Original articles
Exact solitary wave and periodic-peakon solutions of the complex Ginzburg–Landau equation: Dynamical system approach

https://doi.org/10.1016/j.matcom.2021.08.007Get rights and content

Abstract

Using the bifurcation theory of the planar dynamical system, we study the exact solutions of the complex Ginzburg–Landau equation which is a popular model in mathematical physics. All possible exact explicit parametric representations of traveling wave solutions are given under different parameter conditions, including the solitary wave solutions, periodic wave solutions, compacton solutions pseudo-peakon solutions and periodic peakon solutions. In more general parametric conditions, all possible solutions are caught in one dragnet.

Introduction

Ginzburg–Landau equation is a mathematical model describing the phenomena of super-conductivity. It is a kind of nonlinear Schrödinger equation and has a wide range of applications. Many mathematical physicists have done a great deal of work on this equation. More recently, the complex Ginzburg–Landau equation (CGLE) iqt+aqxx+bF(|q|2)q=1|q|2qα|q|2(|q|2)xxβ((|q|2)x)2+γq,has aroused a great deal of research interest among scholars [1], [3], [4], [5], [7], [12], [13], [20], [21], [22], [24], [25], [26], [27], [28], [29], [30]. Where q(x,t) is a complex-valued function representing the soliton molecule, x,t are spatial and temporal coordinates respectively. F is a nonlinear term that describes the general form of dependence on the refractive index of optical fibers. a,α,β,γ are parameters of Eq. (1). This model describes the dynamics of solitary wave propagation through optical fibers for long-distance. It is an interesting and widely applied topic, and many researchers are devoted to applying various powerful methods to construct the exact solution of CGLE. These methods include the traveling wave reduction method [12], [20], [21], the Jacobi elliptic function method [8], [18], the tanh-function method [6], [19], [23], Lie symmetry method [5], Kudryashov method [11], [29], (G/G)- expansion method [21], [29], exp-function expansion method [1], F-expansion scheme [3], and so on [4], [9], [10], [21]. Moreover, in [21], the above methods are integrated to study optical solitons in CGLE with Kerr and power laws of nonlinearity. Four forms of optical solitons are recovered in this paper, and other wave solutions, as a by-product. However, most of these papers only obtain partial solutions of the CGLE, which are usually solitary wave solutions and periodic solutions. There are still many other solutions and abundant dynamic behaviors of the CGLE that remains to be discovered. Recently, Zhu and Xia [30] studied bifurcations and exact solutions of Eq. (1) when the term F(|q|2)=|q|2 via dynamical system approach. Moreover, Kudryashov [12] studied Eq. (1) using traveling wave reduction and obtained the general solution of Eq. (1) when the parameters of the equation meet certain conditions. The dynamics behaviors of the solutions are shown only in the form of solitary wave and periodic wave.

Inspired by this, in this paper, we employ the dynamical systems method to investigate the corresponding traveling wave system of Eq. (1), under very general parametric conditions, to study the exact solutions of the traveling wave system and discuss the dynamical behavior of solutions.

To study the traveling wave solutions of Eq. (1), let q(x,t)=ϕ(xνt)exp(iψ(xνt))=ϕ(ξ)exp(iψ(ξ)),ξ=xνt.Substituting (2) into Eq. (1), separating the imaginary part and real part, yield aϕψξξ+2aψξϕξνϕξ=0,(a2α)ϕξξ+νϕψξaϕψξ2γϕ+2(2βα)ϕξ2ϕ+bϕF(ϕ2)=0, where a2α. Multiplying Eq. (3) on ϕ and integrating the obtained equation once, we get ψξ=ν2agaϕ2,ψ(ξ)=ν2aξgadξϕ2(ξ),where g is an integral constant. If g=0, then ψ=ν2a(xνt).

Substituting (5) into (4), we have (a2α)ϕ+2(2βα)(ϕ)2ϕg2aϕ3+bϕF(ϕ2)γϕ+ν24aϕ=0,where “” denote the differential with respect to ξ.

Remark 1

We notice that the equation (a2α)yzz+2(2βα)yz2yc12ay3+byF(y2)+γy+C0(2C0)y4a+(C01)C1ay=0given by Kudryashov [12] is not correct. The last two items are miscalculated. Therefore, all results in [12] need to be made new studies.

Eq. (6) is equivalent to the following system dϕdξ=y,dydξ=8a(2βα)ϕ2y24g2+4abF(ϕ2)ϕ4+(ν24aγ)ϕ44a(2αa)ϕ3.

If we take F(|q|2)=|q|2n,n(1) is a positive integer, then, when α2β,a4β,a4(αβ),a(2n+4)α4βn+1, system (7) has the first integral H(ϕ,y)=y2(4aγν2)ϕ24a(a4α+4β)+bϕ2n+2(n+1)a2nα+4(βα)g2a(4βa)ϕ2ϕp=h,where p=4(2βα)a2α,h is a constant. Let m be a positive integer. If we take a=1m[2(m+1)α4β], then p=2m.

When α=2β, system (7) has the first integral H1(ϕ,y)=ϕ2y2+1a(a2α)abn+1ϕ2n+414(4aγν2)ϕ4+g2=h.

When a=4β, system (7) has the first integral H2(ϕ,y)=ϕ2y2+18a(a2α)8abn+2ϕ2n+4(4aγν2)ϕ416g2ln(ϕ)=h.

When a=4(αβ), system (7) has the first integral H3(ϕ,y)=y2ϕ2+12a(a2α)2abnϕ2n+g2ϕ4(4aγν2)ln(ϕ)=h.

When a=2(n+2)α4βn+1, we have α2β=12(n+1)(a2α). System (7) has the first integral H4(ϕ,y)=y2ϕ2n+2+1a(a2α)g2ϕ2n+44aγν24nϕ2n+2abln(ϕ)=h.

Especially, when a=4(n1)α+4β2n1, system (7) has the first integral H5(ϕ,y)=y2ϕ4n2+12a(a2α)2ab3nϕ6n4aγν24nϕ4n+g2n1ϕ4(n1)=h.

System (7) is a singular traveling wave system of the first class (see Li and Chen [15], Li [14], Li and Qiao [16]) with the singular straight lines ϕ=0. It depends on seven-parameter group (a,b,α,β,γ,ν,g).

It is known that a wave profile defined by ϕ(ξ) with some phase orbits of a planar dynamical system (7) has the following relationships. A smooth homoclinic orbit to a saddle point of a traveling wave system corresponds to a smooth solitary wave solution of a PDE. All periodic orbits which have segments close to the singular straight line ϕ=0 in a period annulus, correspond to periodic cusp wave solutions (periodic peakon). A homoclinic orbit, which has a segment close to the singular straight line ϕ=0, corresponds to a pseudo-peakon solution. For a family of open orbits, when |y|, which tend to a singular straight line, then this family of orbits corresponds to a family of compactons (see Li, Zhou and Chen [17]).

The present paper is built up as follows. In Section 2, we consider the equilibrium points and phase portraits of the system (7). In Sections 3, 4, 5, we derive all possible exact solutions of ϕ(ξ) under different parameter conditions. Finally, the main theorem and conclusion are given in Section 6.

Section snippets

Phase portraits and equilibrium points of system (7)

Consider the regular system associated with the system (7) as follows, dϕdζ=4a(2αa)ϕ3y,dydζ=8a(2βα)ϕ2y24g2+4abϕ2n+4+(ν24aγ)ϕ4,where dξ=4a(2αa)ϕ3dζ, for ϕ0. Systems (7), (14) have the same first integral, but in the phase plane, two systems define different vector fields in the two sides of the singular straight lines.

Obviously, an equilibrium point Ej(ϕj,0) of system (14) satisfies f(ϕj)=0, where f(ϕ)=4abϕ2n+4(4aγν2)ϕ44g2f0(ϕ)4g2.

When g=0,f0(ϕ)=ϕ4[4abϕ2n(4aγν2)] has three zeros at ϕ

Exact explicit solitary wave solutions and periodic wave solutions of system (7) when p=4(2βα)a2α=4,4 in (8)

Let m be a positive integer. Assume that a=2(m+1)α4βm, then, p=2m. When m=1, then a=4β. and m=1, then a=4(αβ). We discuss these cases in next sections.

Exact explicit solitary wave solutions, periodic wave solutions of system (7) when α=2β in (7) with n=1,2

When α=2β in (7), we see from (9) that y2=hϕ21a(a2α)ϕ2abn+1ϕ2n+414(4aγν2)ϕ4+g2.Thus, applying the first equation of (7), we have ωaξ=ϕ0ϕϕdϕF3(ϕ2,h)=φ0φdφ2F3(φ,h),where ωa=|b(n+1)(a2α)|,φ=ϕ2, and F3(ϕ2,h)=|(n+1)(a2α)b|h(n+1)g2ab(n+1)(4aγν2)4abϕ4ϕ2n+4.

Firstly, assume that a2α>0, we consider the phase portraits in Fig. 2(b).

(i) Corresponding to the two closed orbit families defined by H1(ϕ,y)=h,h(h1,h2). When n=1, (27) can be written as 2ωaξ=φcφdφ(φaφ)(φbφ)(φφc). When n=2, (27)

Exact explicit solitary wave solutions, periodic wave solutions of system (7) when a=2(n+2)α4βn+1,g=0 in (7)

When a=2(n+2)α4βn+1,p=4n2. We see from (13) that if g=0, then y2=1ϕ4n2h12a(a2α)2ab3nϕ6n4aγν24nϕ4n.Under the parameter condition 4ab(4aγν2)>0, system (7) has three equilibrium points O(0,0) and E1(ϕ1,0), where ϕ1=4aγν24ab12n. We have h0=H5(0,0)=0,h1=H5(ϕ1,0)=(4aγν2)3384a3b2n(a2α). The phase portraits of system (7) are shown in Fig. 6.

Taking φ=ϕ2n, then, using the first equation of system (7), we have 2ωbξ=φ0φdφωbh4aγν28abφ2φ3,where ωb=|b3n(a2α)|.

Firstly, assume that a2α>0. we

Main results and conclusion

From the detailed analysis of bifurcations and the explicit expressions of solutions of Eq. (1), we summarize it as a theorem as follows.

Theorem 1

(1) The complex Ginzburg–Landau equation (1) has exact explicit solutions with the form q(x,t)=ϕ(xνt)exp(iψ(xνt))=ϕ(ξ)exp(iψ(ξ)),where when the function F(|q|2)=|q|2n,ϕ(ξ) is a solution of the corresponding traveling wave system (7) and ψ(ξ)=ν2aξgadξϕ2(ξ).

(2) Corresponding to some families of closed orbits of the system (7), ϕ(ξ) has exact explicit

Acknowledgments

The authors wish to express their sincere appreciation to all those who made suggestions for improvements of this paper. This research was supported by the National Natural Science Foundation of China (11871231, 12071162, 11701191, 11371326, 11975145, 11901215).

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