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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 185469, 11 pages
Research Article

Exact Periodic Solutions of the Nonintegrable Kawahara Equation

Department of Applied Mathematics and Informatics, Technical University of Sofia, P.O. Box 384, 1000 Sofia, Bulgaria

Received 26 April 2012; Accepted 19 July 2012

Academic Editors: M. Ehrnström, K. S. Fa, D. Gepner, and W.-H. Steeb

Copyright © 2012 Ognyan Yordanov Kamenov and Anna P. Angova. 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.


In the present paper, we have obtained an exact biperiodic, one-phase solution of the Kawahara evolution equation. Two classes of real periodic waves generated by the biperiodic solution have been analyzed. A modification of the bilinear-transformation method has been applied allowing to provide a single solution of the residual equation derived from the bidifferential reduction of the considered nonintegrable equation. It is shown that the spatial displacements are individual for each separate harmonic of the real periodic solutions.

1. Introduction

The nonlinear evolution equation 𝑢𝑡+𝛼𝑢𝑢𝑥+𝛽𝑢𝑥𝑥𝑥=𝛾𝑢𝑥𝑥𝑥𝑥𝑥(1.1) was introduced by Kawahara [1] as a model describing one-dimensional waves for which the angle between the front and the gradient of the external field tends to the critical angle: 𝜑𝑐=arctg(𝑚1/𝑚0𝑚0/𝑚1). Here, 𝛼, 𝛽, and𝛾 are real parameters (𝛼0), 𝑢=𝑢(𝑥,𝑡) is the elevation of dispersion medium, and 𝑚0, 𝑚1are the ion and electron masses, respectively. Within the zone of the critical angle 𝜑𝑐, the coefficient in front of 𝑢𝑥𝑥𝑥 in (1.1) decreases causing disbalance between the nonlinear and dispersion effects, which is balanced by the member with the highest-order derivative. Therefore, three varieties of the dispersion medium are distinguished: medium with positive dispersion (𝛽<0, 𝛾<0); medium with negative dispersion (𝛽>0, 𝛾>0) and medium with mixed dispersion (𝛽𝛾<0). Yamamoto [2] obtained the same evolution equation (1.1) in describing the dynamics of a nonviscous fluid in the vicinity of the critical depth. In this case, dispersion of the lowest order caused by the gravitation is balanced by the one caused by the surface tension.

The Kawahara evolution equation (1.1) is nonintegrable, which can be easily established by applying the modified criterion of Ablowtz and Segur [3]. Although it does not possess any conservation law and 𝑁-soliton solutions (𝑁2), (1.1) has been a subject of research by a number of authors employing both analytic and numerical methods.

Kano and Nakayama [4] have found elliptic solution of (1.1), from which by means of appropriate phase modulations they obtained periodic cnoidal solutions, as well as M-type and W-type solitary waves. Yamamoto and Takizawa [5] have found a stable progressive-pulse solution 𝑢(𝑥,𝑡)=105𝛽2169𝛾𝛼sec412𝛽13𝛾𝑥36𝛽2𝑡,169𝛾(1.2) which actually represents one-soliton impulse that can be generated only in a medium with positive or negative dispersion (i.e., for sign𝛽=sign𝛾). In the particular case 𝛽=0, (1.1) is called FKdV (Five Korteweg-de Vries) and has been thoroughly studied numerically and analytically by Boyd [6]. Summarizing the numerical analysis accomplished by a number of authors [1, 2, 7], we can say that the Kawahara equation has two types of localized solitary waves: compressed and fictitiously scattered corresponding to a medium with negative and positive dispersion, respectively.

Studies on the existence of periodic solutions of (1.1) are considerably fewer. In 1997 Berloff and Howard [8], by applying the singular manifold method adapted for partial differential equations [9], obtained solitary and approximate periodic solutions of the Kawahara equation.

In the present paper, two families of real periodic solutions of the aforementioned equation are obtained. These exact periodic solutions were proved to be dynamically equivalent. A “spatial” version [10] of the bilinear-transformation method of Hirota [11] and Matsuno [12] has been applied to derive them. In its classic form, this analytic model is inapplicable to nonintegrable partial differential equations.

2. Preliminaries

We could reduce the Kawahara equation in the so-called normal form if instead of the parameter 𝛽 we put 𝜀𝛽, that is, 𝛽𝜀𝛽, where 𝜀=±1, 𝛽>0, and assume that 𝛾>0.

As will be seen below, the last condition does not lead to loss of generality since after the following rescaling: 𝑥𝛾𝛽𝛾𝑥,𝑡𝛽2𝛾𝛽𝛽𝑡,𝑢2𝛼𝛾𝑢(2.1) (1.1) is reduced to its normal form 𝑢𝑡+𝑢𝑢𝑥+𝜀𝑢𝑥𝑥𝑥=𝑢𝑥𝑥𝑥𝑥𝑥,𝜀=±1.(2.2)

Hence, it is easy to realize that for 𝜀=+1, we have a medium with negative dispersion, for 𝜀=+1 along with replacing the variables 𝑢𝑢, 𝑥𝑥, 𝑡𝑡, the medium possesses positive dispersion, while for 𝜀=1, we have a medium with mixed dispersion, that is, we have all available variations of the dispersion medium.

The Kawahara equation (2.2) is invariant with respect to the Galilean transformation 𝑥=𝑥+𝜆𝑢0, 𝑡=𝑡, and 𝑈=𝑢𝑢0, where 𝜆 = const, 𝑢0 = const, that is, if 𝑈(𝑥𝑣0𝑡) is a solution of (2.2) with constant phase velocity 𝑣0, then the function 𝑢(𝑥,𝑡)=𝑢0+𝑈[𝑥(𝑣0+𝜆𝑢0)𝑡] is also a solution of the same equation but with increased phase velocity 𝑣0𝑣0+𝜆𝑢0.

3. Biperiodic Solution

We will represent the solution of (2.2) by means of the Hirota-Satsuma [13] transformation 𝑢(𝑥,𝑡)=𝑎+2𝜇(ln𝑓(𝑥,𝑡))𝑥𝑥,(3.1) where 𝑎, 𝜇 are unknown parameters so far, and 𝑓(𝑥,𝑡) is unknown function, but on the assumption periodic and continuously differentiable to seventh order (with respect to 𝑥) in the semi-infinite domain Ω={(𝑥,𝑡)𝑅2,<𝑥<,𝑡>0}. By substituting 𝑢(𝑥,𝑡) from (3.1) into (2.2) and employing the bidifferential identities for the Hirota operator (see Appendix A) 𝐷𝑛𝑡𝐷𝑚𝑥𝑥𝜑(𝑥,𝑡)𝜓,𝑡=𝜕𝜕𝜕𝑡𝜕𝑡𝑛𝜕𝜕𝜕𝑥𝜕𝑥𝑚𝜑𝑥(𝑥,𝑡)𝜓,𝑡||||𝑡=𝑡𝑥=𝑥,(3.2) we will obtain the following bidifferential form of this equation: 𝑓4𝐷𝑡𝐷𝑥+𝜀𝐷4𝑥𝐷6𝑥𝑓8𝐶𝑓𝑓+22𝐷(𝜇6𝜀)2𝑥𝑓𝑓2+𝐷2𝑥𝑓𝑓15𝑓2𝐷4𝑥𝑓𝑓+𝛼𝑓4𝐷302𝑥𝑓𝑓2=0,(3.3) where 𝐶 is an integration constant assumed to be nonzero. This constant plays a major role in constructing the periodic solutions though it does not have any dynamic features. If in (3.3) we put 𝜇=6𝜀, then a sufficient condition for the function 𝑓(𝑥,𝑡) to be its solution is to satisfy the following two equations accordingly: 𝐷𝑡𝐷𝑥+𝜀𝐷4𝑥𝐷6𝑥4𝐶𝑓𝑓=0,(3.4)𝑎𝑓4+15𝑓2𝐷4𝑥𝐷𝑓𝑓=302𝑥𝑓𝑓2.(3.5)

The first of these equations is called bidifferential (since its structure is such), and the second one will be called residual. We will search for the solution of (3.4) in the form 𝑓(𝑥,𝑡)=𝜃3(𝜉,𝑞)=𝑛=𝑞𝑛2𝑒2𝑖𝑛𝜉,(3.6) where 𝜃3 is the Jacobi biperiodic function [14] with phase variable 𝜉=𝑘𝑥+𝜔𝑡+𝛿. It is possible that the parameters 𝑘, 𝜔, 𝛿 are complex and 𝑞=𝑒𝑖𝜋𝜏, and Im𝜏>0 (i.e., 0<|𝑞|<1) is the perturbation parameter. The function 𝜃3(𝜉,𝑞) has a real period 𝜋/𝑘 and an imaginary one 𝜏/𝑘. We substitute 𝑓(𝑥,𝑡) from (3.6) into the bidifferential (3.4). We obtain an infinite system of algebraic equations 𝑚=𝐹(𝑚)𝑒2𝑖𝑚𝜉=0,(3.7) where 𝐹𝑚(𝑚)=𝑛=4𝑘𝜔(2𝑛𝑚)2+16𝜀𝑘4(2𝑛𝑚)464𝑘6(2𝑛𝑚)6𝑞+8𝐶𝑛22+(𝑛𝑚)(3.8) and having 𝑚=0,±1,±2,.

The bilinear structure of (3.7) makes it possible to apply the index parity principle therein, which means that if in (3.8) we substitute 𝑛𝑛+1, we will have the following relations: 𝐹(𝑚)=𝐹(𝑚2)𝑞2(𝑚1)=𝐹(𝑚4)𝑞2(2𝑚4)==𝐹(0)𝑞𝑚2/2if𝑚isanevennumber𝐹(𝑚)=𝐹(𝑚2)𝑞2(𝑚1)=𝐹(𝑚4)𝑞2(2𝑚4)==𝐹(1)𝑞(𝑚21)/2if𝑚isanoddnumber.(3.9) Summing up separately the even and odd addends in (3.7), we will obtain the following more compact form of this equation: 𝐹(0)𝜃32𝜉,𝑞2+𝑞1/2𝐹(1)𝜃22𝜉,𝑞2=0,(3.10) where 𝜃2(𝑧,𝑞) is the second Jacobi biperiodic function [14] defined by the equality 𝜃2(𝑧,𝑞)=𝑛=𝑞(𝑛+1/2)2𝑒𝑖(2𝑛1)𝑧. Accounting for the linear independence of the functions 𝜃2(2𝜉,𝑞2) and 𝜃3(2𝜉,𝑞2) in (3.10), it is reduced to two equations: 𝐹(0)=0 and 𝐹(1)=0. In the context of equality (3.8) and the functional identities for the 𝜃-functions, given in Appendix B, the last two equations are reduced to the following algebraic linear system: 𝑘𝜃3𝜃𝜔+3𝑞𝐶=8𝜀𝑘4𝜃3+𝑞𝜃364𝑘6𝜃3+3𝑞𝜃3+𝑞2𝜃3,𝑘𝜃2𝜃𝜔+2𝑞𝐶=8𝜀𝑘4𝜃2+𝑞𝜃264𝑘6𝜃2+3𝑞𝜃2+𝑞2𝜃2,(3.11) with respect to 𝜔 and 𝐶. This system is compatible and definite since Δ=(𝑘/𝑞)(𝜃2𝜃3𝜃3𝜃2)=(𝑘/𝑞)𝑊(𝜃2,𝜃3)0, where 𝑊 is the Wronskian determinant, and the wave number is different from zero. We have denoted for convenience 𝜃𝑗=𝜃𝑗(0,𝑞2), 𝑗=2,3, so that all derivatives of these functions in the system (3.11) are with respect to the parameter 𝑞. The solution of system (3.11) is as follows: 𝜔=8𝜀𝑘3𝑊1+𝑞𝜃2,𝜃3𝑊𝜃2,𝜃364𝑘5𝑊𝜃2,𝜃3𝑊𝜃2,𝜃3+3𝑞𝑊𝜃2,𝜃3+𝑞2𝑊𝜃2,𝜃3𝜃𝑊2,𝜃3,(3.12)𝐶=8𝜀𝑘4𝑞2𝑊𝜃2,𝜃3𝑊𝜃2,𝜃3+64𝑘6𝑞2𝑊𝜃2,𝜃3𝑊𝜃2,𝜃3+3𝑞𝑊𝜃2,𝜃3.(3.13)

The residual equation (3.5) does not possess a bilinear structure, which is a typical feature for the nonintegrable partial differential equations. This means that we cannot apply the index parity principle to (3.5), that is, we cannot reduce the infinite system generated by this equation (after substituting 𝑓(𝑥,𝑡) with its equal function 𝜃3(𝜉,𝑡)) to a system of two equations. Consequently, we will represent the parameter 𝑎 in the formal series 𝑎=𝑘4𝑚=𝑎𝑚,(3.14) by the terms 𝑎𝑚, which are real parameters unknown at this stage. Substituting (3.6) and (3.14) into the residual equation (3.5), we will obtain the infinite system 𝑎𝑚𝑛=𝑞2(𝑛𝑚)2+(2𝑛3𝑚)2=240𝑛=2𝑛2(3𝑛𝑚)2(2𝑛𝑚)4𝑞2𝑛2+(2𝑛𝑚)2,𝑚=0,±1,±2,.(3.15)

The infinite series on both sides of the system (3.15) are absolutely convergent for 0<|𝑞|<1. It is easy to deduce that in summation term by term of the equalities from (3.15) for each whole number 𝑚, the series (3.14) is an absolutely and uniformly convergent series under the same condition for the perturbation parameter. Thus, we obtain that the terms of the functional series (3.14) are determined for every integer 𝑚 by the formula 𝑎𝑚(𝑞)=240𝑛=2𝑛2(3𝑛𝑚)2(2𝑛𝑚)4𝑞2𝑛2+(2𝑛𝑚)2𝑛=𝑞2(𝑛𝑚)2+(2𝑛3𝑚)2,𝑚=0,±1,±2,.(3.16)

The terms 𝑎𝑚(𝑞) of the uniformly convergent series (3.14) will be called spatial displacements. This will be explained further.

And finally in this section, we can make the conclusion that the continuously differentiable function 𝑢(𝑥,𝑡)=𝑘4𝑚=𝑎𝑚(𝑞)+12𝜀𝑘2𝑑𝜃𝑑𝜉3(𝜉,𝑞)𝜃3(𝜉,𝑞)(3.17) is a family of localized biperiodic solutions of the Kawahara equation (2.2). For convenience, in (3.17), we have denoted 𝜃3𝑑(𝜉,𝑞)=𝜃𝑑𝜉3(𝜉,𝑞).(3.18)

Figure 1 shows the discrete values of the spatial displacements 𝑎𝑚(𝑞) for a mean value of the perturbation parameter 𝑞=𝑒𝜀𝜋, 𝜀=1, 𝑘=1, and 𝑚=0,1,2,. In addition to the foregoing, the function 𝑢(𝑥,𝑡), expressed by (3.17), is a solution of the considered equation if the phase velocity 𝜔(𝑘,𝑞) satisfies the dispersion relation (3.12), and the integration constant 𝐶 is as in (3.13). The wave number 𝑘0 is complex in the general case, but we have to exclude those values for which the constant 𝐶 is zero, that is, 𝑘2𝜀8𝑊1+3𝑞𝜃20,𝑞2,𝜃30,𝑞2𝑊𝜃20,𝑞2,𝜃30,𝑞21.(3.19)

Figure 1: Discrete values of the spatial displacements 𝑎𝑚(𝑞) for a mean value of the perturbation parameter 𝑞=𝑒𝜀𝜋, 𝜀=1, 𝑘=1, and 𝑚=0,1,2,.

4. Analicity Condition and Real Biperiodic Solutions

The obtained exact biperiodic solution (3.17) of the Kawahara evolution equation (2.2) is generally speaking a rational complex function of the phase variable 𝝃=𝑘𝑥+𝜔𝑡+𝛿. This function has twofold poles in the lattice of complex points 𝜉𝑚𝑛=𝜋(𝑚+1/2)+𝑖𝜏𝜋(𝑛+1/2),𝑚,𝑛𝑍, which are zeros of the function 𝜃3(𝜉,𝑞). As we are interested in the physical relevance of the obtained biperiodic solution, we can suitably choose the free parameters 𝑘, 𝑞, 𝛿, so that we can achieve real biperiodic analogues of the solution and at the same time avoid the twofold poles. For this purpose, let us assume that 𝜏=𝑖𝑠,𝑠>0,thatis,𝑞=𝑒𝜋𝑠<1.(4.1)

Under these conditions, to avoid the poles of the solution 𝑢(𝑥,𝑡) at the points 𝜉𝑚𝑛, it is sufficient to limit the phase variable within the horizontal strip 𝜋𝑠<Im(𝜉)<𝜋𝑠,(4.2) which is actually the analicity condition for the solution. To provide real biperiodic solutions from (3.17), we will consider two possible options for the wave number 𝑘, under the condition of (3.19).(i)The wave number 𝑘 is real. Under this assumption and provided the condition of (4.1) holds, the phase variable 𝜉=𝑘𝑥+𝜔𝑡+𝛿 is real for real phase shift 𝛿. The logarithmic derivative from (3.17) in this case can be expressed in a Fourier series [14] by the equality 𝜃3(𝜉,𝑞)𝜃3(𝜉,𝑞)=4𝑚=(1)𝑚𝑞𝑚1𝑞2𝑚sin(2𝑚𝜉).(4.3) Thus, from (3.17), we obtain a well-defined, real function which is periodical on the real straight line 𝑢(𝑥,𝑡)=𝑘2𝑚=𝑘2𝑎𝑚(𝑞)+48𝜀𝑚(1)𝑚.cosc(𝑚𝜋𝑠)cos(2𝑚𝜉)(4.4) Figure 2 illustrates the forms of the real periodic solution for a real value of the wave number 𝑘, (𝑘=1). It can be seen from the last formula that the spatial displacements 𝑎𝑚(𝑞) contribute to each separate harmonic of the biperiodic solution.(ii)The wave number 𝑘 is imaginary. Let 𝑘𝑖𝑘, and without loss of generality, we can assume that 𝑘>0. Under this hypothesis, it is possible to generate real periodic solutions if the phase velocity is an imaginary number and the phase shift is properly chosen. It is obvious from the dispersion equality (3.12) that the phase velocity 𝜔 is an imaginary number (since 𝜔(𝑘) contains only odd powers of 𝑘). For the phase shift 𝛿, we choose 𝛿𝛿+𝜋𝜏, and hence 𝑖𝜉𝑖𝜉+𝜋𝜏. With this transition, the dispersion relation (3.12) remains the same. Applying the quasiperiodic property for 𝜃3, 𝜃2(𝑖𝜉,𝑞)=𝑞1/4𝑒𝑖𝜉𝜃3𝑖𝜉+𝜋𝜏2,,𝑞(4.5) and from formula (3.17), we have for the logarithmic derivative (see [2]) 𝑑2𝑑𝜉2ln𝜃2(𝑖𝜉,𝑞)=𝑚=sec2[].𝑖(𝑖𝜉𝑖𝑚𝜋𝑠)(4.6) In this case, we also obtain a real periodic function which is well defined in the strip (4.2): 𝑢(𝑥,𝑡)=𝑚=𝑘4𝑎𝑚(𝑞)+12𝜀𝑘2sec2(.𝜉𝑚𝜋𝑠)(4.7)

Figure 2: Real sinusoidal periodic solutions for real values of the wave number: 𝑘=1/2,1 and 𝑞=𝑒𝜋, 𝜀=1.

Figure 3 visualizes the periodic solitary-wave profiles of the sec2 type for imaginary values of the wave number 𝑘𝑖,𝑖/2,𝑖/4, and 𝑞=𝑒𝜋. Despite their structural differences, the two types of real biperiodic solutions of the Kawahara equation obtained in (4.4) and (4.7) are dynamically equivalent. This comes to confirm the nonlinear superposition principle established by Toda [15] for the KdV evolution equation. In conclusion, let us note that the biperiodic real solutions (4.4) and (4.7) actually describe the dynamics of dispersing waves resulting from the dispersion relation (3.12). Indeed, 𝜔(𝑘)0, which means that these waves are propagating in two directions, as periodically repeating groups traveling with group velocity 𝑉=𝜔(𝑘).

Figure 3: Periodic solitary-wave profiles of sec2 type for imaginary values of the wave number: 𝑘=𝑖,𝑖/2,𝑖/4 and 𝑞=𝑒𝜋.

5. Conclusions

Regardless of the purely technical difficulties arising in the application of the bilinear transformation method, it turns out to be a versatile theoretical instrument in analyzing nonintegrable, nonlinear partial differential equations, such as the Kawahara equation (1.1). The major intricacy arises from the residual equation after the bilinear reduction of the initial equation. Harmonization of the bilinear and the residual equation practically means to satisfy an algebraic system with infinite number of equations, but with a finite number of unknowns (usually one or two). The solution of a similar infinite system is feasible if a suitable unknown quantity is presented in the form of a convergent numerical or functional series with unknown terms—a procedure often applied in the Fourier method for linear partial differential equations. What is most surprising here is that the terms of these numerical or functional series have unambiguous physical interpretation. They describe spatial displacements individually for each harmonic. For this reason, we called this modification of the bilinear-transformation method, as well in [10], “spatial.” Note that for some nonintegrable nonlinear evolution equations, such as the Kuramoto-Sivashinsky equation 𝑢𝑡+𝑢𝑢𝑥+𝛼1𝑢𝑥𝑥+𝛼2𝑢𝑥𝑥𝑥+𝛼3𝑢𝑥𝑥𝑥𝑥=0,(5.1) spatial displacements are absent. In this case, their role is taken by the “wave” displacements, that is, the wave number is presented as a convergent infinite series.


A. Logarithmic Derivatives Expressed by the Hirota’s Bilinear Differential Operators 𝐷𝑡, 𝐷𝑥

Consider the following: (ln𝜁)𝑥𝑥=𝐷2𝑥𝜁𝜁2𝜁2;(ln𝜁)𝑡𝑥=𝐷𝑡𝐷𝑥𝜁𝜁2𝜁2,(ln𝜁)𝑥𝑥𝑥𝑥=𝐷4𝑥𝜁𝜁2𝜁2𝐷62𝑥𝜁𝜁2𝜁22,(ln𝜁)𝑥𝑥𝑥𝑥𝑥𝑥=𝐷6𝑥𝜁𝜁2𝜁2𝐷302𝑥𝜁𝜁2𝜁2𝐷4𝑥𝜁𝜁2𝜁2𝐷+1202𝑥𝜁𝜁2𝜁23.(A.1)

B. Identities for the Jacobi 𝜃-Functions

Consider the following: 𝑛=𝑞2𝑛2=𝜃3=𝜃30,𝑞2,𝑛=𝑞𝑛2+(𝑛1)2=𝑞1/2𝜃2=𝑞1/2𝜃20,𝑞2;𝑛=𝑛2𝑞2𝑛2=𝑞𝜃32,𝑛=(2𝑛1)2𝑞𝑛2+(𝑛1)2=2𝑞3/2𝜃2;𝑛=𝑛4𝑞2𝑛2=𝑞𝜃3+𝑞𝜃34,𝑛=(2𝑛1)4𝑞𝑛2+(𝑛1)2=4𝑞3/2𝜃2+𝑞𝜃2;𝑛=𝑛6𝑞2𝑛2=𝑞𝜃3+3𝑞𝜃3+𝑞2𝜃38,𝑛=(2𝑛1)6𝑞𝑛2+(𝑛1)2=8𝑞3/2𝜃2+3𝑞𝜃2+𝑞2𝜃2.(B.1)


The authors would like to thank the Academic Editors for the useful remarks that helped in improving the presentation of the paper.


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