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Advances in Mathematical Physics
Volume 2010 (2010), Article ID 329586, 5 pages
Higher-Order Equations of the KdV Type are Integrable
Department of Civil Engineering, Technological & Educational Institute of Patras, 1 M. Alexandrou Street, Koukouli, 263 34 Patras, Greece
Received 5 January 2010; Accepted 27 January 2010
Academic Editor: Jan Hesthaven
Copyright © 2010 V. Marinakis. 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.
We show that a nonlinear equation that represents third-order approximation of long wavelength, small amplitude waves of inviscid and incompressible fluids is integrable for a particular choice of its parameters, since in this case it is equivalent with an integrable equation which has recently appeared in the literature. We also discuss the integrability of both second- and third-order approximations of additional cases.
The one-dimensional motion of solitary waves of inviscid and incompressible fluids has been the subject of research for more than a century . Probably, one of the most important results in the above study was the derivation of the famous Korteweg-de Vries (KdV) equation 
At first, the equation was difficult to be examined due to the nonlinearity. The first important step was made with the numerical discovery of soliton solutions by Zabusky and Kruskal . Soon thereafter great progress was made by the discovery of the Lax Pair representation  and the Inverse Scattering Transform . The laters results led to a new notion of integrability. More specifically, according to this notion, a partial differential equation is said to be “completely integrable” if it is linearizable through a Lax Pair; thus, it is solvable via the Inverse Scattering Transform (see, e.g., ).
Equation (1.1) represents a first-order approximation in the study of long wavelength, small amplitude waves of inviscid and incompressible fluids. Allowing the appearance of higher-order terms in and , one can obtain more complicated equations. Two such equations, including second- and third-order terms, were proposed in [7, 8] and have, respectively, the forms
Equation (1.2) was first examined both analytically and numerically in . The violation of the Painlevé property in many cases, together with a numerical study of the reduction in the complex -plane, gave strong indications that, in general, this equation is not integrable. Consequently, (1.2) was examined in [10, 11] and it was found that it possesses solitary wave solutions, which, for small values of the parameters and , behave like solitons. New wave solutions of both (1.2) and (1.3) were also examined numerically in  and were also found to behave like solitons.
Equation (1.2) was further examined in [13–20], while (1.3) was examined in [18, 21, 22]. Although an enormous amount of new solutions was presented, no progress has been made regarding the integrability of these equations.
In this paper we show that, for arbitrary and
equation (1.3) is equivalent to an integrable equation recently proposed by Qiao and Liu . We, thus, reveal an integrable case of (1.3) itself. We also discuss the existence of additional integrable cases for both (1.2) and (1.3).
2. An Integrable Case of (1.3)
Recently, Qiao and Liu  proposed a new integrable equation, namely,
The integrability follows directly from the fact that the equation admits a Lax Pair; thus, as mentioned above, is solvable by the Inverse Scattering Transform.
thus, (2.1) becomes
We then set
where are constants; substitute in (1.3), divide by , and write instead of . We thus obtain
Since , relations , , and imply, respectively, that
while relations and imply that
Then, relations , , , and yield, respectively,
We, thus, conclude with relations (1.4), while remains arbitrary.
In  it was shown that (1.2) does not pass the classical Painlevé test [24, 25] for any combination of the parameters, except of course for the cases and in which it reduces to KdV and modified KdV, respectively. However, as also stated in , there are infinitely many cases for which the equation has only algebraic singularities, that is, it admits the so-called weak-Painlevé property . Although this property does not constitute a strong indication for integrability, there are still many integrable equations admitting only algebraic singularities (see, e.g., ).
On the other hand, at least to our knowledge, no results regarding integrable cases of (1.3) have appeared in the bibliography before. Equation (1.3) is highly nonlinear and most probably it is not integrable in general. However, as shown in the previous section, there is at least one nontrivial combination of the parameters, for which it is completely integrable.
Based on the above statements, we believe that it would be interesting to study whether there are any integrable cases for (1.2) or any additional integrable cases for (1.3). We hope to present results in this direction in a future publication.
- G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1974.
- D. J. Korteweg and G. de Vries, “On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary Waves,” Philosophical Magazine, vol. 39, pp. 422–443, 1895.
- N. J. Zabusky and M. D. Kruskal, “Interaction of “solitons” in a collisionless plasma and the recurrence of initial states,” Physical Review Letters, vol. 15, no. 6, pp. 240–243, 1965.
- P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Communications on Pure and Applied Mathematics, vol. 21, no. 5, pp. 467–490, 1968.
- C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg—de Vries equation,” Physical Review Letters, vol. 19, no. 19, pp. 1095–1097, 1967.
- M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, vol. 4 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1981.
- A. S. Fokas, “On a class of physically important integrable equations,” Physica D, vol. 87, no. 1–4, pp. 145–150, 1995.
- A. S. Fokas and Q. M. Liu, “Asymptotic integrability of water waves,” Physical Review Letters, vol. 77, no. 12, pp. 2347–2351, 1996.
- V. Marinakis and T. C. Bountis, “On the integrability of a new class of water wave equations,” in Proceedings of the Conference on Nonlinear Coherent Structures in Physics and Biology, D. B. Duncan and J. C. Eilbeck, Eds., Heriot-Watt University, Edinburgh, UK, July 1995.
- V. Marinakis and T. C. Bountis, “Special solutions of a new class of water wave equations,” Communications in Applied Analysis, vol. 4, no. 3, pp. 433–445, 2000.
- E. Tzirtzilakis, M. Xenos, V. Marinakis, and T. C. Bountis, “Interactions and stability of solitary waves in shallow water,” Chaos, Solitons & Fractals, vol. 14, no. 1, pp. 87–95, 2002.
- E. Tzirtzilakis, V. Marinakis, C. Apokis, and T. Bountis, “Soliton-like solutions of higher order wave equations of the Korteweg—de Vries type,” Journal of Mathematical Physics, vol. 43, no. 12, pp. 6151–6165, 2002.
- S. A. Khuri, “Soliton and periodic solutions for higher order wave equations of KdV type. I,” Chaos, Solitons & Fractals, vol. 26, no. 1, pp. 25–32, 2005.
- W.-P. Hong, “Dynamics of solitary-waves in the higher order Korteweg—de Vries equation type (I),” Zeitschrift für Naturforschung, vol. 60, no. 11-12, pp. 757–767, 2005.
- J. Li, W. Rui, Y. Long, and B. He, “Travelling wave solutions for higher-order wave equations of KdV type. III,” Mathematical Biosciences and Engineering, vol. 3, no. 1, pp. 125–135, 2006.
- Y. Long, J. Li, W. Rui, and B. He, “Traveling wave solutions for a second order wave equation of KdV type,” Applied Mathematics and Mechanics, vol. 28, no. 11, pp. 1455–1465, 2007.
- V. Marinakis, “New solutions of a higher order wave equation of the KdV type,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 4, pp. 519–525, 2007.
- V. Marinakis, “New solitary wave solutions in higher-order wave equations of the Korteweg—de Vries type,” Zeitschrift für Naturforschung, vol. 62, no. 5-6, pp. 227–230, 2007.
- J. Li, “Exact explicit peakon and periodic cusp wave solutions for several nonlinear wave equations,” Journal of Dynamics and Differential Equations, vol. 20, no. 4, pp. 909–922, 2008.
- W. Rui, Y. Long, and B. He, “Some new travelling wave solutions with singular or nonsingular character for the higher order wave equation of KdV type (III),” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 11, pp. 3816–3828, 2009.
- J. Li, J. Wu, and H. Zhu, “Traveling waves for an integrable higher order KdV type wave equations,” International Journal of Bifurcation and Chaos, vol. 16, no. 8, pp. 2235–2260, 2006.
- J. Li, “Dynamical understanding of loop soliton solution for several nonlinear wave equations,” Journal Science in China Series A, vol. 50, no. 6, pp. 773–785, 2007.
- Z. Qiao and L. Liu, “A new integrable equation with no smooth solitons,” Chaos, Solitons & Fractals, vol. 41, no. 2, pp. 587–593, 2009.
- J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522–526, 1983.
- J. Weiss, “The Painlevé property for partial differential equations. II. Bäcklund transformation, Lax pairs, and the Schwarzian derivative,” Journal of Mathematical Physics, vol. 24, no. 6, pp. 1405–1413, 1983.
- A. Ramani, B. Dorizzi, and B. Grammaticos, “Painlevé conjecture revisited,” Physical Review Letters, vol. 49, no. 21, pp. 1539–1541, 1982.
- C. Gilson and A. Pickering, “Factorization and Painlevé analysis of a class of nonlinear third-order partial differential equations,” Journal of Physics A, vol. 28, no. 10, pp. 2871–2888, 1995.