MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 462957 10.1155/2013/462957 462957 Research Article Existence Analysis of Traveling Wave Solutions for a Generalization of KdV Equation Long Yao Chen Can Alfonzetti Salvatore College of Mathematics, Honghe University Mengzi Yunnan 661100 China uoh.edu.cn 2013 4 2 2013 2013 12 09 2012 16 11 2012 2013 Copyright © 2013 Yao Long and Can Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By using the bifurcation theory of dynamic system, a generalization of KdV equation was studied. According to the analysis of the phase portraits, the existence of solitary wave, cusp wave, periodic wave, periodic cusp wave, and compactons were discussed. In some parametric conditions, exact traveling wave solutions of this generalization of the KdV equation, which are different from those exact solutions in existing references, were given.

1. Introduction

In 1995, Fokas  derived a generalization of KdV equation (1)ut+ux+νuxxt+βuxxx+αuux+13αν(uuxxx+2uxuxx)=0, which from physical considerations via a methodology introduced by Fuchssteiner and Fokas . Equation (1) can also be deriven by the approaches described in . By using the bifurcation theory of dynamic system (see  and references cited therein), it is easy to investigate exact traveling wave solutions of (1) under different kinds of parametric conditions. For example, in , by using the bifurcation theory of dynamic system, some subsection-function and implicit function solutions such as compactons, solitary waves, smooth periodic waves, and nonsmooth periodic waves with peaks as well as the existence conditions have been presented by Bi. By using the same method, Li and Zhang  studied a generalization form of the modified KdV equation, which is more complex than (1). In , the existence of solitary wave, kink and antikink wave solutions, and uncountably infinitely many smooth and nonsmooth periodic wave solutions is discussed. By using the improved method named integral bifurcation method [10, 11], Rui et al.  obtain all kinds of soliton-like or kink-like wave solutions, periodic wave solutions with loop or without loop, smooth compacton-like periodic wave solution, and nonsmooth periodic cusp wave solution for (1). The integral bifurcation method possessed some advantages of the bifurcation theory of the planar dynamic system [6, 13] and auxiliary equation method (see [14, 15] and references cited therein), it is easily combined with computer method  and useful for many nonlinear partial differential equations (PDEs) including some PDEs with high-power terms, such as K(m,n) equation . However, we cannot discuss the existence of traveling wave solutions by using this method. Therefore, the bifurcation theory of the planar dynamic system bulks large with the existence analysis of traveling wave solutions of nonlinear PDEs.

In this paper, by using the bifurcation theory of dynamic system and a translation transformation, we will restudy (1). According to the analysis of the phase portraits, the existence of solitary wave, cusp wave, periodic wave, periodic cusp wave, and compactons will be discussed. The obtained results are different from those in [8, 12].

2. Bifurcations of Phase Portraits of the System and the Existence of Traveling Wave Solutions for (<xref ref-type="disp-formula" rid="EEq1.1">1</xref>) 2.1. Bifurcations of Phase Portraits for the 2-Dimensional Planar System of (<xref ref-type="disp-formula" rid="EEq1.1">1</xref>)

In this subsection, we discuss the first integral and bifurcations of phase portraits of (1).

Making a traveling wave transformation u=ϕ(ξ)+λ with ξ=x-ct, (1) can be reduced to the following ordinary differential equation: (2)(1-c)ϕ+(β-cν)ϕ+α(ϕ+λ)ϕ+13αν[(ϕ+λ)ϕ+2ϕϕ′′]=0. We call the solution ϕ(ξ)+λ traveling wave solution.

Integrating (2) with respect to ξ, we have (3)(β-cν+13ανλ+13ανϕ)ϕ′′=g+(c-1-αλ)ϕ-12αϕ2-16αν(ϕ)2, where g is an integral constant. Let dϕ/dξ=y. Thus, (3) can be rewritten as the following two-dimensional planar system: (4)dϕdξ=y,dydξ=g+(c-1-αλ)ϕ-(1/2)αϕ2-(1/6)ανy2β-cν+(1/3)ανλ+(1/3)ανϕ, which is a singular system. Equation (4) has the following first integral: (5)H(ϕ,y)(3cν-3β-ανλ-ανϕ)y2-αϕ3+3(c-1-αλ)ϕ2+6gϕ=h. Because the dy/dξ is not defined when β-cν+(1/3)ανλ+(1/3)ανϕ=0, so we make a transformation as follows: (6)dξ=(β-cν+13ανλ+13ανϕ)dτ, where τ is a parameter. Under the transformation (6), (4) becomes the following regular system: (7)dϕdτ=(β-cν+13ανλ+13ανϕ)y,dydτ=g+(c-1-αλ)ϕ-12αϕ2-16ανy2. Obviously, (7) has the same first integral as (4) which is (5).

Obviously, system (7) has two equilibrium points M1,2  (ϕ1,2,0) on ϕ-axes, where ϕ1,2=((c-1-αλ)±(c-1-αλ)2+2αg)/α. System (7) has two equilibrium points S±(ϕs,Y1,2) on singular line ϕ=ϕs, where ϕs=(3cν-3β-ανλ)/αν, Y1,2=±3αν(2g+2(c-1-αλ)ϕs-αϕs2)/αν. By using (5), we obtain Hamiltonian of these equilibrium points as follows: (8)h1,2H(ϕ1,2,0)=-αϕ1,23+3(c-1-αλ)ϕ1,22+6gϕ1,2,h3hs=H(ϕs,Y1,2)=(3cν-3β-ανλ-ανϕs)Y1,22-αϕs3+3(c-1-αλ)ϕs2+6gϕs. According to the characteristic and the relation of Hamiltonian of each equilibrium points, we obtain three bifurcations of parameters as follows: (9)g1=-c(c-2-2αλ)+(1+αλ)22α,g2=(3cν-3β-ανλ)(2ν+cν-3β+ανλ)2αν2,g3=(3ν+2ανλ-3β)(4cν-ν-2ανλ-3β)8αν2.

Under different parametric conditions, that is to say, according to different bifurcations of parameters, in the different area of parameters and on the different bifurcation curves, we derive all kinds of bifurcations of phase portraits for the system (7), which are shown in Figures 1 and 2.

The phase portraits of (7) under different parametric conditions for ν<0.

g = g 1

g 1 < g < g 3

g = g 3

g 3 < g < g 2

g = g 2

g > g 2

The phase portraits of (7) under different parametric conditions for ν<0.

c < 0 , g > g 2

c < 0 , g = g 2

c < 0 , g 1 < g < g 2

c < 0 , g = g 1

c > 0 , g = g 1

c > 0 , g 1 < g < g 2

c > 0 , g = g 2

c > 0 , g > g 2

Suppose that u(x,t)=ϕ(ξ)+λ is a continuous traveling wave solution of (1) for ξ(-,), and limξϕ(ξ)=a, limξ-ϕ(ξ)=b. It is well known that (i) ϕ(x,t) is called a solitary wave solution if a=b; (ii) ϕ(x,t) is called a kink or antikink solution if ab. Usually, a solitary wave solution of a nonlinear wave equation corresponds to a homoclinic orbit of its traveling wave equation; a kink (or antikink) wave solution corresponds to a heteroclinic orbit (or connecting orbit). Similarly, a periodic orbit of a traveling wave equation corresponds to a periodic travelling wave solution of the nonlinear wave equation. To find all possible bifurcations of solitary waves, and periodic waves, kink and antikink wave of a nonlinear wave equation, we need to investigate the existence of all homoclinic, heteroclinic orbits, and periodic orbits for its traveling wave equation in the parameter space. The solitary wave solution is also called soliton solution. Recently, some new phenomena of soliton are revealed by many works, see  and the references cited therein.

2.2. Existence of Traveling Wave Solutions for (<xref ref-type="disp-formula" rid="EEq1.1">1</xref>)

In this subsection, according to the bifurcations of phase portraits in Figures 1 and 2, by using analysis of the phase portraits, we will discuss the existence of smooth solitary traveling wave solutions, kink wave solutions, peakon solutions, compacton solutions, and periodic traveling wave solutions of (1).

Theorem 1.

Existence theorem of traveling wave solution.

Suppose that ν<0. Then consider the following.

Equation (1) has infinitely many periodic wave solutions corresponding to a family of close orbits H(ϕ,y)=h under the following parametric conditions:

g1<g<g3, h1<h<h2, see Figure 1(b);

g=g3, h1<h<h2, see Figure 1(c);

g3<g<g2, h3<h<h2, see Figure 1(d).

Equation (1) has a solitary wave solution corresponding to the homoclinic orbit H(ϕ,y)=h1 under the parametric conditions g1<g<g3, see Figure 1(b).

Equation (1) has a solitary cusp wave (peakon) solution corresponding to two heteroclinic orbits H(ϕ,y)=hs under the following parametric conditions g=g3, h1=hs, see Figure 1(c).

Equation (1) has a periodic cusp wave solution corresponding to a semilunar orbits H(ϕ,y)=h3 under the following parametric conditions g3<g<g2, h3=h4, see Figure 1(c).

Equation (1) has infinitely many compacton solutions corresponding to a  family of bounded open orbits H(ϕ,y)=h under the following parametric conditions:

g3<g<g2, h1<h<h3, see Figure 1(d);

g=g2, h1<h<h3, see Figure 1(e);

g>g2, h1<h<h3, see Figure 1(f).

Equation (1) has two-half kink wave (generalized kink wave) solutions  corresponding to two heteroclinic orbits H(ϕ,y)=h under the following  parametric conditions:

g3<g<g2, h=h1, see Figure 1(d);

g>g2, h=h1  or  h=h2, see Figure 1(f).

Theorem 2.

Existence theorem of traveling wave solution.

Suppose that ν>0. Then consider the following.

Equation (1) has infinitely many periodic wave solutions corresponding to a family of close orbits H(ϕ,y)=h under the following parametric conditions:

c<0, g>g2, h1<h<h3 and h3<h<h2, see Figure 2(a);

c<0, g=g2, h3<h<h2, see Figure 2(b);

c<0, g1<g<g2, h1<h<h2, see Figure 2(c);

c>0, g1<g<g2, h1<h<h2, see Figure 2(f);

c>0, g=g2, h1<h<h2, see Figure 2(g);

c>0, g>g2, h1<h<h3 and h3<h<h2, see Figure 2(g).

Equation (1) has a solitary wave solution corresponding to the homoclinic orbit H(ϕ,y)=h1 or H(ϕ,y)=h2 under the parametric conditions:

c<0, g1<g<g2, h=h1, see Figure 2(c);

c>0, g1<g<g2, h=h2, see Figure 2(f).

Equation (1) has a periodic cusp wave solution corresponding to a semilunar orbits H(ϕ,y)=h3 under the following parametric conditions:

c<0, g>g2, h2=h3, see Figure 2(a);

c>0, g>g2, h2=h3, see Figure 2 (h).

Equation (1) has infinitely many compacton solutions corresponding to a family of bounded open orbits H(ϕ,y)=h under the following parametric conditions:

c<0, g>g2, h>h2 and -<h<h1, see Figure 2(a);

c<0, g=g2, h>h3 and 0<h<h3, see Figure 2(b);

c<0, g1<g<g2, h>h3 and -<h<h1, see Figure 2(c);

c<0, g=g1, h>h3 and -<h<h2, see Figure 2(d);

c>0, g=g1, h1<h<+ and -<h<h1, see Figure 2(e);

c>0, g1<g<g2, h2<h<+ and -<h<h1, see Figure 2(f);

c>0, g=g2, h2<h<+ and h1<h<h2, see Figure 2(g);

c>0, g>g2, h2<h<+ and -<h<h1, see Figure 2(g).

Equation (1) has two-half kink wave (generalized kink wave) solutions corresponding to two heteroclinic orbits H(ϕ,y)=h under the following parametric conditions:

c<0, g=g1, h=h1=h2, see Figure 2(d);

c>0, g=g1, h=h1=h2, see Figure 2(e).

3. The Exact Travelling Wave Solutions of (<xref ref-type="disp-formula" rid="EEq1.1">1</xref>)

In this section, we discuss exact travelling wave solutions of (1) under some parametric conditions.

Case 1.

When  h=hs, g=g3, system (7) has two heteroclinic orbits, see Figure 1(c); by using (5), their expressions can be written as (10)y=±1-ν(ϕ-3β-3ν-2ανλ2αν). Corresponding to these two heteroclinic orbits, (1) has a peakon solution as follows: (11)u=λ+(3(2cν-ν-3β)2αν)e-1/-ν  |ξ|.

Case 2.

When h=h1, g1<g<g3, λ=(c-1)/α, g=α, c=-(2  αν-3β+ν)/2ν, system (7) has a homoclinic orbit, see Figure 1(b); by using (5), the expression can be written as (12)y=±-1ν(-22+ϕ)(2+ϕ). Corresponding to the homoclinic orbit, (1) has a solitary wave solution as follows: (13)u=λ+2tanh2((1/2)-1/νξ)+22sech2((1/2)-1/νξ).

Case 3.

When h=h1, g=g2, system (7) has two heteroclinic orbits, see Figure 1(e); by using (5), the expression can be written as (14)y=±-1ν(ϕ-ϕ1)z1-ϕϕs-ϕ(ϕ1<ϕ<ϕs<z1), where z1=(5cν+ν-6β-ανλ)/αν. Corresponding to these two heteroclinic orbits, taking (0,ϕi0), (i=1,2) as the initial values and substituting (14) into the first expression of (4), we have (15)ϕi0ϕ1φ-ϕ1ϕs-φz1-φdφ=±0ξ-1νdξ~. In order to compute the left integral of (15), we let (ϕs-φ)/(z1-φ)=ψ. First, using the integration by substitution, we compute the indefinite integral (1/(φ-ϕ1))(ϕs-φ)/(z1-φ)dφ, see the following computation: (16)1φ-ϕ1ϕs-φz1-φdφ=2(ϕs-z1)z1-ϕ1ψ2(ψ2-1)(ψ2-(ϕs-ϕ1)/(z1-ϕ1))dψ=(1ψ+1-1ψ+1)dψ+δ(1ψ-δ-1ψ+δ)dψ=ln[ψ+1ψ-1(ψ-δψ+δ)δ]=ln[ϕs-ϕ+z1-ϕϕs-ϕ-z1-ϕ(ϕs-ϕ-δz1-ϕϕs-ϕ+δz1-ϕ)δ], where δ=(ϕs-z1)/(z1-ϕ1). Second, by using (16), completing the integral equation (15), we obtain (17)ln[ϕs-ϕ+z1-ϕϕs-ϕ-z1-ϕ(ϕs-ϕ-δz1-ϕϕs-ϕ+δz1-ϕ)δ]=lnΩi±-1νξ, where (18)Ωi=(ϕs-ϕi0+z1-ϕi0ϕs-ϕi0-z1-ϕi0)(ϕs-ϕi0-δz1-ϕi0ϕs-ϕi0+δz1-ϕi0)δi=1,2, with ϕi0(ϕ1,ϕs). Finally, substituting ϕ=u-λ into (18), and then simplifying them, we obtain two generalized kink wave solutions of (1) as follows: (19)(ϕs-u+λ+z1-u+λϕs-u+λ-z1-u+λ)(ϕs-u+λ-δz1-u+λϕs-u+λ+δz1-u+λ)δ=Ω1e-1/νξ,ξ(-,l1),(20)(ϕs-u+λ+z1-u+λϕs-u+λ-z1-u+λ)(ϕs-u+λ-δz1-u+λϕs-u+λ+δz1-u+λ)δ=Ω2e--1/νξ,ξ(l2,+), where li=--νlnΩi, (i=1,2).

Case 4.

When g=g2, h=hs, system (7) has a close orbit, see Figure 2(b); by using (5), the expression can be written as (21)y=±1ν(z2-ϕ)(ϕ-ϕs), where z2=(6β-3ν-3cν-ανλ)/αν. Corresponding to this close orbit, (1) has a periodic wave solution as follows: (22)u=(λ+3ν+2ανλ-3β2αν)+3ν+6cν-6β2ανcos(1ν  ξ).

Case 5.

When g=g2, h=0, system (7) has a open orbit, see Figure 2(g); by using (5), the expression can be written as (23)y=±1ν(ϕ-z3)(ϕ-z4)(0-ϕ)ϕ-ϕs(z3<z4<ϕs<ϕ0), where z3,4=(3ν(c-1-αλ)Δ)/2αν, Δ=3(5cν+ν-6β-ανλ)(3cν+3ν-6β+ανλ). Corresponding to this open orbit, substituting (23) into the first equation of (4) yields (24)ϕsϕφ-ϕs(φ-z3)(φ-z4)(0-φ)  dφ=±0ξ1ν  dξ~. By using the formulas 256.02th and 400th in , we integrate (ϕ)sϕ(φ-ϕs)/[(φ-z3)(φ-z4)(0-φ)]dφ of (24), thus we obtain a compacton solution of (1) as follows: (25)Π(sn-1(z4(u-λ-ϕs)ϕs(u-λ-z4),k2),α22)-sn-1(z4(u-λ-ϕs)ϕs(u-λ-z4),k2)=Ω3(ξ2*-|ξ|), where Π(φ,α~2,k) is the incomplete elliptic integral of the third kind in Legendre’s canonical form, and α22=ϕs/z4, k22=ϕs(z4-z3)/[z4(ϕs-z3)], Ω3=-z4(ϕs-z3)/(2ν(ϕs-z4)), ξ2*=(Π(α22,k2)-K(k2))/Ω3.

4. Conclusion

In this work, by using the bifurcation theory of dynamic system, the generalization of KdV equation (1) was studied. According to the analysis of the phase portraits, the existing theorems on solitary wave, cusp wave, periodic wave, periodic cusp wave and compactons are given. In some parametric conditions, exact peakon solution (11), solitary wave solution (13), generalized kink wave solutions (15) and (16), smooth periodic wave solution (19), and compacton solution (21) were obtained. These exact solutions are different from those exact solutions in [8, 12].

Acknowledgments

The authors thank reviewers very much for their useful comments and helpful suggestions. This research is supported by the National Natural Science Foundation of China (11161020) and the Natural Science Foundation of Yunnan Province (2011FZ193).

Fokas A. S. On a class of physically important integrable equations Physica D 1995 87 1–4 145 150 10.1016/0167-2789(95)00133-O 1361680 ZBL1194.35363 Fuchssteiner B. Fokas A. S. Symplectic structures, their Bäcklund transformations and hereditary symmetries Physica D 1982 4 1 47 66 10.1016/0167-2789(81)90004-X 636470 Li Z. Sibgatullin N. R. An improved theory of long waves on the water surface Journal of Applied Mathematics and Mechanics 1997 61 177 482 Fuchssteiner B. Application of hereditary symmetries to nonlinear evolution equations Nonlinear Analysis 1979 3 6 849 862 10.1016/0362-546X(79)90052-X 548956 ZBL0419.35049 Rosenau P. Hyman J. M. Compactons: solitons with finite wavelength Physical Review Letters 1993 70 5 564 567 2-s2.0-12044253199 10.1103/PhysRevLett.70.564 Li J. Liu Z. Smooth and non-smooth travelling waves in a nonlinearly dispersive equation Applied Mathematical Modelling 2000 25 41 56 Long Y. He B. Rui W. Chen C. Compacton-like and kink-like waves for a higher-order wave equation of Korteweg-de Vries type International Journal of Computer Mathematics 2006 83 12 959 971 10.1080/00207160601170007 2304964 ZBL1134.35096 Bi Q. Wave patterns associated with a singular line for a bi-Hamiltonian system Physics Letters A 2007 369 5-6 407 417 Li J. Zhang J. Bifurcations of travelling wave solutions for the generalization form of the modified KdV equation Chaos, Solitons & Fractals 2004 21 4 889 913 10.1016/j.chaos.2003.12.026 2042808 ZBL1045.37045 Rui W. He B. Long Y. Chen C. The integral bifurcation method and its application for solving a family of third-order dispersive PDEs Nonlinear Analysis: Theory, Methods & Applications 2008 69 4 1256 1267 10.1016/j.na.2007.06.027 2426689 ZBL1144.35461 Weiguo R. Yao L. Bin H. Zhenyang L. Integral bifurcation method combined with computer for solving a higher order wave equation of KdV type International Journal of Computer Mathematics 2010 87 1–3 119 128 10.1080/00207160801965321 2598729 ZBL1182.65161 Rui W. Chen C. Yang X. Long Y. Some new soliton-like solutions and periodic wave solutions with loop or without loop to a generalized KdV equation Applied Mathematics and Computation 2010 217 4 1666 1677 10.1016/j.amc.2009.09.036 2727410 ZBL1203.35242 Li J. Liu Z. Traveling wave solutions for a class of nonlinear dispersive equations Chinese Annals of Mathematics. Series B 2002 23 3 397 418 2-s2.0-0036033675 10.1142/S0252959902000365 Junqi H. An algebraic method exactly solving two high-dimensional nonlinear evolution equations Chaos, Solitons & Fractals 2005 23 2 391 398 10.1016/j.chaos.2004.02.044 2089566 ZBL1069.35065 Yomba E. The extended F-expansion method and its application for solving the nonlinear wave, CKGZ, GDS, DS and GZ equations Physics Letters, Section A 2005 340 1–4 149 160 2-s2.0-19444369705 10.1016/j.physleta.2005.03.066 Wu X. Rui W. Hong X. Exact traveling wave solutions of explicit type, implicit type, and parametric type for K(m,n) equation Journal of Applied Mathematics 2012 2012 23 236875 10.1155/2012/236875 2904530 Biswas A. Yildirim A. Hayat T. Aldossary O. M. Sassaman R. Soliton perturbation theory for the generalized Klein-Gordon equation with full nonlinearity Proceedings of the Romanian Academy. Series A 2012 13 1 32 41 2911674 Triki H. Yildirim A. Hayat T. Aldossary O. M. Biswas A. Topological and non-topological soliton solutions of the Bretherton equation Proceedings of the Romanian Academy. Series A 2012 13 2 103 108 Ebadi G. Kara A. H. Petkovic M. D. Yildirim A. Biswas A. Solitons and conserved quantities of the Ito equation Proceedings of the Romanian Academy. Series A 2012 133 215 224 Johnpillai A. G. Yildirim A. Biswas A. Chiral solitons with Bohm potential by Lie group analysis and traveling wave hypothesis Romanian Journal of Physics 2012 57 3-4 545 554 2924868 Triki H. Crutcher S. Yildirim A. Hayat T. Aldossary O. M. Biswas A. Bright and dark solitons of the modified complex Ginzburg Landau equation with parabolic and dual-power law nonlinearity Romanian Reports in Physics 2012 64 2 357 366 Triki H. Yildirim A. Hayat T. Aldossary O. M. Biswas A. Shock wave solution of Benney-Luke equation Romanian Journal of Physics 2012 57 7-8 1029 1034 Girgis L. Milovic D. Konar S. Yildirim A. Jafari H. Biswas A. Optical Gaussons in birefringent fibers and DWDM systems with intermodal dispersion Romanian Reports in Physics 2012 64 3 663 671 Triki H. Yildirim A. Hayat T. Aldossary O. M. Biswas A. Topological and non-topological solitons of a generalized nonlinear Schrodinger's equation with perturbation terms Romanian Reports in Physics 2012 64 3 672 684 Crutcher S. Oseo A. Yildirim A. Biswas A. Oscillatory parabolic law spatial optical solitons Journal of Optoelectronics and Advanced Materials 2012 14 1-2 29 40 Biswas A. Khan K. Rahaman A. Yildirim A. Hayat T. Aldossary O. M. Bright and dark optical solitons ion birefringent fibers with Hamiltonian perturbations and Kerr law nonlinearity Journal of Optoelectronics and Advanced Materials 2012 14 7-8 571 576 Byrd P. F. Friedman M. D. Handbook of Elliptic Integrals for Engineers and Physicists 1954 Berlin, Germany Springer xiii+355 Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete 0060642