Bifurcations of travelling wave solutions in the discrete NLS equations

We show that there exists a continuous two-parameter family of single-humped travelling wave solutions in the third-order derivative NLS equation, when it is derived from the integrable Ablowitz-Ladik lattice. On the contrary, there are no single-humped solutions in the third-order derivative NLS equation, when it is derived from the cubic NLS equation with on-site interactions. Nevertheless, we show that there exists an infinite discrete set of one-parameter families of double-humped travelling wave solutions in the latter case. Our results are valid in the neighborhood of the zero-dispersion point on the two-parameter plane of travelling wave solutions.