Qualitative analysis and explicit traveling wave solutions for the Davey–Stewartson equation

Abstract

In this paper, we use the bifurcation method of dynamical systems to study the traveling wave solutions for the Davey-Stewartson equation. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow-up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended.where u is the (complex valued) envelope of the wave packet associated with the fast oscillations, and v is the induced mean flow, x and y are the slow, horizontal scales parallel and perpendicular to the fast oscillations, respectively, whereas t is the slow time in the group velocity frame. As usual, irrotational flow of an inviscid fluid is studied, and u is connected with the velocity potential. Of particular significance is the parameter a, and a D 1 and a D 1 is called the DS I and DS II equations, respectively. The constant b measures the cubic nonlinearity. The DS equation is the two-dimensional generalization of the nonlinear Schrödinger (NLS) equation.The solutions to the DS equation have been studied previously in various aspects. The N-soliton solution of the DS equation was studied by Anker and Freeman [3] in the inverse scattering transform method and later by Satsuma and Ablowitz [4] in the Hirota method. Satsuma and Ablowitz also revealed that the solutions describing multiple collisions of rational solitons are constructed from the N-soliton solutions for the equation. The interaction of two dark line solitons, which are skewed with respect to each other, has been studied by Anker and Freeman [5]. They found the existence of soliton resonance. The dromion solutions to the DS equation, which decay exponentially in all directions, were obtained in various aspects [6][7][8][9]. The DS equation is known to have solutions called solitoffs, and solitoff-solitoff and solitoff-dromion resonant behaviors have been studied [10]. Watanabe and Tajiri [11] and Tajiri et al. [12] investigated the interactions between y-periodic soliton and line soliton and between y-periodic soliton and algebraic soliton. Recently, doubly periodic wave patterns were studied by Chow and Lou [13] and Li et al. [14]. Tajiri and Arai [15] studied the existence of long-range interaction between two quasi-line solitons through a periodic soliton.The aim of this paper is to study the traveling wave solutions and their relations for Equation (1.1) by using the bifurcation method and qualitative theory of dynamical systems [16][17][18][19][20]. Through some special phase orbits, we obtain many smooth periodic wave solutions and periodic blow-up solutions. Their limits contain kink profile solitary wave solutions, unbounded wave solutions, periodic blow-up solutions, and solitary wave solutions.

Some new nonlinear wave solutions and their convergence for the (2+1)‐dimensional Broer–Kau–Kupershmidt equation

“…Hence, m 4˛, 0 is a saddle point when mˇ< 0, and m 4˛, 0 is a center when mˇ> 0. According to the qualitative theory of dynamical systems [16], we obtain the bifurcation phase portraits of system (3.5) as Figures 11,12,13,and 14. Using the information provided by the phase portraits, we give derivations to our main results as follows.…”

“…For the parametric conditions, see [8]. In this paper, by employing the bifurcation method of dynamical systems [9][10][11][12][13][14][15], we obtain some new nonlinear wave solutions for Equation (1.1). These solutions contain solitary wave solutions, blow-up wave solutions, periodic smooth wave solutions, periodic blowup wave solutions, and kink wave solutions.…”

Soliton solution and bifurcation analysis of the KP–Benjamin–Bona–Mahoney equation with power law nonlinearity

“…Letting (g i , 0) be one of the singular points of system (10), then the characteristic values of the linearized system of system (10) at the singular points (g i , 0) are…”

“…The study of nonlinear evolution equations (NLEEs) has been going on for quite a few decades now [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. There has been several improvements that are noticed.…”

Infinitely many solitary waves of an integrable equation with singularity

“…For this reason, the scope of this paper is to explore the relationship between the existence of the infinitely many solitary waves and the wave speed for Eq. (1.5) by adopting the bifurcation method of dynamical systems [6,7,[16][17][18][19][20][21][22][23]. We list the other approaches for solving the solitary waves for comparison [5,[8][9][10]24,25].…”