Integral bifurcation method combined with computer for solving a higher order wave equation of KdV type

Some new travelling wave solutions with singular or nonsingular character for the higher order wave equation of KdV type (III)

“…The integral bifurcation method possessed two kinds of strongpoint of the bifurcation theory of dynamic system (see [12][13][14][15][16] and references cited therein), extended F-expansion method and auxiliary equation method (see [17][18][19]and references cited therein), and easily combined with computer method [20]. So, by using this method, we shall obtain some peculiar travelling wave solutions with singular or nonsingular character of Eq.…”

Frobenius’ Idea Together with Integral Bifurcation Method for Investigating Exact Solutions to a Water Wave Model of the Generalized mKdV Equation

“…where is an integral constant. Employing direct Frobenius' idea, we need not change (5) into a 2-dimensional planar system as the method in [33][34][35][36]. we can directly integrate (5) again; see the following calculus.…”

“…Finding their exact solutions has extensive applications in many scientific fields such as hydrodynamics, condensed matter physics, solid-state physics, nonlinear optics, neurodynamics, crystal dislocation, model of meteorology, water wave model of oceanography, and fibre-optic communication. The research methods for solving nonlinear PDEs deal with the inverse scattering transformation [1,2], the Darboux transformation [3][4][5], the Bäcklund transformation [5][6][7][8], the bilinear method and multilinear method [9,10], the classical and nonclassical Lie group approaches [11,12], the Clarkson-Kruskal direct method [13][14][15], the mixing exponential method [16], the geometrical method [17,18], the truncated Painlev́expansion [19,20], the function expansion method (including tanh expansion method [21,22], sine-cosine expansion method [23,24], exp-function method [25], and multiple exp-function method [26]), the bifurcation theory of planar dynamical system [27,28], the F-expansion type method [29,30], / method [31,32], and the integral bifurcation method [33][34][35][36]. Among these available methods for solving nonlinear PDEs, some of them employed Frobenius' idea directly or indirectly.…”

A singular limit problem for the Kudryashov-Sinelshchikov equation

“…Particularly, when (i.e., ), a kind of new periodic wave solutions which is called meandering solutions was obtained. In , the authors studied by using the integral bifurcation method (see ). They found some new traveling wave solutions of , which extends the results in .…”