Traveling waves in a generalized Camassa–Holm equation involving dual-power law nonlinearities

The solitary wave, kink and anti-kink solutions coexist at the same speed in a perturbed nonlinear Schrödinger equation

“…The geometric singular perturbation theory (GSPT) (see Fenichel [15] and Jones [20]) is an effective tool to deal with this issue (e.g. see Li et al [24][25][26], Du et al [11,13,14], Chen et al [8], Wang and Zhang [39], Qiu et al [31], Zhu et al [46], Mansour [29]). Ogawa [30] and Du et al [12] studied the so-called 'Saddle loop' case (see figure 1(a)) corresponding to the unperturbed Kortewegde Vries (KdV) equation and Nizhnik-Novikov-Veselov equation, it can be seen that the Hamiltonian system with an elliptic Hamiltonian of degree three is H(U, V) = 1 2 V 2 + f(U), where f (U) is a polynomial in U of degree three.…”

Geometric singular perturbation analysis to a perturbed (1 + 1)-dimensional dispersive long wave equation

“…There exists an extensive literature on the use of geometric singular perturbation theory [22,23] to deal with these issues including KdV equation [24,25], generalized KdV equation [26][27][28], delayed CH equation [29], generalized CH equation [30], CH Kuramoto-Sivashinsky equation [31], perturbed BBM equation [32], generalized BBM equation [33],…”

“…Except for detecting the MRAI of perturbed equations, another effective method is to directly compute the explicit expression of Melnikov function which, in turn, the existence of traveling wave solution is proved. Qiu et al [30], Zhu et al [47] used the explicit expression of Melnikov function to discuss the existence of solitary waves in a perturbed generalized BBM and generalized CH equation, respectively. Cheng and Li [48] computed the expression of Melnikov function for the delayed DP equation.…”