Families of Surface Gap Solitons and Their Stability via the Numerical Evans Function Method

The nonlinear Schr\"{o}dinger equation with a linear periodic potential and a nonlinearity coefficient $\Gamma$ with a discontinuity supports stationary localized solitary waves with frequencies inside spectral gaps, so called surface gap solitons (SGSs). We compute families of 1D SGSs using the arclength continuation method for a range of values of the jump in $\Gamma$. Using asymptotics, we show that when the frequency parameter converges to the bifurcation gap edge, the size of the allowed jump in $\Gamma$ converges to 0 for SGSs centered at any $x_c\in \R$. Linear stability of SGSs is next determined via the numerical Evans function method, in which the stable and unstable manifolds corresponding to the 0 solution of the linearized spectral ODE problem need to be evolved. Zeros of the Evans function coincide with eigenvalues of the linearized operator. Far from the SGS center the manifolds are spanned by exponentially decaying/increasing Bloch functions. Evolution of the manifolds suffers from stiffness but a numerically stable formulation is possible in the exterior algebra formulation and with the use of Grassmanian preserving ODE integrators. Eigenvalues with positive real part above a small constant are then detected using the complex argument principle and a contour parallel to the imaginary axis. The location of real eigenvalues is found via a straightforward evaluation of the Evans function along the real axis and several complex eigenvalues are located using M\"{u}ller's method. The numerical Evans function method is described in detail. Our results show the existence of both unstable and stable SGSs (possibly with a weak instability), where stability is obtained even for some SGSs centered in the domain half with the less focusing nonlinearity. Direct simulations of the PDE for selected SGS examples confirm the results of Evans function computations.