Painleve ´ analysis and new analytical solutions for compound KdV–Burgers equation with variable coefficients A.M. Abourabia, K.M. Hassan, and E.S. Selima Abstract: We consider the solutions of the compound Korteweg–de Vries (KdV)–Burgers equation with variable coeffi- cients (vccKdV–B) that describe the propagation of undulant bores in shallow water with certain dissipative effects. The Weiss–Tabor–Carnevale (WTC)–Kruskal algorithm is applied to study the integrability of the vccKdV–B equation. We found that the vccKdV–B equation is not Painleve ´ integrable unless the variable coefficients satisfy certain constraints. We used the outcome of the truncated Painleve ´ expansion to construct the Ba ¨cklund transformation, and three families of new analytical solutions for the vccKdV –B equation are obtained. The dispersion relation and its characteristics are illus- trated. The stability for the vccKdV–B equation is analyzed by using the phase portrait method. PACS Nos: 47.35.Bb, 47.35.Fg, 91.30.Fn Re ´sume ´: Nous e ´tudions les solutions de l’e ´quation compose ´e KdV–Burgers avec coefficients variables (vccKdV–B) qui de ´crit la propagation de mascarets ondulants dans l’eau de faible profondeur en pre ´sence de certains effets dissipatifs. Nous utilisons l’algorithme Weiss–Tabor–Carnevale (WTC)–Kruskal de fac ¸on a ` ve ´rifier l’inte ´grabilite ´ de l’e ´quation vccKdV–B. Nous trouvons que l’e ´quation vccKdV–B n’est pas inte ´grable au sens de Painleve ´, a ` moins que les coefficients variables satisfassent certaines contraintes. Nous utilisons l’expansion tronque ´e de Painleve ´ pour construire la transforma- tion de Ba ¨cklund et nous obtenons trois familles de nouvelles solutions analytiques de l’e ´quation vccKdV–B. Nous de ´cri- vons la relation de dispersion et ses caracte ´ristiques et e ´tudions la stabilite ´ de l’e ´quation vccKdV–B par la me ´thode du portrait de phase. [Traduit par la Re ´daction] 1. Introduction Integrable nonlinear partial differential equations (PDEs) have remarkable properties such as infinitely many general- ized symmetries, infinitely many conservation laws, the Painleve ´ property, the Ba ¨cklund transformation, the bilinear form, etc. Integrable equations model physically interesting wave phenomena in population and molecular dynamics, nonlinear networks, chemical reactions, and waves in mate- rial science. By investigating the integrability of a nonlinear PDE, one gains important insight into the structure of the equation and the nature of its solutions [1–3]. In the last few decades, a group of scientists such as phys- icists, engineers, and applied mathematicians have been at- tracted to two contrasting themes: first, the theory of dynamical systems popularly associated with the study of chaos. Second, the theory of integrable (or nonintegrable) systems associated, among other things, with the study of solitary waves [1–5]. The compound Korteweg–de Vries (KdV)–Burgers type equation with nonlinear terms of any order reads [6] u t þ au p u x þ bu 2p u x þ cu xx þ du xxx ¼ 0 ð1Þ where a, b, c, d are constants, c = 0 and p ‡ 1. This equa- tion is considered a combination of many equations such as KdV, mKdV, Burgers, and KdV–Burgers equations, invol- ving nonlinear dispersion and dissipation effects. We see that, if a = a(t), b =–b(t), c = c(t), d = d(t) are arbitrary functions of t and p = 1, (1) will be converted to the following equation: u t þ aðtÞuu x  bðtÞu 2 u x þ cðtÞu xx þ dðtÞu xxx ¼ 0 ð2Þ which is called the compound KdV–Burgers equation with time-dependent coefficients and it has many important appli- cations. In ref. 7, Lu et al. reduced (2) to a new simplified form based on the homogeneous balance principle, and in ref. 8, Han and Xie studied the same problem in a white- noise environment. This equation was studied by many authors in the case of constant coefficients (see refs. 9–12). The first term in (2) describes the time evolution of the wave propagating in one direction. The second and third terms are nonlinear convective terms with different orders, which account for the steepening of the wave. The fourth is the so-called viscous dissipative term. The last linear disper- sive term describes the spreading of the wave. The traveling wave keeps its appearance all through its propagation, the balance is required between the nonlinear steepening of the water wave and the dispersive term. Received 31 October 2009. Accepted 21 December 2009. Published on the NRC Research Press Web site at cjp.nrc.ca on 9 April 2010. A.M. Abourabia, 1 K.M. Hassan, and E.S. Selima. Department of Mathematics, Faculty of Science, Menoufiya University, Shebin El-koom 32511, Egypt. 1 Corresponding author (e-mail: am_abourabia@yahoo.com). 211 Can. J. Phys. 88: 211–221 (2010) doi:10.1139/P10-003 Published by NRC Research Press