CONSTRUCTION OF NEW TRAVELING WAVE SOLUTIONS FOR THE (2+1) DIMENSIONAL EXTENDED KADOMTSEV-PETVIASHVILI EQUATION ∗

This work is interested in constructing traveling wave solutions for the (2+1)-dimensional extended Kadomtsev–Petviashvili equation that is utilized as a model for the surface waves and internal waves in straits or channels. Based on the bifurcation analysis of the traveling wave system, we use the conserved quantity to construct some new bounded traveling wave solutions such as periodic and solitary solutions in addition to some unbounded novel wave solutions. Some of the new solutions and their corresponding orbits are clarified graphically. Moreover, we examine numerically the dynamical behaviour for the perturbed (2+1)-dimensional extended Kadomtsev–Petviashvili equation by adding a perturbed periodic term.


Introduction
In recent few decades, the wide applications of nonlinear evolution equations(NLEEs) have become an important research area for the interested in the study of applied sciences aiming to seek exact soliton solutions.These applications mostly appear in various fields such as fluid dynamics, optical fibers, water waves, plasma, nuclear physics and biological systems.NLEEs are considered in finite dimensions such as (1+1), (2+1) and (3+1)-dimensions.Kadomtsev-Petviashvili equation (KPe) is a significant one of those NLEEs.The KPe appears in various fields especially in the nonlinear dynamical systems, such as ferromagnetic media, water waves and multi-component plasmas.There are several techniques of integration that were established in literature.In [9] the (2+1)-dimensional Camassa-Holm KPe has been studied utilizing of the G ′ /G method, the exponential function method and the ansatz method.In [10,11,36,37] the (3+1)-dimensional extended KPe has been studied with power law nonlinearity appearing in the multi-component plasmas study.Based on a computerized symbolic computation and extended variable coefficient homogeneous balance method, the dynamics of solitons and nonlinear waves in plasmas and superfluids have been described for the (3+1)-dimensional KPe in [25].In [45] a modified three-soliton method has been proposed and applied to the (2+1)-dimensional KPe.In [6] the multiple Exp-function method has been applied to construct multiple wave solutions for the (3 +1)-dimensional generalized B-type KPe.A Shallow water wave equation in [3,24], has been modeled by (2+1)-dimensional Gardner-KPe to obtain soliton and other solutions.The multiple exp-function method has been applied to explore the shape nature of anti-kink solutions of a (3+1)-dimensional B-type KPe in [8].The Lie symmetry method has been applied to the logarithmic-KP equation in [38].A lattice Boltzmann method has been proposed in [39] to simulate the solitons of the KPe.In [46] a Hirota bilinear differential equation method has been proposed to a (3+1)-dimensional generalized KPe and two classes of lump solutions have been derived.In [33] the authors used F-expansion method and the generalized extended tanh method to develop solitary waves solutions of both KPe and modified KPe.Lump solutions to a (2+1)-dimensional extended KPe have been presented in [30].More recently, generalized bilinear method has been used for (3+1)-dimensional KP-Boussinesqlike equation to study the dynamics of lump solutions in [34].The Hirota bilinear method has been investigated in [1,22] for (2+1)-dimensional extended KPe modeling the surface and internal waves.In [35] a modification form of extended auxiliary equation mapping method has been applied in a generalized KP modified equal width-Burgers.By a symbolic calculation technique, the lump-kink solutions and multiple-soliton have been obtained in [23] for generalized (3+1)-dimensional KPe.In [12] multi-wave solutions for Hirto equation are obtained utilizing three powerful function techniques.The extension exponential rational function method has been applied successfully in [13] to build some novel traveling wave solution for the higher Sharma-Tasso-Olver equation.Some traveling wave solutions nonlinear propagation of Kundu-Eckhaus dynamical model were introduced in [14] using using logarithmic transformation and function method.In Ref [15] authors introduced multi-wave solutions for the hydrodynamics Zakharov-Kuznetsov model.Extra papers dealing with these types of equations can be found in [4,5,7,29,[40][41][42][43][44].
In this paper, we consider a (2+1)-dimensional extended KPe: where α, β are arbitrary constants.We will apply the bifurcation method which has been successfully utilized to construct traveling wave solutions for different problems (see, e.g., [16-21, 26-28, 31, 32, 47]).Several reasons are motivating us to apply the bifurcation method to construct traveling wave solutions for a given PDE.One of the most important reasons is the ability to determine the type of the solutions through the kind of the corresponding orbits before finding the solutions.For instance, the existence of periodic orbits, homoclinic orbits, and smooth heteroclinic orbits for traveling wave systems refers to periodic wave solutions, smooth solitary solutions, and smooth kink wave solutions, or oscillatory traveling wave solutions for partial differential equations, respectively.Also, it has some advantages which will be outlined in the text for suitability.The work in the following is organized as: Section 2 presents analysis of bifurcation and phase portrait of the planar dynamical system corresponding to Eq. (1.1).Section 3 is concerned with constructing traveling wave solutions for Eq.(1.1) by applying the bifurcation analysis.Graphical representations are given in Section 4. Section 5 contains the study of the dynamical behaviour for the perturbed (2+1)dimensional extended Kadomtsev-Petviashvili equation due to the adding a perturbed periodic term.Section 6 is a conclusion of the work.

Bifurcation and phase portrait
We consider the traveling wave transformation where k, l are constants referring the direction cosine of the propagation of traveling wave velocity ω in the plane xy.Inserting the transformation (2.1) into Eq.(1.1), we get where ′ refers to the derivative with respect to η. Integrating Eq. (2.2) twice with respect to η and setting the first integration constant equals to zero, we obtain where ρ is an arbitrary constant and γ is given by Eq. (2.3) can be presented as a planar dynamical system in the form (2.5) System (2.5) is named a traveling wave system.System (2.5) is a conservative system because ∂ψ ′ ∂ψ + ∂z ′ ∂z = 0.Moreover, it is a Hamilton system because is derived from the canonical Hamilton equations ψ ′ = ∂H ∂ψ , z ′ = − ∂H ∂ψ , where H is a Hamiltonian function and it takes the form In physics, the Hamiltonian shown in Eq. (2.6) describes one dimension motion of a particle taking place as a result of the potential forces that are derived from the potential function Due to the Hamiltonian shown in Eq. (2.6) does not explicitly rely on the independent variable η that plays the role of the time in mechanics and so, the Hamiltonian (2.6) is a conserved quantity.Thus, we have where h is a constant.The study of the bifurcation of the traveling wave system (2.5) represents significant role in constructing and explaining the traveling wave solution for Eq.(1.1).Assuming u = ψ(η) is a continuous solution for Eq.(1.1) for η ∈ R with lim η→+∞ ψ(η) = ν 1 and lim η→−∞ ψ(η) = ν 2 .Hence, u is named a solitary wave solution if ν 1 = ν 2 , while it is a kink(or anti-kink) wave solution if ν 1 ̸ = ν 2 .Moreover, there is a relation between the type of the orbits and the classification of the wave solution.For instance, the solitary wave solution and kink (anti-kink) wave solutions for Eq.(1.1) are associated with the homoclinic and hetroclinic orbits for the system (2.5), respectively.Also, the existence of periodic orbit for system (2.5) refers the presence of periodic traveling wave solution.Consequently, this motivates us to study the bifurcation of system (2.5) in order to be able to classify the solutions into solitary waves, breaking waves, kink (anti-kink) waves and periodic waves depending on the type of the orbits for the system (2.5).
To investigate the bifurcation and phase portrait for the dynamical system shown in Eq. (2.5), we find its equilibrium points and study their nature.Clearly, we set ψ ′ = 0 and z ′ = 0 to find the equilibrium points and consequently, the equilibrium points lie on ψ− axis and satisfies the quadratic equation 3ψ 2 k 2 + γψ − ρ = 0. Thus, we have the following possibilities: , 0 ) and ) .
Without loss of generality we can assume hencforth that k is positive.To specify the nature of the equilibrium points E i in view of the theory of planar dynamical systems, we calculate the eigenvalues of the Jacobian matrix of system (2.5) evaluated at the equilibrium (ψ 0 , 0) and they are Therefore, the equilibrium point E 1 will be a cusp since the eigenvalues (2.9) at E 1 are zeros and the Poincaré index of that point is 0. The phase portrait for this case is illustrated by Fig. 1(a) for different values of the parameter h and the point E 1 is indicated as a black point.On the other side, the eigenvalues (2.9) at E 2 and E 3 respectively.So, the equilibrium point E 2 is a center point while E 3 is a saddle point and thus, the phase portrait for this case is clarified by Fig. 1(b) for different values of the parameter h and equilibrium points E 2 and E 3 are outlined by two solid points.
It is clear that the space surface C 0 = {(ρ, k, ρ) ∈ R 3 , γ 2 k 2 + 12ρ = 0} splits the 3D parameter space into two zones.The related parameter bifurcation sets are To clarify the parameter bifurcation sets intuitively, let us fix the parameter γ to a fixed value γ 0 and hence the bifurcation boundary takes the form This boundary curve splits the kρ-plane into two disjoint regions as it is outlined by Fig. 2.

Constructing traveling wave solutions using a conserved quantity
Now, we are going to construct some traveling wave solutions for Eq.(1.1) taking into account the bifurcation analysis.Entering the first equation in Eq. (2.5) into Eq.(2.8) and separating the variables, we obtain where To integrate both sides of Eq. (3.1), the range of the parameters γ, ρ, k, and h is demanded due to the different values of these parameters will imply to different wave solutions.Hence, the key step for integrating Eq. (3.1) is finding this range of those parameters.There are two methods used to detect these ranges of parameters such as a bifurcation analysis and the complete discrimination system function for R 3 (ψ).The application of bifurcation theory is more significant and has some advantages over the other method.It gives the range of the parameters and can also be employed to classify the types of traveling wave solutions through the type of the the orbits without calculating the solutions.Based on the physical meaning of the problem under consideration, we restrict ourselves in construction real traveling wave solutions for Eq.(1.1).Hence, we only consider certain intervals for ψ for which R 3 (ψ) > 0. Furthermore, the construction of all possible traveling wave solutions for Eq.(1.1) requires to specify the type of the orbits for system (2.5) either it is bounded or unbounded for different values of the energy h.Hence, it is more suitable to introduce the proposition Proposition 3.1.The traveling wave system (2.5) has only two types of bounded orbits which are the periodic orbit in green and homoclinic orbit in red as illustrated by Fig. 1(b).While the other orbits are unbounded orbits as outlined in Fig. 1.
Now, we calculate the values of the energy h at the equilibrium points E i , i = 1, 2, 3, we have Thus, we integrate Eq. (3.1) along these bounded and unbounded orbits individually.

Expressions of bounded traveling waves of system (2.5)
According to the proposition 3.1, only when (ρ, k, γ) ∈ C + , there exist bounded traveling wave solutions.These solutions are periodic waves and solitary wave because they corresponds periodic orbits and homoclinic orbit, respectively.

Periodic waves
On the level h ∈]h 2 , h 3 [, there is a family of periodic orbits in green around the center point E 2 which is surrounded by the homoclinic orbit in red as outlined by Fig. 1(b).Any orbit of this family cuts ψ−axis in three points and this shows the polynomial R 3 (ψ) has three simple real roots.Thus, R 3 (ψ) = (ψ − r 1 )(ψ − r 2 )(r 3 − ψ), where r 1 , r 2 , r 3 are real parameters with r 1 < r 2 < ψ < r 3 .Let us assume the period of this closed curve is 2T and ψ(0) = r 2 .Thus, Eq. (1.1) becomes Eq. (3.4) gives the periodic wave solution in the form where The solution (3.5) is a new periodic wave solution for Eq.(1.1) with period 2 k1 K(k 1 ), where K(k 1 ) is a complete elliptic integral of the first kind [2].

Solitary waves
When the value of the energy of the system (2.5) equals the value of the energy at the saddle point E 3 , i.e., h = h 3 , there is an orbit in red passing through the saddle point E 3 , see Fig. 1(b).This orbit cuts the ψ− axis in two points and this proves the polynomial R 3 (ψ) has three real roots; one is simple and the other is double at E 3 .Thus, the polynomial R 3 (ψ) can be written in the form and r 4 < ψ < r 5 .Let us assume ψ(0) = r 5 and Insert R 3 (ψ) into Eq.(3.1) and integrate both sides, we obtain Eq. (3.6) gives where

Expressions for unbounded traveling waves of system (2.5)
In this section, we construct unbounded traveling waves of the system (2.5) along energy curves for distinct values of the energy.We consider the following cases, individually Case 3.1: Let us assume (ρ, k, γ) ∈ C 0 , there are unbounded orbits according to the value of the energy h.
1.If the value of the energy does not equal to the value of the energy at the cusp point, i.e., h ̸ = h 1 , the system (2.5) has two families of unbounded orbits in blue and in brown {H(ψ, z) = h, h ̸ = h 1 }, see Fig. 1(a).A single orbit of these families whether it is in blue or brown intersects ψ−axis in one point and this proves the the polynomial R 3 (u) has one real root and two conjugate complex roots.Thus, it can be written as where −∞ < ψ < r 6 and let ψ(−∞) = 0. We use the last formula for R 3 (ψ) and Eq.(3.1), we get for η > 0. Eq. (3.8) gives where 2. If the energy of the system equals the value of the energy at the cusp, the system has another unbounded orbit in red as outlined by Fig. 1(a).This orbit intersects the ψ−axis in a single point which is the equilibrium point E 1 .This proves the polynomial R 3 (ψ) has three multiple roots and so it is , where −∞ < ψ < − k 2 γ 6 .Let us assume ψ(0) = −∞.Taking into account the upper branch of this orbit and Insert this formula for R 3 (ψ) into Eq.(3.1) and integrating both sides, we get (3.10)

Case 3.2:
We consider the case (ρ, k, γ) ∈ C + .The phase portrait for the system (2.5) in this case is described by Fig. 1(b).The system has some family of unbounded orbits for different values of h.Based on different values of the energy h, we construct unbounded traveling wave solutions for system (2.5).Let us consider them separately: 1. On the energy level which is greater than the energy at the saddle point E 3 , there is a family of orbits, H(ψ, z) = h, h > h 3 }, in blue as outlined by Fig. 1(b).Any member of this family cuts ψ−axis in one point and this shows the polynomial R 3 (ψ) has only one real zero and the others are two conjugate complex.Thus, it can be written in the form where −∞ < ψ < r 7 .We postulate ψ(0) = −∞.Inserting the last formula for R(ψ) into Eq.(1.1), we obtain for η > 0. Eq. (3.11) gives where there is two families of orbits in green as outlined by Fig. 1(b).
One of them is a periodic family around the center point E 2 , it is bounded and it is considered in sub subsection 3.1.1.While the other one has the parabolic shape and it is unbounded and so it is consider here.The polynomial R 3 (ψ) Inserting the last formula for R 3 (ψ) into the differential form (3.1) and integrating both sides, we obtain for η > 0. Thus, we have where r3−r1 , and η 3 = 4 Ω5 K(k 4 ), where K(k 4 ) is a complete elliptic integral of the first kind [2].
there is a family of unbounded orbits with hyperbolic shape in brown color for the dynamical system (2.5).A single orbit of this family cuts ψ− axis in one point and this proves the polynomial (3.2) has only one real root.Thus, the polynomial (3.2) can be written in the form , where −∞ < ψ < r 8 and we assume ψ(0) = −∞.
Entering the last expression for R 3 (ψ) into Eq.(3.1) and integrating both sides in a similar computation with the first item in case to, we get where , where K(k 5 ) is a complete elliptic integral of the first type [2].4. On the level of the energy corresponding to the saddle point E 3 , i.e., h = h 3 , there is an orbit in red as clarified by Fig. 1(b).This orbit consists of two parts; the first is a homoclinic orbit which is considered in sub subsection 3.1.2.While the the second is its extension and it will be considered here.The polynomial R 3 (ψ) takes the form R 3 (ψ) = (ψ − r 4 ) 2 (r 5 − ψ), where −∞ < ψ < r 4 < r 5 and assume ψ(0) = −∞.Using the last expression for R 3 and Eq.(3.1) and integrating both sides, we get for η > 0. It follows that where 5. On the energy level which equals the value of the energy at the center point E 2 , i.e. h = h 2 , the system (2.5) has a unbounded orbit in black in addition to Equilibrium point E 2 .Thus, the polynomial R 3 (ψ) takes the form and −∞ < ψ < r 9 .Assume ψ(0) = −∞, entering the last expression for R 3 (ψ) into Eq.(3.1), and integrate both sides, we get for η > 0. It follows that where then the dynamical system (2.5) has no equilibrium point and its phase portrait is clarified by Fig. 1(c).All orbits are unbounded and each one of them intersects ψ− axis in one point.This shows R 3 (ψ) has one real root and two complex roots and hence, it takes the form where −∞ < ψ < r 11 .We assume ψ(−∞) = 0, insert the last expression for R 3 (ψ) into Eq.(3.1), and integrate both sides, we obtain where

Graphic representation
This section pursues to illustrate some wave solutions by displaying 3D graphics to them, the corresponding orbit of this solution, in addition to the density diagram.
For k = 1, γ = 2, ρ = 1, the dynamical system (2.5) has two equilibrium points E 2 = ( 13 , 0), E 3 = (−1, 0) and notice the type of the orbit depends on the value of h as it was discussed in section 2. Let us clarify that for different values of h:   • When h = 1 which is the energy at the equilibrium point E 3 = (−1, 0), the dynamical system ( 5) has an orbit consisting of two parts: The first one is a homoclinic orbit connecting the saddle point E 3 , around the center E 2 , with itself as in Fig. 5   • On any other level of energy, the system (2.5) has two types of orbits depending on whether h > 1 (blue orbit) or h < 1 (brown orbit), see Fig.

Quasi-Periodic solutions
This section aims to examine the dynamical behaviour of the perturbed form of Eq. (1.1) as a result of adding the perturbed periodic term A cos(q(kx + ly − ωt)), where A, q are arbitrary constants.Hence, Eq. (1.1) takes the form Applying the transformation (2.1) to Eq. (1.1), integrating twice with respect to η and setting the first integration constant equals zero, we obtain where p = A q 2 .Eq. (5.2) can be written as a 2D-non autonomous dynamical system in the form In order to look for some distinct dynamical behaviours of the perturbed Eq. (1.1), we investigate several values of the physical parameters.Fig. 9 illustrates the 2D and 3D phase portrait for the perturbed dynamical system (5.3) when the system's parameters takes the values k = 1, p = −0.05,q = 2, ρ = −1, γ = 1 with initial conditions ψ(0) = 0.05, z(0) = 0.05.While, Fig. 10 clarifies the 2D and 3D phase portrait for the perturbed dynamical system (5.3) when the parameters of the system admits the values k = 0.04, p = 5, q = 0.05, ρ = 4, γ = 2 with initial conditions ψ(0) = 0.05, z(0) = 1.Consequently, the perturbed dynamical system (5.3) is observed to have quasi-periodic behaviours as outlined by the two Figs.9 and 10.

Conclusion
We have interested in studying a (2+1)-dimensional extended Kadomtsev-Petviashvili equation.We have applied a well-known wave transformation to convert Eq. (1.1) into a second order differential equation which is rewritten as a two dimensional dynamical system.This system has been proven to be a conservative Hamiltonian system having a conserved quantity.The bifurcation analysis and phase portrait for the Hamilton system have been investigated.We have been motivated by the applying the bifurcation method due to the following reasons: The second part is unbounded and hence, the corresponding solution is (3.14) which is singular solution.Thus, for the same conditions, we have two completely different solutions from the mathematical and physical point of view, see Table .1.Employing the conserved quantity, new traveling wave solutions that are classified into periodic, solitary, and singular solutions have been constructed and explained graphically.Finally, the dynamical behavior of a perturbed form of the equation has been investigated by adding the perturbed periodic cosine term.

Figure 1 .
Figure 1.Phase portrait for the planar dynamical system (2.5).The solid circles indicate to the equilibrium points.

Figure 2 .
Figure 2. Bifurcation boundary on the kρ-plane for a fixed value of γ.

5
which lies between the values of the energy at the two equilibrium points E 2 and E 3 , then the system (2.5) has an orbit as outlined by Fig.3(a).This orbit consists of two parts; the first one is closed periodic orbit and corresponds to the periodic solution (3.5) clarified by Fig.3(b) with the density diagram explained by Fig. 3(c).The second one, which takes the shape of a parabola, corresponds to the solution (3.14) which is clarified by Fig. 4(a) with the density diagram outlined by Fig. 4(b).
(a).This orbit corresponds to the solitary solution (3.7) outlined by Fig. 5(b) with the density diagram shown by Fig. 5(c).The second one is the extension of the homoclinic orbit and it corresponds to the singular solution (3.17).This solution and its density diagram are shown by Fig. 6(a) and (b).

Figure 5 .
Figure 5. Graphic representations for the solitary solution (3.7).For k = 1, γ = −6, ρ = −3, the dynamical system (2.5) has a unique equilibrium point E 1 = (1, 0) which is a cusp.We clarify the wave solution and its corresponding orbit for different values of h: • On the energy h = 1 of the cusp point, the dynamical system (2.5) has an orbit shown by Fig. 7(a).This orbit corresponds to the solution (3.10) outlined by Fig. 7(b) with the density diagram displayed by Fig. 7(c).

Table 1 .
Conditions on the parameters and permitted interval of real propagation