### Abstract

We investigate travelling solitary waves in the 1D Salerno model, which interpolates between the cubic discrete nonlinear Schrödinger (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.

Original language | English |
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Publication status | Published - 2008 |

### Bibliographical note

Sponsorship: UK EPSRC### Keywords

- asymptotics
- stokes constant
- solitons
- dnls

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## Cite this

Melvin, TRO., Champneys, AR., & Pelinovsky, Dmitr, E. (2008).

*Discrete travelling solitons in the Salerno model*. http://hdl.handle.net/1983/1071