Spectral Stability of Constrained Solitary Waves for the Generalized Singular Perturbed KdV Equation

This paper is systematically concerned with the solitary waves on the constrained manifold preserved the \(L^2\)-momentum conservation for the generalized singular perturbed KdV equation with \(L^2\)-subcritical, critical and supercritical nonlinearities, which is a long-wave approximation to the capillary-gravity waves in an infinitely long channel with a flat bottom. First, using the profile decomposition in \(H^2\) and the optimal Gagliardo–Nirenberg inequality, we prove the existence of subcritical ground state solitary waves and describe their asymptotic behavior. Second, we obtain some sufficient conditions for the existence and non-existence of critical ground states, and then prove the existence of critical and supercritical ground state solitary waves on the Derrick–Pohozaev manifold by utilizing the new minimax argument and the numerical simulation of the best Gagliardo–Nirenberg embedding constant. Meanwhile, we use the moving plane method to obtain the existence of positive and radially symmetric solutions. Furthermore, we study the concentration behavior of the critical ground state solutions. Finally, the spectral stability of the ground state solitary wave solutions is discussed by using the instability index theorem.