Series solutions and bifurcation of traveling waves in the Benney–Kawahara–Lin equation

Wave solutions of the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity

“…In addition, when referring to dynamical behaviour of wave solutions, people still do not clearly know how these solutions evolve with variation of parameters. In order to solve these problems, we introduce the dynamical system method to study equation (1.1), which has been shown to be a powerful and efficient method to find traveling wave solutions [17][18][19]41,42]. By this method, we convert equation (1.1) into corresponding traveling wave system and discuss two cases (α = 2β and α ̸ = 2β ).…”

Generalized variational principles of the Benney-Lin equation arising in fluid dynamics

“…( 1) reduces to the generalized Kuramoto-Sivashinsky equation that describes the waves in the vertical and inclined falling film, in liquid films that are subjected to interfacial stress from adjacent gas flow [15]. First and last, the BLE is an important general equation in the field of fluid dynamics and its solution has always been a research hotspot, many different effective method have been obtained such as variational iteration method [15], homotopy perturbation method [16], residual power series method [17], Lie symmetry analysis [18], Adomian decomposition method [19] and so on [20,21]. However its variational principle has not been studied and reported yet.…”

On the solutions for a Benney-Lin type equation

“…In particular, [16] show that the Cauchy problem of ( 6) is ill-posed in some energy spaces. Explicit and exact solutions for (6) are analyzed in [92], while the existence of the traveling-wave solutions is studied in [78,96].…”

“…Moreover, in [55], the author deduced (8) to describe describing one-dimensional propagation of smal-amplitude long waves in various problems of fluid dynamics and plasma physics. Mathematical properties of (8) were studied recently in many detail, including the local and global well-posedness in Bourgain spaces [41,40,53,54,90], the local and global well-posedness in energy space [43,45,50,88,93,94], the existence of solitary wave solution [9,51], the stability of periodic traveling wave solutions see [2,1,73,87,96], the well-posedness of the initial-boundary value problem on a bounded domain [10,13,44,58], the initial-boundary value problem on the half-line [12,11], periodic solutions [6,79], and numerical solutions [80,4,42,52,72,91]. In [74], the authors prove that the solution of (8) converges to the solution of (7), while, following [26,19,20,27,66,82], in [23,24], the convergence of the solution of…”