Exact Solutions of Space Dependent Korteweg–de Vries Equation by The Extended Unified Method Hamdy I. ABDEL-GAWAD  , Nasser S. ELAZAB y , and Mohamed OSMAN z Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt (Received November 17, 2012; accepted February 7, 2013; published online March 15, 2013) Recently the unified method for finding traveling wave solutions of nonlinear evolution equations was proposed by one of the authors (HIAG). It was shown that, this method unifies all the methods being used to find these solutions. In this paper, we extend this method to find a class of formal exact solutions to Korteweg–de Vries equation with space dependent coefficients. KEYWORDS: exact solution, extended unified method, Korteweg–de Vries equation, variable coefficients 1. Introduction We consider the following evolution equation f x; t; u; @u @t ; @u @x ; @ 2 u @x@t ; @ 2 u @x 2 ; ... ; @ m u @x m   ¼ 0; m > 1; ð1Þ where f is a polynomial in its arguments. With relevance to the Korteweg–de Vries (KdV) equation we write H x; t; u; @u @x ; ...    @u @t þ f 0 ðx; tÞ @ m u @x m þ f 1 x; t; u; @u @x   ¼ 0: ð2Þ When Eq. (1) does not depend explicitly on x and t, it can be reduced to a subclass of ordinary differential equations by using the Lie groups for partial differential equations 1) or by using similarity transformations. Among them, the equation for traveling waves is gðu; u 0 ;u 00 ; ... u ðmÞ Þ¼ 0; u 0 ¼ du dz ; z ¼ x  ct; ð3Þ which results due to the translation symmetry of (1). The Painleve’ analysis was used to testing the integr- ability of partial differential equations, which has been developed in Ref. 2 The exact solutions of (2) for completely or partially integrable were dealt with them by the auto-Ba ¨cklund transformation. This was done by truncating Painleve’ expansion. 3–9) Recently the auto- Ba ¨cklund transformation that was extrapolated in Refs. 10–14 and the homogeneous balance method in Refs. 15–19 were used to find solutions for evolution equations with variable coefficients in the form uðx; tÞ¼ @ m2 @x m2 ðaðÞ x Þþ u ð0Þ ðx; tÞ; where  is the base function. 2. The Extended Unified Method Explicit solutions of Eq. (2) are, in fact, particular solutions. In this respect, these solutions can be mapped to polynomial or rational solutions through an ‘‘auxiliary’’ function and with appropriate auxiliary equations. The later equations may be solved to elementary of elliptic functions. In Ref. 20 the notion of unified method was implemented and detailed necessary conditions for the existence of polynomial or rational solutions were established. In the present paper, we extend this method to handle the evolution equations with variable coefficients of the type (2). 2.1 Polynomial solutions Here, we assume that, we search for solutions of (2) which are in C s ðR  R þ Þ (the class of continuously partially differentiable functions up to order s), and we define the set of functions S ¼f : R  R þ ! K  R; q t ¼ P t k ðÞ; ð x Þ p ¼ P x k ðÞg; P t k ðÞ¼ X k i¼0 b i ðx; tÞ i ðx; tÞ; P x k ðÞ¼ X k i¼0 c i ðx; tÞ i ðx; tÞ: ð4Þ Indeed the set S contains elementary or elliptic functions for some particular values of q, p and k when p  q ¼ 1 or p ¼ q  2 respectively. We mention here that when p ¼ q ¼ 1, the set S is closed under addition and multi- plication by a real number. Here, we shall confine ourselves to the case p ¼ q ¼ 1. In the present case, the mapping method asserts that there exist a positive integer n and a mapping M : C s ðR  R þ Þ! ;  ¼ v; v ¼ X s 0 i¼0 a i ðx; tÞ i ; 2 S;s 0  s ( ) such that MðuÞ¼ P n ðÞ, n<s 0 <s which satisfies the properties Mð 1 u 1 þ  2 u 2 Þ¼  1 Mðu 1 Þþ  2 Mðu 2 Þ; Mðu 1 u 2 Þ¼ Mðu 1 ÞMðu 2 Þ; Mðu t Þ¼ðMðuÞÞ t ; Mðu x Þ¼ðMðuÞÞ x : Thus M is a ring homomorphism that conserves differentia- tion. By using (4) we find that, Mðu t Þ¼ P t ðn1þkÞ ðÞ2 ;Mðu x Þ¼ P x ðn1þkÞ ðÞ2 : By using the properties of M and the last results and as H  f ðx; t; u; u t ; ...Þ is a polynomial in its arguments, we find that MðHÞ is a polynomial. So that there exists s 0  s such that MðHÞ¼ P s 0 ðÞ2 . It is worthy to notice that all these polynomials have different coefficients. More simply the mapping M assigns to u and H with two auxiliary equations, the polynomials P n ðÞ and P s 0 ðÞ respectively. In the case of Eq. (2) s 0 ¼ n  m þ mk. The utility of the above presentation is that it helps us to give Journal of the Physical Society of Japan 82 (2013) 044004 044004-1 FULL PAPERS #2013 The Physical Society of Japan http://dx.doi.org/10.7566/JPSJ.82.044004