L. Blank and T. Dohnal
Surface Gap Solitons are solitary waves in periodic structures localized to
an interface. We consider the 1D periodic nonlinear Schroedinger model with
a cubic nonlinearity and a nonlinearity interface represented by a discontinuity
in the nonlinearity coeffcient. As shown in [1], this model supports families
of surface gap solitons. These solutions `physical' and are observable in experiment
only if they are stable with respect to perturbations of the intial data. We
study, therefore, their linear stability. Direct numerical eigenvalue computations
of the linearization eigenvalue problem often lead to spurious eigenvalues and
are hard to perform systematically as the support of localized eigenfunctions
is a-priori unknown. We select an alternative approach based on the Evans function
method analogously to [2], where the eigenvalue problem is reduced to an ODE
evolution of stable and unstable manifolds corresponding to the zero solution of the linearized problem and
an evaluation of a Wronskian. The evolution needs to be carried out only over the effective support of the surface gap soliton. Reformulation
of the problem in exterior algebra turns out necessary for removing stuffness of the ODE.
Results of this work have recently been submitted in [3].
[1] T. Dohnal and D. Pelinovsky, ``Surface Gap Solitons at a Nonlinearity Interface," SIAM J. Appl. Dyn. Syst. 7, 249-264 (2008). (arXiv:0704.1742)
[2] D. E. Pelinovsky, A. A. Sukhorukov and Y. S. Kivshar, "Bifurcations and stability of gap solitons in periodic potentials," Phys. Rev. E 70, 036618 (2004 ).
[2] E. Blank and T. Dohnal, "Families of Surface Gap Solitons and their Stability via the Numerical Evans Function Method," submitted to SIAM J. Appl. Dyn. Syst., 2009. (arXiv:0910.4858)