Existence of solitary waves and periodic waves for a perturbed generalized BBM equation

On existence of kink and antikink wave solutions of singularly perturbed Gardner equation

“…[2][3][4][5][6][7] Especially, Chen and Liu 4 found the heteroclinic orbits and corresponding kink and antikink wave solutions by the method of dynamical systems. [8][9][10][11][12][13][14][15][16][17][18][19][20][21] Note that Equation (1) can be viewed as perturbation of Equation (2). To study the kink and antikink wave solutions of Equation (1), we first give some necessary results about Equation (2).…”

Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms

“…and found its compaction of dispersive structures. More recently, Chen et al [7] investigated a perturbed generalized BBM equation,…”

“…Both of the works [47] and [7] studied the perturbation problems restricted on manifolds, by using geometric singular perturbation theory. In [7], the authors applied Picard-Focus equations to determine the existence of periodic waves, and developed a good approach to prove that the dominating factor of the Melnikov function is monotonic, see Lemma 4.10 in [7]. Using the same approach, they also proved that the perturbed generalized defocusing mKdV equation,…”

The existence, uniqueness and asymptotic estimates of solutions for third-order full nonlinear singularly perturbed vector boundary value problems

“…Lodhi and Mishra [12] discussed second order singularly perturbed nonlinear boundary value problems by using the quintic B-spline method. Recently, the geometric singular perturbation theory has also received a great deal of interests in studying the Burgers-KdV equation [13], the vector-disease model [14], the perturbed BBM equation [15], the perturbed Camassa-Holm equation [16] and the perturbed shallow water wave model [17] etc.…”