Topological Soliton Solution and Bifurcation Analysis of the Klein-Gordon-Zakharov Equation in (1+1)-Dimensions with Power Law Nonlinearity

Abstract

This paper addresses the Klein-Gordon-Zakharov equation with power law nonlinearity in (1+1)-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.

1. Introduction

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 (1+1)-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 (1+1)-dimensions that are going to be studied in this paper is given by [6] 𝑞_𝑡𝑡−𝑘²𝑞_𝑥𝑥+𝑎𝑞+𝑏𝑟𝑞+𝑐𝑞^2𝑛𝑞=0,𝑟_𝑡𝑡−𝑘²𝑟_𝑥𝑥=𝑑𝑞^2𝑛_𝑥𝑥, (1)(2) where 𝑎,  𝑏,  𝑐,  𝑑, and 𝑘 are real valued constants. Additionally 𝑞(𝑥,𝑡) is a complex valued dependent variable and 𝑟(𝑥,𝑡) 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 𝑞(𝑥,𝑡)=𝐴₁tanh^𝑝1𝜏𝑒^𝑖𝜙,𝑟(𝑥,𝑡)=𝐴₂tanh^𝑝2𝜏, (3)(4) where 𝜏=𝐵(𝑥−𝑣𝑡). (5) Here, in (3) and (4) 𝐴₁,  𝐴₂ and 𝐵 are free parameters, while 𝑣 is the velocity of the soliton. The unknown exponents 𝑝₁ and 𝑝₂ will be determined, in terms of 𝑛 by the aid of balancing principle. The phase component of (3) is given by 𝜙=−𝜅𝑥+𝜔𝑡+𝜃, (6) where 𝜅 represents the soliton frequency, 𝜔 is the soliton wave number, and 𝜃 is the phase constant. Substituting the hypothesis (3) and (4) into (1) and (2) yields 𝑝₁𝑝₁−1𝑣²−𝑘²𝐵²tanh^𝑝1−2𝜏−2𝑖𝑝₁𝑣𝜔−𝜅²𝑘²𝐵tanh^𝑝1−1𝜏−2𝑝²₁𝑣²−𝑘²𝐵²+𝜔²+𝐾²𝜅²tanh^𝑝1𝜏+2𝑖𝑝₁𝑣𝜔−𝜅²𝑘²𝐵tanh^𝑝1+1𝜏+𝑝₁𝑝₁+1𝑣²−𝑘²𝐵²tanh^𝑝1+2𝜏+𝑎tanh^𝑝1𝜏+𝑏𝐴₂tanh^𝑝1+𝑝2𝜏+𝑐𝐴₁^2𝑛tanh^{(2𝑛+1)𝑝1}𝜏=0,𝑝₂𝑝₂−1𝑣²−𝑘²𝐴₂𝐵²tanh^𝑝2−2𝜏−2𝑝²₂𝑣²−𝐾²𝐴₂𝐵²tanh^𝑝2𝜏+𝑝₂𝑝₂+1𝑣²−𝑘²𝐴₂𝐵²tanh^𝑝2+2𝜏−𝑑𝐴₁^2𝑛𝐵²2𝑛𝑝₁2𝑛𝑝₁−1tanh^{2𝑛𝑝1−2}𝜏−8𝑛²𝑝²₁tanh^2𝑛𝑝1𝜏+2𝑛𝑝₁2𝑛𝑝₁+1tanh^2𝑛𝑝1+2𝜏=0, (7)(8) respectively. Now, splitting (7) into two real and imaginary parts gives 𝑝₁𝑝₁−1𝑣²−𝑘²𝐵²tanh^𝑝1−2𝜏−2𝑝²₁𝑣²−𝑘²𝐵²+𝜔²+𝑘²𝜅²tanh^𝑝1𝜏+𝑝₁𝑝₁+1𝑣²−𝑘²𝐵²tanh^𝑝1+2𝜏+𝑎tanh^𝑝1𝜏+𝑏𝐴₂tanh^𝑝1+𝑝2𝜏+𝑐𝐴₁^2𝑛tanh^{(2𝑛+1)𝑝1}𝜏=0,2𝑖𝑝₁𝑣𝜔−𝜅²𝑘²𝐵tanh^𝑝1−1𝜏−2𝑖𝑝₁𝑣𝜔−𝜅²𝑘²𝐵tanh^𝑝1+1𝜏=0. (9)(10) From (9), equating the exponents 𝑝₁+𝑝₂ and 𝑝₁+2 gives 𝑝₂=2, (11) and then equating (2𝑛+1)𝑝₁ with 𝑝₁+2 gives 𝑝₁=1𝑛. (12) Finally, equating the exponent pairs (2𝑛+1)𝑝₁ and 𝑝₁+𝑝₂ gives 𝑝₂=2𝑛𝑝₁. (13) Now the values of 𝑝₁ and 𝑝₂ from (11) and (12) satisfy (13).

Finally, equating the coefficients of the linearly independent functions tanh^𝑝1±𝑗𝜏, 𝑗=±1,0,±2 in (9) and (10) to zero gives 𝑣𝜔−𝜅²𝑘²2𝑣𝐵=0,₂−𝑘²𝐵²+𝑛²𝜔²+𝜅²𝑘²()𝑣−𝑎=0,𝑛+1²−𝑘²𝐵²+𝑛²𝑏𝐴₂+𝑐𝐴₁^2𝑛=0. (14) Again, equating the coefficients of the linearly independent functions tanh^𝑝2±𝑗𝜏, 𝑗=±1,±2 in (8) to zero implies 𝑣²−𝑘²𝐴₂−𝑑𝐴₁^2𝑛=0. (15) Solving (14)-(15) we get 𝑘𝜔=²𝜅²𝑣,𝐵=𝑛𝑎−𝑘²𝜅²−𝜔²2𝑣²−𝑘²,𝐴₁=(𝑛+1)𝑣²−𝑘²𝜔²−𝑎−𝜅²𝑘²𝑑2𝑐𝑣²−𝑘²+𝑏𝑑^1/2𝑛,𝐴₂=𝑑(𝑛+1)𝜔²−𝑎−𝑘²𝜅²2𝑐𝑣²−𝑘².+𝑏𝑑 (16)(17)(18)(19) The relations (17), (18), and (19) introduce the restrictions given by 𝑎−𝑘²𝜅²−𝜔²𝑣²−𝑘²𝑑𝑣>0,₂−𝑘²𝜔²−𝑎−𝐾²𝜅²𝑐𝑣²−𝑘²𝑐𝑣+𝑏𝑑>0,₂−𝑘²+𝑏𝑑≠0. (20) Thus the topological solution of the 𝑞 and 𝑟 wave functions are given: 𝑞(𝑥,𝑡)=𝐴₁tanh^1/𝑛[𝐵(𝑥−𝑣𝑡)]𝑒^{𝑖(−𝜅𝑥+𝜔𝑡+𝜃)},𝑟(𝑥,𝑡)=𝐴₂tanh²[𝐵(𝑥−𝑣𝑡)]. (21)

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

We assume that the traveling wave solutions of (1) and (2) are of the form 𝑞(𝑥,𝑡)=𝑒^𝑖𝜂𝜑(𝜉),𝑟(𝑥,𝑡)=𝜓(𝜉),𝜂=𝑚𝑥+𝑙𝑡,𝜉=𝑝𝑥−𝑣𝑡, (22)(23) where 𝜑(𝜉) and 𝜓(𝜉) are real functions, 𝑚, 𝑙, 𝑝, and 𝑣 are real constants.

Substituting (22) and (23) into (1) and (2), we find that 𝑝=−𝑣𝑙/𝑚𝑘², 𝜑 and 𝜙 satisfy the following system: 𝑣²−𝑘²𝑝²𝜑^′′−𝑙²−𝑘²𝑚²−𝑎𝜑+𝑏𝜑𝜓+𝑐𝜑^2𝑛+1=0,𝑣²+𝑘²𝑝²𝜓^′′−𝑑𝑝²𝜑^2𝑛′′=0. (24)(25) Integrating (25) twice and letting the first integral constant be zero, we have 𝜓=𝑑𝑝²𝜑^2𝑛𝑣²−𝑘²𝑝²+𝑔,𝑣²−𝑘²𝑝²≠0, (26) where 𝑔 is the second integral constant.

Substituting (26) into (24), we have 𝑣²−𝑘²𝑝²𝜑^′′−𝑙²−𝑘²𝑚²𝜑+−𝑎−𝑏𝑔𝑐+𝑏𝑑𝑝²𝑣²−𝑘²𝑝²𝜑^2𝑛+1=0. (27) To facilitate discussions, we let 𝛿=𝑏𝑑𝑝²+𝑐𝑣²−𝑘²𝑝²𝑣²−𝑘²𝑝²²,𝑙𝜃=²−𝑘²𝑚²−𝑎−𝑏𝑔𝑣²−𝑘²𝑝². (28)(29) Letting 𝜑^′=𝑧, then we get the following planar system: d𝜑d𝜉=𝑧,d𝑧d𝜉=−𝛿𝜑^2𝑛+1+𝜃𝜑. (30) Obviously, the above system (30) is a Hamiltonian system with Hamiltonian function 𝐻𝜑,𝑧=𝑧²+𝛿𝜑𝑛+1^2𝑛+2−𝜃𝜑². (31) In order to investigate the phase portrait of (30), set 𝑓𝜑=−𝛿𝜑^2𝑛+1+𝜃𝜑. (32) Obviously, when 𝛿𝜃>0, 𝑓(𝜑) has three zero points, 𝜑₋, 𝜑₀, and 𝜑₊, which are given as follows: 𝜑₋𝜃=−𝛿^1/2𝑛,𝜑₀=0,𝜑₊=𝜃𝛿^1/2𝑛. (33) When 𝛿𝜃⩽0, 𝑓(𝜑) has only one zero point 𝜑₀=0. (34) Letting (𝜑_𝑖,0) be one of the singular points of system (30), then the characteristic values of the linearized system of system (30) at the singular points (𝜑_𝑖,0) are 𝜆_±=±𝑓^′𝜑_𝑖. (35) From the qualitative theory of dynamical systems, we know the following.(I) If 𝑓^′(𝜑_𝑖)>0, (𝜑_𝑖,0) is a saddle point.(II) If 𝑓^′(𝜑_𝑖)<0, (𝜑_𝑖,0) is a center point.(III) If 𝑓^′(𝜑_𝑖)=0, (𝜑_𝑖,0) is a degenerate saddle point.

Therefore, we obtain the bifurcation phase portraits of system (30) in Figure 1.

Figure 1 

The bifurcation phase portraits of system (30). (I) 𝛿>0, 𝜃>0, (II) 𝛿<0,  𝜃⩾0, (III) 𝛿<0,  𝜃<0, (IV) 𝛿>0, 𝜃⩽0.

Let 𝐻𝜑,𝑧=ℎ, (36) where ℎ is Hamiltonian.

Next, we consider the relations between the orbits of (30) and the Hamiltonian ℎ.

Set ℎ^∗=𝐻𝜑₊,0=𝐻𝜑₋,0. (37)

According to Figure 1, we get the following propositions.

Proposition 1. Suppose that 𝛿>0 and 𝜃>0, one has the following.(I)When ℎ⩽−ℎ^∗, system (30) does not have any closed orbits. (II)When −ℎ^∗<ℎ<0, system (30) has two periodic orbits Γ₁  and Γ₂.(III)When ℎ=0, system (30) has two homoclinic orbits Γ₃  and Γ₄.(IV)When ℎ>0, system (30) has a periodic orbit Γ₅.

Proposition 2. Suppose that 𝛿<0 and 𝜃<0, one has the following.(I)When ℎ<0or ℎ>ℎ^∗, system (30) does not have any closed orbits. (II)When 0<ℎ<ℎ^∗, system (30) has three periodic orbits Γ₆, Γ₇, and Γ₈.(III)When ℎ=0, system (30) has two periodic orbits Γ₉  and Γ₁₀.(IV)When ℎ=ℎ^∗, system (30) has two heteroclinic orbits Γ₁₁  and Γ₁₂.

Proposition 3. (I) When 𝛿>0, 𝜃⩾0 and ℎ>0, system (30) has a periodic orbits.
(II) When 𝛿<0, 𝜃⩽0, system (30) does have not any closed orbits.

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 𝛿>0 and 𝜃>0, one has the following.(I)When −ℎ^∗<ℎ<0, (1) and (2) have two periodic wave solutions (corresponding to the periodic orbits Γ₁  and Γ₂ in Figure 1). (II)When ℎ=0, (1) and (2) have two solitary wave solutions (corresponding to the homoclinic orbits Γ₃  and Γ₄ in Figure 1). (III)When ℎ>0, (1) and (2) have two periodic wave solutions (corresponding to the periodic orbit Γ₅  in Figure 1).

Proposition 5. If 𝛿<0 and 𝜃<0, one has the following.(I)When 0<ℎ<ℎ^∗, (1) and (2) have two periodic wave solutions (corresponding to the periodic orbit Γ₇  in Figure 1) and two periodic blow-up wave solutions (corresponding to the periodic orbits Γ₆  and Γ₈  in Figure 1). (II)When ℎ=0, (1) and (2) have periodic blow-up wave solutions (corresponding to the periodic orbits Γ₉  and Γ₁₀  in Figure 1). (III)When ℎ=ℎ^∗, (1) and (2) have two kink profile solitary wave solutions. (corresponding to the heteroclinic orbits Γ₁₁  and Γ₁₂  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 𝛿>0 and 𝜃>0. From the phase portrait, we see that there are two symmetric homoclinic orbits Γ₃ and Γ₄ connected at the saddle point (0,0). In (𝜑,𝑧)-plane the expressions of the homoclinic orbits are given as 𝑧=±𝛿𝜑𝑛+1−𝜑^2𝑛+(𝑛+1)𝜃𝛿. (38) Substituting (38) into d𝜑/d𝜉=𝑧 and integrating them along the orbits Γ₃ and Γ₄, we have ±^𝜑_𝜑11𝑠−𝑠^2𝑛+(𝑛+1)𝜃/𝛿d𝑠=𝛿𝑛+1^𝜉₀d±𝑠,_𝜑𝜑21𝑠−𝑠^2𝑛+(𝑛+1)𝜃/𝛿d𝑠=𝛿𝑛+1^𝜉₀d𝑠, (39) where 𝜑₁=−((𝑛+1)𝜃/𝛿)^1/2𝑛 and 𝜑₂=((𝑛+1)𝜃/𝛿)^1/2𝑛.

Completing the above integrals we obtain 𝜑=(𝑛+1)𝜃𝛿sech𝑛√𝜃𝜉^1/𝑛𝜑=−(𝑛+1)𝜃𝛿sech𝑛√𝜃𝜉^1/𝑛. (40) Noting (22), (23), and (26), we get the following solitary wave solutions: 𝑞₁𝑥,𝑦,𝑡=𝑒^𝑖𝜂(𝑛+1)𝜃𝛿sech𝑛√𝜃𝜉^1/𝑛,𝑟₁=−()𝑥,𝑦,𝑡𝑛+1𝛽𝜃sech𝑛√𝜃𝜉²𝛿𝑝²+𝑚²𝑞+𝑔,₂𝑥,𝑦,𝑡=−𝑒^𝑖𝜂(𝑛+1)𝜃𝛿sech𝑛√𝜃𝜉^1/𝑛,𝑟₂=−()𝑥,𝑦,𝑡𝑛+1𝛽𝜃sech𝑛√𝜃𝜉²𝛿𝑝²+𝑚²+𝑔, (41) where 𝛿 is given by (28), 𝜃 is given by (29), 𝜂=𝑚𝑥+𝑙𝑡, and 𝜉=𝑝𝑥−𝑣𝑡.

Secondly, we will obtain the explicit expressions of traveling wave solutions for (1) and (2) when 𝛿<0 and 𝜃<0. From the phase portrait, we note that there are two special orbits Γ₉ and Γ₁₀, which have the same Hamiltonian with that of the center point (0,0). In (𝜑,𝑧)-plane the expressions of the orbits are given as 𝑧=±−𝛿𝜑𝑛+1𝜑^2𝑛−(𝑛+1)𝜃𝛿. (42) Substituting (42) into d𝜑/d𝜉=𝑧 and integrating them along the two orbits Γ₉ and Γ₁₀, it follows that ±_𝜑^+∞1𝑠𝑠^2𝑛−(𝑛+1)𝜃/𝛿d𝑠=−𝛿𝑛+1^𝜉₀d±𝑠,_𝜑𝜑41𝑠𝑠^2𝑛−(𝑛+1)𝜃/𝛿d𝑠=−𝛿𝑛+1^𝜉₀d𝑠, (43) where 𝜑₄=((𝑛+1)𝜃/𝛿)^1/2𝑛.

Completing the above integrals we obtain 𝜑=±(𝑛+1)𝜃𝛿csc𝑛√−𝜃𝜉^1/𝑛,𝜑=±(𝑛+1)𝜃𝛿sec𝑛√−𝜃𝜉^1/𝑛. (44) Noting (22), (23), and (26), we get the following periodic blow-up wave solutions: 𝑞₃𝑥,𝑦,𝑡=±𝑒^𝑖𝜂(𝑛+1)𝜃𝛿csc𝑛√−𝜃𝜉^1/𝑛,𝑟₃=−()𝑥,𝑦,𝑡𝑛+1𝛽𝜃csc𝑛√−𝜃𝜉²𝛿𝑝²+𝑚²𝑞+𝑔,₄𝑥,𝑦,𝑡=±𝑒^𝑖𝜂(𝑛+1)𝜃𝛿sec𝑛√−𝜃𝜉^1/𝑛,𝑟₄=−()𝑥,𝑦,𝑡𝑛+1𝛽𝜃sec𝑛√−𝜃𝜉²𝛿𝑝²+𝑚²+𝑔, (45) where 𝛿 is given by (28), 𝜃 is given by (29), 𝜂=𝑚𝑥+𝑙𝑡, and 𝜉=𝑝𝑥−𝑣𝑡.

4. Numerical Simulation

We decompose the function 𝑞 in (1) in the form 𝑞=𝑢+𝑖𝑣 (46) Substituting in (1) and (2) we have 𝑢_𝑡𝑡−𝑘²𝑢_𝑥𝑥+𝑎𝑢+𝑏𝑟𝑣+𝑐𝑢²+𝑣^2𝑛𝑣𝑢=0,_𝑡𝑡−𝑘²𝑣_𝑥𝑥+𝑎𝑣+𝑏𝑟𝑣+𝑐𝑢²+𝑣^2𝑛𝑟𝑣=0,_𝑡𝑡−𝑘²𝑢_𝑥𝑥−𝛼𝑢²+𝑣^2𝑛_𝑥𝑥=0. (47) We assume that 𝑢^𝑛_𝑚, 𝑣^𝑛_𝑚, 𝑟^𝑛_𝑚 is the exact solution and 𝑈^𝑛_𝑚, 𝑉^𝑛_𝑚, 𝑅^𝑛_𝑚 is the approximate solution at the grid point (𝑥_𝑚,𝑡_𝑛). The proposed scheme can be displayed as 1𝑘²𝛿²_𝑡𝑈^𝑛_𝑚−𝑘²ℎ²𝛿²_𝑥𝑈^𝑛_𝑚+𝑎𝑈^𝑛_𝑚+𝑏𝑅_𝑚,𝑛𝑈^𝑛_𝑚+𝑐𝑈^𝑛_𝑚²+𝑈^𝑛_𝑚^2𝑛𝑈^𝑛_𝑚1=0,𝑘²𝛿²_𝑡𝑉^𝑛_𝑚−𝑘²ℎ²𝛿²_𝑥𝑉^𝑛_𝑚+𝑎𝑉^𝑛_𝑚+𝑏𝑅_𝑚,n𝑉^𝑛_𝑚+𝑐𝑈^𝑛_𝑚²+𝑉^𝑛_𝑚^2𝑛𝑉^𝑛_𝑚1=0,𝑘²𝛿²_𝑡𝑅^𝑛_𝑚−𝑘²ℎ²𝛿²_𝑥𝑅^𝑛_𝑚+𝛼𝑈^𝑛_𝑚²+𝑉^𝑛_𝑚^2𝑛_𝑥𝑥=0, (48) where 𝛿²_𝑡𝑈^𝑛_𝑚=𝑈_𝑚^𝑛+1−2𝑈^𝑛_𝑚+𝑈_𝑚^𝑛−1,𝛿²_𝑥𝑈^𝑛_𝑚=𝑈^𝑛_𝑚+1−2𝑈^𝑛_𝑚+𝑈^𝑛_𝑚−1. (49) The similar notation for 𝛿²_𝑡𝑉^𝑛_𝑚,𝛿²_𝑥𝑉^𝑛_𝑚and 𝛿²_𝑡𝑅^𝑛_𝑚,𝛿²_𝑥𝑅^𝑛_𝑚 are 𝛿²_𝑡𝑉^𝑛_𝑚=𝑉_𝑚^𝑛+1−2𝑉^𝑛_𝑚+𝑉_𝑚^𝑛−1,𝛿²_𝑥𝑉^𝑛_𝑚=𝑉^𝑛_𝑚+1−2𝑉^𝑛_𝑚+𝑉^𝑛_𝑚−1,𝛿²_𝑡𝑅^𝑛_𝑚=𝑅_𝑚^𝑛+1−2𝑅^𝑛_𝑚+𝑅_𝑚^𝑛−1,𝛿²_𝑥𝑅^𝑛_𝑚=𝑅^𝑛_𝑚+1−2𝑅^𝑛_𝑚+𝑅^𝑛_𝑚−1, (50) 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.

To get the numerical solution the initial conditions are taken from the exact solution (21). Figure 2 displays the numerical solutions of |𝑞(𝑥,𝑡)| and 𝑟(𝑥,𝑡) at 𝑛=1, respectively. We choose the parameter 𝑛=1,𝐾=1,𝜅=0.5,𝑎=0.5,𝑏=0.5,𝑐=−0.5,𝑑=0.5,𝑐=0.5,𝑣=0.6. (51)

Figure 2 

Topological solution for the Klein-Gordon-Zakharov equations.

5. Conclusions

This paper studied the KGZ equation in (1+1)-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 (2+1)-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 𝐺^′/𝐺-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.