Bifurcation structure and stability of solitary waves in the cubic–quintic nonlinear Schrödinger equation with self-steepening

We numerically investigate how self-steepening and quintic nonlinearity influence the bifurcation structure and stability of solitary waves in the cubic–quintic nonlinear Schrödinger equation with self-steepening and a symmetric double-well potential. In the regime with self-focusing cubic and self-defocusing quintic nonlinearities, the interplay between competing nonlinearities gives rise to intricate bifurcation patterns, including double pitchfork and saddle–node bifurcations. Increasing the self-steepening strength modifies the bifurcation topology by eliminating certain branches and reducing multistability through the suppression of saddle–node bifurcations. As the defocusing strength of the quintic term increases, symmetry breaking is progressively suppressed, and the bifurcation structure reduces to a continuous symmetric branch that folds at a saddle–node bifurcation, where both the lower and upper segments are stable. In contrast, when both nonlinearities are self-focusing, the bifurcation structure exhibits a single supercritical pitchfork bifurcation accompanied by stable asymmetric branches and remains qualitatively unchanged under variations in either the quintic or self-steepening parameters. These results provide valuable insight into how higher-order nonlinearities shape the existence and stability of localized states, with potential applications in ultrafast optics and nonlinear wave phenomena.