The Soliton Wave Solutions and Bifurcations of the (2 + 1)-Dimensional Dissipative Long Wave Equation

With the help of the bifurcation theory of dynamical differential system and maple software, we shall devote ourselves to research travelling wave solutions and bifurcations of the (2 + 1)-dimensional dissipative long wave equation. The study of travelling wave solutions for long wave equation derives a planar Hamiltonian system. Based on phase portraits, we obtain exact explicit expressions of some bounded traveling wave solutions and some important singular traveling wave solutions, under different parametric conditions.


Introduction
Nonlinear phenomena exist in many branches of science and engineering, such as plasma physics, fluid mechanics, gas dynamics, elasticity, relativity, chemical reactions, ecology, fiber optics, solid state physics, biomechanics, etc., which are essentially governed by nonlinear partial differential equation. With the increasing appearance of traveling waves in many physical phenomena, it becomes more and more important to seek the exact traveling wave solutions, which are helpful to provide information about the structure of complex physical phenomena and reveal the profound nature of physics. Therefore, it is an important task to learn the exact traveling wave solution of the nonlinear partial differential equation. It is worth noting that there is no one unique method that can solve the exact traveling wave solutions of all types of nonlinear evolution equations led to appearing a number of methods, such as the Jacobi elliptic function method [1], the homogeneous balance method [2], improved simple equation method [3], G ′ G -expansion method [4][5][6], improved G ′ G -expansion method [7], Truncation Painleve expansion method [8], homotopy perturbation method [9], variational method [10].
In past few years, the (2 + 1)-dimensional dissipative long wave equation [11] has played an important role in many fields of Physics. Such a significant nonlinear model has attracted many people's interest. The (2 + 1)-dimensional dissipative long wave equation as follows: In [12], the exp(− ( )) expansion method [13] and the variable separation method [14] have been applied to system (1.1) to learn the structure of their local soliton wave and the evolution of soliton wave with time.
In this paper, we consider travelling wave solutions and bifurcations of system (11), by using the bifurcation theory of dynamical differential system [15][16][17][18][19][20] and maple software. Under fixed parameter conditions, the exact explicit parametric representations of some bounded traveling wave solutions are obtained. In addition, we obtain some important singular wave solutions.

Traveling Wave Transformation and First Integral
To find the traveling wave solutions of system (1.1), we first assume that and where is the traveling coordinate, u( ) is the real amplitude function to be determined, c is the wave velocity. Substituting (2.1) into system (11), system (1.1) is equivalent to the following ordinary differential system: where u � ( ) = du( ) d . The first expression of (2.2) integrated once, we get The second expression of (2. The phase orbits of system (2.6) determine all traveling wave solutions of system (11). Suppose that v = v( ) is a continuous solution of system (2.6) for ∈ (−∞, +∞) and lim homoclinic orbit corresponds to a solitary wave solution of system (2.5). If P ≠ Q , v( ) is called a heteroclinic orbit corresponds to a kink(anti-kink) wave solution of system (2.5). Thus, in order to investigate traveling wave solutions of system (1.1), we shall try to find as many bounded traveling wave solutions as possible and some singular wave solutions of system (2.6) depending on the parameters.

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, there exist two equilibrium points of system (2.6) on the v-axis, one of them is a double equilibrium point. When > 0 , Based on the above analysis, according to the bifurcation theory of dynamical differential system [15][16][17], we obtain the phase portraits and bifurcations of system (2.6) as follows:

The Travelling Wave Solutions of System (11)
In Fig. 1(1)- Fig. 1(16), there is no bounded traveling wave solution of system (2.6) as we can see from Fig. 1. In Fig. 1(17)-Fig. 1(37), there exist periodic waves, homoclinic orbit waves, and heteroclinic orbit waves of system (2.6), corresponding to periodic wave solutions, solitary wave solutions, and kink(anti-kink) wave solutions of system (2.5), respectively. Next we shall calculate the exact expressions for these travelling wave solutions. Note that the relation between the solutions of system (2.5) and the solutions of system (1. In order to state conveniently, we shall omit the expressions of the component u(x, y, t) of system (11) in the following results. For the convenience of expression, we define From (2.7), we obtain Substituting (4.1) into the first equation of system (2.6), we get We discuss three cases as follows: Case I: c 2 > 4 , = 0.

Fig. 3 A family of periodic wave solutions defined by
Journal of Nonlinear Mathematical Physics (2022) 29:659-677 From (4.4), we get the following a family of singular periodic wave solutions, see Fig. 4.

Fig. 5 The curves defined by
ds.

Fig. 9 A solitary wave solution defined by H(v, ) = h 3 for
, the level curves defined by H(v, ) = h include a family of periodic wave orbit and a family of singular wave orbit, as shown in Fig. 11. The Fig. 11 The curves defined by H(v, ) = h for c 2 > 4 , parametric representation of these periodic wave solutions are the same as (4.5). The parametric representation of these singular wave solutions are the same as (4.6).
From (4.10), we obtain a solitary wave solution as follows, see Fig. 14. (4.14) ds, ds, parametric representation of these periodic wave solutions are the same as (4.5). The parametric representation of these singular wave solutions are the same as (4.6).

Conclusion
In the paper, with the help of the bifurcation theory of dynamical differential system and maple software, we study the nonlinear wave solutions to the (2+1)-dimensional dissipative long wave equation (1.1) and obtain exact explicit expressions of some bounded traveling wave solutions, including solitary wave solutions, periodic wave solutions, and kink(anti-kink) wave solutions. In addition, We get some singular periodic wave solutions and singular wave solutions.

Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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