Hybrid solitary waves for the generalized Kuramoto-Sivashinsky equation

Multi-form solitary wave solutions of the KdV-Burgers-Kuramoto equation

“…As an example, in figure 1 Top row (Left), when fixed the parameters b ; c ; θ; γ; β to −0.02;−0.5; 0.02; 0.01; 0.01, respectively, then derived a=−0.000 186 05; α=−0.226 0; ν=16.044 1 from equations (25), (26), (27) respectively. These profiles are different from those obtained in [17,18,[24][25][26][27][28]34].…”

“…We also use the notion of solitary wave family because in a common way, the sech analytic sequence is known as that which indicates the fundamental solitary wave of bright type and the analytical sequence tanh is known as that which indicates the fundamental solitary wave of dark type. In contrast to the previous work and the different representations of solitary wave profiles obtained in the past [17,18,[24][25][26][27][28]34], the analytical sequence sech tanh 2 4 qx qx is that of a family solitary waves of bright type or of the large family n sech 0 n qx  ( )and therefore the representative is sechqx and the analytic representation sech tanh 2 5 qx qx that of a dark type solitary wave or the great family n sinh sech 0 n qx qx  ( )and therefore the representative is tanh qx . Figures 1 and 2 show the hybrid solitary wave profiles resulting from the combination of the solitary wave packets of the dark family and the bright family, the analytical sequence of which is given by equation (28).…”

“…From equation (42), it is easy that α, β and θ satisfy the following condition 4 , 43 2 a q g = -( ) One notices that this result is the same as in equation (32). The resolution of equation (38) and equation ( It should be noted that all families of analytical solutions obtained in this section are different from those found in [11,17,18,[24][25][26][27][28]34]…”

“…It is a method developed on the theory of construction of nonlinear partial differential equation solutions [31][32][33][34][35][36][37][38][39][40][41][42]. This method is interested particularly the NPDEs of the form [31][32][33][34][35][36][37][38][39][40][41][42]…”

“…We focus on the case 0 gba ¹ in equation (1). Based on the BDKm largely explained in [31][32][33][34][35][36][37][38][39][40][41][42], equation (6) may have the following ansatz(which can be supported by equation (1)…”