Traveling wave solutions for the generalized (2+1)-dimensional Kundu-Mukherjee-Naskar equation

The paper’s Theorem 1 (for system (8)) and Theorem 2 (for system (14)) state exactly the same existence and type (center/saddle/cusp) classifications as the candidate solution, including the degenerate cases aκω − r = 0 and the cusp at the double root threshold for system (14). The paper justifies these via Hamiltonian structure and standard normal-form arguments, e.g., p' = y, y' = −(2κb/(am))p^3 in the degenerate case for system (8), and a k=2, bn=0 normal form for system (14) at the double root, concluding a cusp; the candidate solution uses the same Hamiltonian and linearization facts and the same cusp normal form, differing only in exposition detail. See Theorem 1 and its proof for system (8) and the treatment of the degenerate case aκω − r = 0, as well as the construction and energy function for system (14), and Theorem 2 for the cusp and center/saddle partition by parameters .