Research Article | Open Access
Topological Soliton Solution and Bifurcation Analysis of the Klein-Gordon-Zakharov Equation in -Dimensions with Power Law Nonlinearity
This paper addresses the Klein-Gordon-Zakharov equation with power law nonlinearity in ()-dimensions. The integrability aspect as well as the bifurcation analysis is studied in this paper. The numerical simulations are also given where the finite difference approach was utilized. There are a few constraint conditions that naturally evolve during the course of derivation of the soliton solutions. These constraint conditions must remain valid in order for the soliton solution to exist. For the bifurcation analysis, the phase portraits are also given.
The theory of nonlinear evolution equations (NLEEs) has come a long way in the past few decades [1–20]. Many of the NLEEs are pretty well known in the area of theoretical physics and applied mathematics. A few of them are the nonlinear Schrödinger's equation, Korteweg-de Vries (KdV) equation, sine-Gordon equation which appear in nonlinear optics, fluid dynamics, and theoretical physics, respectively. It is also very common to come across several combo NLEEs such as the Schrödinger-KdV equation, Klein-Gordon-Zakharov (KGZ) equation, and many others that are also studied in the context of applied mathematics and theoretical physics. This paper is going to focus on the KGZ equation that will be studied with power law nonlinearity in -dimensions.
The integrability aspects and the bifurcation analysis will be the main focus of this paper. The ansatz method will be applied to obtain the topological 1-soliton solution, also known as the shock wave solution, to this equation. The constraint conditions will be naturally formulated in order for the soliton solution to exist. Subsequently, the bifurcation analysis will be carried out for this paper. In this context, the phase portraits will be given. Additionally, other traveling wave solutions will be enumerated. Finally, the numerical simulation to the equation will be given. The finite difference scheme will also be given.
2. Mathematical Analysis
The KGZ equation with power law nonlinearity in -dimensions that are going to be studied in this paper is given by  where , , , , and are real valued constants. Additionally is a complex valued dependent variable and is a real valued dependent variable. This section will focus on extracting the shock wave solutions to the KGZ equation (1) and (2) that are also known as topological soliton solution. Therefore the starting hypothesis will be where Here, in (3) and (4) , and are free parameters, while is the velocity of the soliton. The unknown exponents and will be determined, in terms of by the aid of balancing principle. The phase component of (3) is given by where represents the soliton frequency, is the soliton wave number, and is the phase constant. Substituting the hypothesis (3) and (4) into (1) and (2) yields respectively. Now, splitting (7) into two real and imaginary parts gives From (9), equating the exponents and gives and then equating with gives Finally, equating the exponent pairs and gives Now the values of and from (11) and (12) satisfy (13).
Finally, equating the coefficients of the linearly independent functions , in (9) and (10) to zero gives Again, equating the coefficients of the linearly independent functions , in (8) to zero implies Solving (14)-(15) we get The relations (17), (18), and (19) introduce the restrictions given by Thus the topological solution of the and wave functions are given:
3. Bifurcation Analysis
This section will carry out the bifurcation analysis of the Klein-Gordon-Zakharov equation with power law nonlinearity. Initially, the phase portraits will be obtained and the corresponding qualitative analysis will be discussed. Several interesting properties of the solution structure will be obtained based on the parameter regimes. Subsequently, the traveling wave solutions will be discussed from the bifurcation analysis.
3.1. Phase Portraits and Qualitative Analysis
Substituting (22) and (23) into (1) and (2), we find that , and satisfy the following system: Integrating (25) twice and letting the first integral constant be zero, we have where is the second integral constant.
Substituting (26) into (24), we have To facilitate discussions, we let Letting , then we get the following planar system: Obviously, the above system (30) is a Hamiltonian system with Hamiltonian function In order to investigate the phase portrait of (30), set Obviously, when , has three zero points, , , and , which are given as follows: When , has only one zero point Letting be one of the singular points of system (30), then the characteristic values of the linearized system of system (30) at the singular points are From the qualitative theory of dynamical systems, we know the following.(I) If , is a saddle point.(II) If , is a center point.(III) If , is a degenerate saddle point.
Let where is Hamiltonian.
Next, we consider the relations between the orbits of (30) and the Hamiltonian .
According to Figure 1, we get the following propositions.
Proposition 1. Suppose that and , one has the following.(I)When , system (30) does not have any closed orbits. (II)When , system (30) has two periodic orbits and .(III)When , system (30) has two homoclinic orbits and .(IV)When , system (30) has a periodic orbit .
Proposition 2. Suppose that and , one has the following.(I)When or , system (30) does not have any closed orbits. (II)When , system (30) has three periodic orbits , , and .(III)When , system (30) has two periodic orbits and .(IV)When , system (30) has two heteroclinic orbits and .
From the qualitative theory of dynamical systems, we know that a smooth solitary wave solution of a partial differential system corresponds to a smooth homoclinic orbit of a traveling wave equation. A smooth kink wave solution or a unbounded wave solution corresponds to a smooth heteroclinic orbit of a traveling wave equation. Similarly, a periodic orbit of a traveling wave equation corresponds to a periodic traveling wave solution of a partial differential system. According to the above analysis, we have the following propositions.
Proposition 4. If and , one has the following.(I)When , (1) and (2) have two periodic wave solutions (corresponding to the periodic orbits and in Figure 1). (II)When , (1) and (2) have two solitary wave solutions (corresponding to the homoclinic orbits and in Figure 1). (III)When , (1) and (2) have two periodic wave solutions (corresponding to the periodic orbit in Figure 1).
Proposition 5. If and , one has the following.(I)When , (1) and (2) have two periodic wave solutions (corresponding to the periodic orbit in Figure 1) and two periodic blow-up wave solutions (corresponding to the periodic orbits and in Figure 1). (II)When , (1) and (2) have periodic blow-up wave solutions (corresponding to the periodic orbits and in Figure 1). (III)When , (1) and (2) have two kink profile solitary wave solutions. (corresponding to the heteroclinic orbits and in Figure 1).
3.2. Exact Traveling Wave Solutions
Firstly, we will obtain the explicit expressions of traveling wave solutions for (1) and (2) when and . From the phase portrait, we see that there are two symmetric homoclinic orbits and connected at the saddle point . In -plane the expressions of the homoclinic orbits are given as Substituting (38) into and integrating them along the orbits and , we have where and .
Secondly, we will obtain the explicit expressions of traveling wave solutions for (1) and (2) when and . From the phase portrait, we note that there are two special orbits and , which have the same Hamiltonian with that of the center point . In -plane the expressions of the orbits are given as Substituting (42) into and integrating them along the two orbits and , it follows that where .
4. Numerical Simulation
We decompose the function in (1) in the form Substituting in (1) and (2) we have We assume that , , is the exact solution and , , is the approximate solution at the grid point . The proposed scheme can be displayed as where The similar notation for and are The proposed scheme is implicit and can be easily solved by the fixed point method. The scheme is second order in space and time directions.
This paper studied the KGZ equation in -D with power law nonlinearity from three different avenues. First, the topological 1-soliton solution to the equation was determined by the aid of ansatz method. The by-product of this solution is a couple of constraint conditions that must remain valid in order for the solitons to exist. Subsequently, the bifurcation analysis is carried out for this equation that leads to the phase portraits and several other solutions to the equation, using the traveling wave hypothesis. This leads to the solitary waves and periodic singular waves. Finally, the numerical simulation that was conducted using finite difference scheme leads to the simulations for the topological soliton solutions.
These results are pretty complete in analysis. They are going to be extended in the future. An obvious way to expand or generalize these results is going to extend to -D. These results will be reported soon. Another avenue to look into this equation further is to consider the perturbation terms and then obtain exact solution, and additionally study the perturbed KGZ equation using other tools of integrability. These include mapping method, Lie symmetries, exp-function method, and the -expansion method. These will lead to a further plethora of solutions. Such results will be reported in the future. That is just a foot in the door.
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