investigate travelling solitary waves in the 1D Salerno model, which interpolates between the cubic discrete nonlinear Schrodinger (DNLS) equation and the integrable Ablowitz-Ladik (AL) model. In a travelling frame the model becomes an advance delay equation to which we analyse existence of homoclinic orbits to the rest state. The method of beyond all orders asymptotics is used to compute the so-called Stokes constant that measures the splitting of the stable and unstable manifolds. Through computing zeros of the Stokes constant, we find that there exists a number of solution families for parameter values between the DNLS and AL limits of the Salerno model. Using a pseudo-spectral method, we numerically continue these solution families and show that their parameters approach the curves of the zero level of the Stokes constant as the soliton amplitude approaches zero. An interesting topological structure of solutions occurs in parameter space. As the AL limit is approached solutions sheets of single-hump solutions undergo folds and become double-hump solitons. Numerical simulation suggest the single-humps to be stable and to interact almost inelastically.

discrete nonlinear Schrodinger equation, traveling wave solutions, differential advance-delay equations, Stokes constants, beyond-all-order asymptotic expansions, stability of nonlinear waves