Topological Solitons and Bifurcation Analysis of the PHI-Four Equation

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1 BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Bull. Malays. Math. Sci. Soc. () 37(4) (4), 9 9 Topological Solitons Bifurcation Analysis of the PHI-Four Equation JUN CAO, MING SONG AND 3 ANJAN BISWAS, Department of Mathematics, Faculty of Sciences, Yuxi Normal University, Yuxi 653, China Department of Mathematics, Shaoxing University, Shaoxing, 3, China 3 Department of Mathematical Sciences, Delaware State University, Dover, DE 99-77, USA 3 Department of Mathematics, King Abdulaziz University, Jeddah-589, Saudi Arabia caoj@yxnu.net, songming 5@63.com, 3 biswas.anjan@gmail.com Abstract. This paper studies the PHI-four equation that arises in Quantum Mechanics. The topological -soliton solution or kink solution is obtained by the ansatz method. The bifurcation analysis is then subsequently carried out several other solutions are retrieved from the analysis. These solutions include the solitary wave solutions, periodic waves periodic singular waves. The constraint conditions also fall out from the analysis that must exist in order for the soliton solutions to exist. Thus various previous list of solutions for this equation are exped. Mathematics Subject Classification: 35Q5, 35Q55, 37K. Introduction Keywords phrases: Bifurcation method, PHI-four equation, traveling wave solutions. The PHI-four equation is a very important nonlinear evolution equation (NLEE) in the area of Mathematical Physics, in particular Quantum Mechanics. This equation was studied extensively by several Mathematical Physicists across the globe. It is about time to take a look at this equation from a different perspective in order to extract several other solutions. In order to stay focussed, this paper will concentrate on the ansatz method the bifurcation analysis to reveal the several other solutions. The integrability studies of these NLEEs is a big deal in this area of Physics Mathematics [-5]. However, one must exercise extreme caution in carrying out the integration of these NLEEs as pointed out in 9 [4]. Without this cautionary approach, the results would be flawed. The ansatz approach will be first used to carry out the integration of the PHI-four equation. This will reveal a topological -solition solution that is also known as the kink solution. This will lead to a couplre of constraint conditions that must remain valid in order for the kink solution to exist. Subsequently, the paper will address the bifurcation analysis of the problem where the phase portraits of this equation of study will be obtained. Additionally by the traveling wave approach, several other solutions will be obtained. They are the Communicated by Norhashidah Mohd. Ali. Received: June 5, ; Revised: October,.

2 J. Cao, M. Song A. Biswas cnoidal waves, snoidal waves, solitary waves, periodic waves, singular periodic waves others.. Topological -Soliton Solution Or Kink Solution The PHI-four equation that is going to be studied in this paper is given by (.) u tt k u xx = au + bu 3 where in (.), the dependent variable is u(x, t) while the spatial temporal independent variables are x t respectively. The other parameters k, a b are all real-valued constants. In order to extract the topological -soliton solution of this equation, it is necessary to bear in mind that the solitons are the outcome of a delicate balance between dispersion nonlinearity. This leads to the balancing principle that will be applied to obtain the soliton solution. In order to get started, the -soliton solution ansatz is taken to be [6, ]: (.) u(x,t) = Atanh p τ where (.3) τ = B(x vt) Here in (.) (.3), the parameters A B are known as free parameters of the soliton or the kink v is the velocity of the soliton. The value of the unknown exponent p will fall out during the course of derivation of the soliton solution. Substituting (.) into (.) simplifying leads to (.4) p(p )AB ( v k ) tanh p τ p AB ( v k ) tanh p τ + p(p + )AB ( v k ) tanh p+ τ = aatanh p τ + ba 3 tanh 3p τ By the balancing principle, equation the exponents 3p p +, gives (.5) 3p = p + which gives (.6) p = This shows that the first term on the left h side gets knocked off. From the remaining terms, the linearly independent functions are tanh p+ j τ for j =,. Therefore, setting its respective coefficients to zero, yields (.7) A = a b a (.8) B = (v k ) These values of the free parameters immediately pose the constraint conditions (.9) ab < (.) a ( v k ) <

3 Topological Solitons Bifurcation Analysis of the PHI-Four Equation respectively. Thus, finally, the -soliton solution to the PHI-four equation is given by (.) u(x,t) = Atanh[B(x vt)] with the free parameters given by (.7) (.8). This kink solution will hold as long as the constraint conditions given by (.9) (.) remains valid. 3. Bifurcation analysis In this section, the Phi-four equation will be rewritten as (3.) u tt αu xx λu + βu 3 =. In this section, the aim is to study the traveling wave solutions their relations for Eq. (3.) by using the bifurcation method qualitative theory of dynamical systems [5 ]. Through some special phase orbits, we obtain many smooth periodic wave solutions periodic blow-up solutions. Their limits contain kink profile solitary wave solutions, unbounded wave solutions, periodic blow-up solutions solitary wave solutions. 3.. Phase portraits qualitative analysis We assume that the traveling wave solutions of (3.) is of the form (3.) u(x,t) = ϕ(ξ ), ξ = x ct, we have (3.3) (c α)ϕ λϕ + βϕ 3 =. To relate conveniently, let (3.4) = β c α, (3.5) µ = λ c α. Letting ϕ = y, then we get the following planar system { dϕ dξ (3.6) = y, dy dξ = ϕ3 + µϕ. Obviously, the above system (3.6) is a Hamiltonian system with Hamiltonian function (3.7) H(ϕ,y) = y + ϕ4 µϕ. In order to investigate the phase portrait of (3.6), set (3.8) f (ϕ) = ϕ 3 + µϕ. Obviously, f (ϕ) has three zero points, ϕ, ϕ ϕ +, which are given as follows µ µ (3.9) ϕ =, ϕ =, ϕ + =. Letting (ϕ i,) be one of the singular points of system (3.6), then the characteristic values of the linearized system of system (3.6) at the singular points (ϕ i,) are (3.) λ ± = ± f (ϕ i ).

4 J. Cao, M. Song A. Biswas From the qualitative theory of dynamical systems, we know that () If f (ϕ i ) >, (ϕ i,) is a saddle point. () If f (ϕ i ) <, (ϕ i,) is a center point. (3) If f (ϕ i ) =, (ϕ i,) is a degenerate saddle point. Therefore, we obtain the phase portraits of system (3.6) in Figure. Figure. The phase portraits of system (3.6), (a) <, µ <, (b) >, µ > Let (3.) H(ϕ,y) = h, where h is Hamiltonian. Next, we consider the relations between the orbits of (3.6) the Hamiltonian h. Set (3.) h = H(ϕ +,) = H(ϕ,) = µ. According to Figure, we get the following propositions. Proposition 3.. Suppose that < µ <, we have () When h < or h > h, system (3.6) does not any closed orbit. () When < h < h, system (3.6) has three periodic orbits Γ, Γ Γ 3. (3) When h =, system (3.6) has two periodic orbits Γ 4 Γ 5. (4) When h = h, system (3.6) has two heteroclonic orbits Γ 6 Γ 7. Proposition 3.. Suppose that > µ >, we have () When h h, system (3.6) does not any closed orbit. () When h < h <, system (3.6) has two periodic orbits Γ 8 Γ 9. (3) When h =, system (3.6) has two homoclinic orbits Γ Γ. (4) When h >, system (3.6) has a periodic orbit Γ. 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 above analysis, we have the following propositions.

5 Topological Solitons Bifurcation Analysis of the PHI-Four Equation 3 Proposition 3.3. If < µ <, we have () When < h < h, (3.) has two periodic wave solutions (corresponding to the periodic orbit Γ in Fig. ) two periodic blow-up wave solutions(corresponding to the periodic orbits Γ Γ 3 in Figure ). () When h =, (3.) has periodic blow-up wave solutions(corresponding to the periodic orbits Γ 4 Γ 5 in Figure ). (3) When h = h, (3.) has two kink profile solitary wave solutions two unbounded wave solutions (corresponding to the heteroclinic orbits Γ 6 Γ 7 in Figure ). Proposition 3.4. If > µ >, we have () When h < h <, (3.) has two periodic wave solutions(corresponding to the periodic orbits Γ 8 Γ 9 in Figure ). () When h =, (3.) has two solitary wave solutions(corresponding to the homoclinic orbits Γ Γ in Figure ). (3) When h >, (3.) has two periodic wave solutions(corresponding to the periodic orbit Γ in Figure ). 3.. Traveling wave solutions their relations Firstly, we will obtain the explicit expressions of traveling wave solutions for the (3.) when < µ <. () From the phase portrait, we note that there are three periodic orbits Γ, Γ Γ 3 passing the points (ϕ,),(ϕ,), (ϕ 3,) (,). In (ϕ,y)-plane the expressions of the orbits are given as (3.3) y = ± ϕ (ϕ ϕ )(ϕ ϕ )(ϕ ϕ 3 )(ϕ ), µ µ where ϕ = µ+ µ, ϕ = µ+ µ, ϕ 3 = µ µ, = < h < h. Substituting (3.3) into dϕ/dξ = y integrating them along Γ, Γ Γ 3, we have (3.4) ± (s ϕ )(s ϕ )(s ϕ 3 )(s ) ds = ξ ds, (3.5) ± ϕ (s ϕ )(s ϕ )(s ϕ 3 )(s ) ds = (3.6) ϕ = ± ), sn ( ξ, ϕ 3 (3.7) ϕ = ±ϕ 3 sn ( ξ, ϕ ) 3. Noting that (3.), we get the following periodic wave solutions (3.8) u (x,t) = ± ), sn ( (x ct), ϕ 3 ξ ds.

6 4 J. Cao, M. Song A. Biswas (3.9) u (x,t) = ±ϕ 3 sn ( (x ct), ϕ 3 () From the phase portrait, we note that there are two special orbits Γ 4 Γ 5, which have the same hamiltonian with that of the center point (,). In (ϕ,y)-plane the expressions of the orbits are given as (3.) y = ± ϕ (ϕ ϕ 5 )(ϕ ϕ 6 ), where ϕ 5 = µ/ ϕ 6 = /µ. Substituting (3.) into dϕ/dξ = y integrating them along the two orbits Γ 4 Γ 5, it follows that + (3.) ± ϕ s (s ϕ 5 )(s ϕ 6 ) ds = ξ ds. (3.) ϕ = ± µ csc µξ. Noting that (3.), we get the following periodic blow-up wave solutions µ (3.3) u 3 (x,t) = ± csc µ(x ct). (3) From the phase portrait, we see that there are two heterclinic orbits Γ 6 Γ 7 connected at saddle points (ϕ,) (ϕ +,). In (ϕ,y)-plane the expressions of the heterclinic orbits are given as (3.4) y = ± (ϕ ϕ ) (ϕ ϕ + ). Substituting (3.4) into dϕ/dξ = y integrating them along the heterclinic orbits Γ 6 Γ 7, it follows that ϕ (3.5) ± (s ϕ )(ϕ + s) ds = ξ ds, + (3.6) ± ϕ (s ϕ )(s ϕ + ) ds = ξ ds. µ (3.7) ϕ = ± tanh µ ξ, µ (3.8) ϕ = ± coth µ ξ. ).

7 Topological Solitons Bifurcation Analysis of the PHI-Four Equation 5 Noting that (3.), we get the following kink profile solitary wave solutions µ (3.9) u 4 (x,t) = ± tanh µ (x ct), unbounded wave solutions µ (3.3) u 5 (x,t) = ± coth µ (x ct). Secondly, we will obtain the explicit expressions of traveling wave solutions for the (3.) when > µ >. () From the phase portrait, we see that there are two closed orbits Γ 8 Γ 9 passing the points (ϕ 7,), (ϕ 8,), (ϕ 9,) (ϕ,). In (ϕ,y)-plane the expressions of the closed orbits are given as (3.3) y = ± (ϕ ϕ7 )(ϕ ϕ 8 )(ϕ ϕ 9 )(ϕ ϕ), where ϕ 7 = µ+ µ, ϕ 8 = µ µ µ µ, ϕ 9 = µ+ µ, ϕ = h < h <. Substituting (3.3) into dϕ/dξ = y integrating them along Γ 8 Γ 9, we have (3.3) ± ϕ (ϕ s)(ϕ 9 s)(ϕ 8 s)(s ϕ 7 ) ds = ϕ 7 ξ ds, ϕ ξ (3.33) ± ϕ (s ϕ7 )(s ϕ 8 )(s ϕ 9 )(ϕ s) ds = ds. ( )) (ϕ ϕ 8 )ϕ 7 + (ϕ 8 ϕ 7 )ϕ (sn ω ξ,κ (3.34) ϕ = ( ( )), ϕ ϕ 8 + (ϕ 8 ϕ 7 ) sn ω ξ,κ (3.35) ϕ = ϕ (ϕ ϕ 9 ) sn ϕ ξ, ϕ ϕ 9, ϕ (ϕ ϕ 8 )(ϕ 9 ϕ 7 ) where ω = κ = (ϕ ϕ 9 )(ϕ 8 ϕ 7 ) (ϕ ϕ 8 )(ϕ 9 ϕ 7 ). Noting that (3.), we get the following periodic wave solutions ( ( )) ) ((ϕ ϕ 8 )ϕ 7 + (ϕ 8 ϕ 7 )ϕ sn ω (x ct),κ (3.36) u 6 (x,t) = ( ( )), ϕ ϕ 8 + (ϕ 8 ϕ 7 ) sn ω (x ct),κ

8 6 J. Cao, M. Song A. Biswas (3.37) u 7 (x,t) = ϕ (ϕ ϕ 9 ) sn ϕ (x ct), ϕ ϕ 9. () From the phase portrait, we see that there are two symmetric homoclinic orbits Γ Γ connected at the saddle point (,). In (ϕ,y)-plane the expressions of the homoclinic orbits are given as (3.38) y = ± ϕ (ϕ ϕ )(ϕ ϕ), where ϕ = µ/ ϕ = µ/. Substituting (3.38) into dϕ/dξ = y integrating them along the orbits Γ Γ, we have (3.39) ± ϕ ϕ s (s ϕ )(ϕ s) ds = ϕ (3.4) ± ϕ s (s ϕ )(ϕ s) ds = (3.4) ϕ = (3.4) ϕ = µ sech µξ, µ sech µξ. ξ ds, ξ ds. Noting that (3.), we get the following solitary wave solutions µ (3.43) u 8 (x,t) = sech µ(x ct), (3.44) u 9 (x,t) = µ sech µ(x ct). (3) From the phase portrait, we see that there are a closed orbit Γ passing the points (ϕ 3,) (,). In (ϕ,y)-plane the expressions of the closed orbits are given as (3.45) y = ± (ϕ4 ϕ)(ϕ ϕ 3 )(ϕ c )(ϕ c ), µ+ µ where ϕ 3 = h µ+ µ, = h µ µ, c = i h µ µ, c = i h h >. ϕ

9 Topological Solitons Bifurcation Analysis of the PHI-Four Equation 7 Substituting (3.45) into dϕ/dξ = y integrating them along the orbit Γ, we have ϕ ξ (3.46) ± ϕ 3 (ϕ4 s)(s ϕ 3 )(s c )(s c ) ds = ds, ϕ4 ξ (3.47) ± ϕ (ϕ4 s)(s ϕ 3 )(s c )(s c ) ds = ds. (3.48) ϕ = ϕ 3 cn( µξ, ϕ3 (3.49) ϕ = cn( µξ,ϕ4 ), µ ). µ Noting that (3.), we get the following periodic wave solutions ) µ(x (3.5) u (x,t) = ϕ 3 cn( ct), ϕ3, µ ) µ(x (3.5) u (x,t) = cn( ct),ϕ4. µ (a) (b) (c) Figure. The imaginary part of the periodic wave solution u + (x,t) evolute into the kink wave solutions u 4+ (x,t) at t = with the conditions (3.5). (a) h =.8; (b) h =.; (c) h =.5. Thirdly, we will give that relations of the traveling wave solutions. () Letting h h, it follows that µ/, ϕ 3 µ/, ϕ 3 / sn( µ(x ct),) = tanh µ(x ct). Therefore, we obtain u (x,t) u 5 (x,t) u (x,t) u 4 (x,t). () Letting h +, it follows that µ/, ϕ 3, ϕ 3 / sn( µ(x ct),) = sin µ(x ct). Therefore, we obtain u (x,t) u 3 (x,t). (3) Letting h, it follows that ϕ µ/, ϕ 9, ϕ 8, ϕ 7 µ/, ω µ/, k sn( µ/(x ct),) = tanh µ/(x ct). Therefore, we obtain u 6 (x,t) u 8 (x,t)...4.6

10 8 J. Cao, M. Song A. Biswas (4) Letting h, it follows that ϕ µ/, ϕ 9, ϕ 8, ϕ 7 µ/, ϕ ϕ 9 ϕ sn( µ(x ct),) = tanh µ(x ct). Therefore, we obtain u 7 (x,t) u 9 (x,t). (5) Letting h +, it follows that µ/, ϕ 3 µ/, ϕ 3 /µ, /µ cn( µ(x ct),) = sech µ(x ct). Therefore, we obtain u (x,t) u 8 (x,t) u (x,t) u 9 (x,t). Finally, we will show that the periodic wave solutions u + (x,t) evolute into the kink profile solitary wave solutions u 4+ (x,t) when the Hamiltonian h h (corresponding to the changes of phase orbits of Figure as h varies). We take some suitable choices of the parameters, such as (3.5) α =, β =, c =, λ =, as an illustrative sample draw their plots (see Figure ). 4. Conclusions This paper studies the PHI-four equation by the aid of ansatz method bifurcation analysis. These approaches allowed to reveal several solution of this equation. They are cnoidal waves, snoidal waves, solitary waves, kinks, periodic waves, periodic singular waves others. The constraint conditions imposes the restrictions on the choice of the parameters coefficients of the governing equations. There are several other NLEES where, particularly, the bifurcation method, can be applied to obtain these interesting solutions to them. The results of these research will be available in due course of time. Acknowledgement. Research is supported by the National Natural Science Foundation of China (No. 3669) Natural Science Foundation of Educational Department of Yunnan Province (No.3Y48). References [] K. C. Basak, P. C. Ray R. K. Bera, Solution of non-linear Klein-Gordon equation with a quadratic nonlinear term by Adomian decomposition method, Commun. Nonlinear Sci. Numer. Simul. 4 (9), no. 3, [] A. Biswas, C. Zony E. Zerrad, Soliton perturbation theory for the quadratic nonlinear Klein-Gordon equation, Appl. Math. Comput. 3 (8), no., [3] A. Biswas, A. Yildirim, T. Hayat, O. M. Aldossary R. Sassaman, Soliton perturbation theory for the generalized Klein-Gordon equation with full nonlinearity, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 3 (), no., 3 4. [4] N. A. Kudryashov, Seven common errors in finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simul. 4 (9), no. 9-, [5] R. Sassaman A. Biswas, Soliton perturbation theory for phi-four model nonlinear Klein-Gordon equations, Commun. Nonlinear Sci. Numer. Simul. 4 (9), no. 8, [6] R. Sassaman A. Biswas, Topological non-topological solitons of the generalized Klein-Gordon equations, Appl. Math. Comput. 5 (9), no.,. [7] R. Sassaman, A. Heidari, F. Majid A. Biswas, Topological non-topological solitons of the generalized Klein-Gordon equations in + dimensions, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 7 (), no., [8] R. Sassaman, A. Heidari A. Biswas, Topological non-topological solitons of nonlinear Klein-Gordon equations by He s semi-inverse variational principle, J. Franklin Inst. 347 (), no. 7,

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