Travelling wave solutions for the generalized Burgers–Huxley equation

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Trav the first-integral method which is based on the ring theory of commutative algebra, has been proposed and developed to of va ution than many and p the t uðx; 0Þ ¼ ½c2 þ c2 tanhðrcxÞ� 1 n; ð2Þ and derived that the exact solution to (2) was given by uðx; tÞ ¼ c 2 þ c 2 tanh rc x� ca 1þ n� ð1þ n� cÞðq� aÞ 2ð1þ nÞ � � t � �� �� �1 n ð3Þ 0096-3003/$ - see front matter � 2008 Elsevier Inc. All rights reserved. E-mail address: [email protected] Applied Mathematics and Computation 204 (2008) 733–737 Contents lists available at ScienceDirect Applied Mathematics and Computation doi:10.1016/j.amc.2008.07.020 ut þ aunux � uxx ¼ buð1� unÞðun � cÞ: ð1Þ where a; b;n P 0 and c can be either sign. Eq. (1) can be regarded as a model to describe the interaction between reaction mechanisms, convection effects and diffusion transports; see [25]. This equation also includes as particular case several known evolution equations: when n ¼ 1; b ¼ 0, it reduces to the Burgers equation; when n ¼ 1;a ¼ 0, it reduces to the Fitz- Hugh–Nagumo equation [26,27]; when n ¼ 1;a ¼ 0; c ¼ �1, it is the Newwell–Whitehead equation [28]; Wang et al. [29] considered Eq. (1) with the Cauchy condition, i.e., ut þ aunux � uxx ¼ buð1� unÞðun � cÞ; 0 6 x 6 1; t P 0; ( study the travelling wave solutions the exact travelling solitary wave sol ing the first-integral method, rather suffering so many difficulties. It has complicated and tedious calculation In the present work, we consider rious nonlinear evolution equations [21–24]. Unlike the traditional techniques, s for some nonlinear evolution equations can be solved easily and quickly by apply- the traditional methods for exact solutions as well as numerical solutions, without advantages over other traditional techniques, and it mainly avoids a great deal of rovides more exact and explicit travelling solitary solutions with a high accuracy. ravelling wave solutions of the following generalized Burgers–Huxley equation a r t i c l e i n f o Keywords: Solitary wave The first-integral method The ring theory Burgers–Huxley equation a b s t r a c t In this paper, travelling wave solutions for the generalized Burgers–Huxley equation are studied. By using the first-integral method, which is based on the ring theory of commuta- tive algebra, we obtain a class of travelling solitary wave solutions for the generalized Bur- gers–Huxley equation. A minor error in the literature is clarified. � 2008 Elsevier Inc. All rights reserved. 1. Introduction As is well known that investigating the exact travelling wave solutions to nonlinear evolution equations plays an impor- tant role in the study of nonlinear physical phenomena. In order to obtain the exact solutions, a number of methods have been proposed, such as the homogeneous balance method [1], the hyperbolic tangent expansion method [2–4], the Jacobi elliptic function expansion method [5–9], F-expansion method [10–12], sine–cosine method [13–18], tanh function method [19,20] and so on. A feature common to all the above methods is that when solving the solutions of nonlinear evolution equa- tions, they are all using the Computer Algebra systemMaple, or Mathematica to calculate. Recently, a new powerful method, Xijun Deng School of Information Science and Mathematics, Yangtze University, Jingzhou, Hubei 434023, China elling wave solutions for the generalized Burgers–Huxley equation journal homepage: www.elsevier .com/ locate/amc ring th sions first i sional throu ducib Qðx; zÞ ¼ Pðx; zÞGðx; zÞ: Hib (ii) 734 X. Deng / Applied Mathematics and Computation 204 (2008) 733–737 (iv) For a polynomial Q of k½X1;X2; . . . ; Xn� to be zero on the set of zeros in L of an ideal c of k½X1;X2; . . . ; Xn�, it is necessary and sufficient that there exists an integer m > 0 such that Qm 2 c. Now, we are applying the above Divisor Theorem to look for the first-integral of system (8). Suppose that / ¼ /ðnÞ and z ¼ zðnÞ are the nontrivial solutions to (8), and Xð/; yÞ ¼Pmi¼0aið/Þyi is an irreducible polynomial in Cð/; yÞ such that Xð/ðnÞ; yðnÞÞ ¼ Xm i¼0 aið/Þyi ¼ 0; ð9Þ where aið/Þði ¼ 1;2; . . . ; mÞ are polynomials of / and amð/Þ–0. We start our study withm ¼ 1. Note that dXdn is a polynomial of / and y, and Xð/ðnÞ; yðnÞÞ ¼ 0 implies that dXdn jð8Þ ¼ 0. According to the Divisor Theorem, there exists a polynomial Hð/; yÞ ¼ pð/Þ þ qð/Þy in Cð/; yÞ such that set of polynomials of k½X1;X2; . . . ; Xn� zero at x. n k-automorphisms s of L such that yi ¼ sðxiÞ for 1 6 i 6 n. (iii) For an ideal a of k½X1;X2; . . . ; Xn� to be maximal, it is necessary and sufficient that there exists x in Ln such that a is the Let x ¼ ðx1; x2; . . . ; xnÞ; y ¼ ðy1; y2; . . . ; ynÞ be two elements of Ln. For the set of polynomials of k½X1;X2; . . . ; Xn� zero at x to be identical with the set of polynomials of k½X1;X2; . . . ; Xn� zero at y, it is necessary and sufficient that there exists a (i) Every ideal c of k½X1;X2; . . . ;Xn� not containing 1 admits at least one zero in Ln. ert–Nullstellensatz. Let k be a field and L an algebraic closure of k. Then It follows immediately from the following theorem in commutative algebra [32]: isor Theorem. Suppose that Pðx; zÞ and Qðx; zÞ are polynomials of two variablesx and z in C½x; z� and Pðx; zÞ is irre- le in C½x; z�. If Qðx; zÞ vanishes at all zero points of Pðx; zÞ, then there exists a polynomial Gðx; zÞ in Cðx; zÞ such that easily. Next, let us recall the Divisor Theorem for two variables in the complex domain C: Div ntegrals to system (8) under various parameter conditions. Then using these first integrals, the above two-dimen- autonomous system (8) can be reduced to some different first-order integrable differential equations. Finally, gh solving these first-order differential equations directly, travelling wave solutions for Eq. (1) can be established �c//0 þ a/2/0 þ 1� n /02 � //00 ¼ bn/2ð1� /Þð/� cÞ; ð7Þ where /0 and /00 denote d/dn and d 2/ dn2 , respectively. If we let /0/ ¼ y, then Eq. (7) can be written as the two-dimensional autonomous system d/ dn ¼ /y; dy dn ¼ �cyþ a/y� 1n y2 � bnð1� /Þð/� cÞ: ( ð8Þ In order to find the travelling wave solutions of Eq. (1), we are now applying the first-integral method, the key idea of which is to utilize the so-called Divisor theorem which is based on the ring theory of commutative algebra and to obtain 1 � � 2. Exact travelling wave solutions for Eq. (1) In this section, we start out our study for Eq. (1). Firstly, making the following transformation / ¼ un; ð4Þ we can obtain an equation for / as /t þ a//x þ ð1� 1nÞð/xÞ2 / � /xx ¼ bn/ð1� /Þð/� cÞ: ð5Þ Assume Eq. (5) has the travelling wave solution as follows /ðx; tÞ ¼ /ðnÞ; n ¼ x� ct; ð6Þ where c is wave velocity. Substituting (6) into Eq. (5) yields eory of commutative algebra, we establish several travelling solitary wave solutions for Eq. (1). Finally, some conclu- are given in Section 3. where r ¼ nðq�aÞ4ð1þnÞ and q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ 4bð1þ nÞ p . Very recently, Ismail et al. [30,31] obtained the analytical solution of Eq. (1) in the series form, by applying the adomian decomposition method. The remainder of this paper is organized as follows. In Section 2, Using the first-integral method which is based on the that is and where Since c0 is in putati which Fro Solvin In this Using � �� �� �1 where X. Deng / Applied Mathematics and Computation 204 (2008) 733–737 735 2 2 4ðnþ 1Þ 2ðnþ 1Þ x0 is an arbitrary constant. uðx; tÞ ¼ 1� 1 coth nðq� aÞ x� a� qþ ða� qÞðnþ 1Þc t þ x0 n ; ð20Þ uðx; tÞ ¼ 1 2 � 1 2 tanh nðq� aÞ 4ðnþ 1Þ x� a� qþ ða� qÞðnþ 1Þc 2ðnþ 1Þ t þ x0 � �� �� �1 n ; ð19Þ Solving Eq. (18) and changing to the original variables, we obtain travelling solitary wave solutions to Eq. (1) dn 2ðnþ 1Þ d/ ¼ nða� qÞ/ð/� 1Þ: ð18Þ this first-integral, the second-order ordinary differential equation (8) reduces to y ¼ 2ðnþ 1Þ ð/� 1Þ: ð17Þ c ¼ 2ðnþ1Þ : case, (9) becomes nða� qÞ A1 ¼ 2ðnþ1Þ ; B1 ¼ a�q2 ; ða�qÞþða�qÞðnþ1Þc >>>< >>>: ð16Þ g Eq. (15), we can obtain that �nða�qÞ8 ð1þ 1nÞA1 þ a� B1 ¼ 0; �c ¼ 1n A1 � B1c: >>: ð15Þ A1B1 ¼ bn;>> comes the hy sider t erally 3. Con Re equat by app wave We be Riccat Refer [1] M.L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A 213 (1996) 79–287. [3] E.J. Parkes, B.R. Duffy, Travelling solitary wave solutions to a compound KdV-Burgers equation, Phys. Lett. A 229 (1997) 217–220. [4] E.G [5] Z.T (2 [6] E. [7] E. 736 X. Deng / Applied Mathematics and Computation 204 (2008) 733–737 463–476. . Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212–218. . Fu, S.K. Liu, S.D. Liu, Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A 290 001) 72–76. Fan, J. Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Phys. Lett. A 305 (2002) 83–392. 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It is shown that the first-integral method some special ansatze, for example, the Riccati equation [33–35] are widely used as an ansatze for various nonlinear evolution clusions cently, many researchers often study and seek for the exact solutions of nonlinear differential equations by proposing more complicated and involves the irregular singular point theory, and the elliptic integrals of the second kind and per-elliptic integrals. Some solutions in the functional form can not be expressed explicitly. One does not need to con- he cases mP 5 because it is well known that an algebraic equation with the degree greater than or equal to 5 is gen- not solvable. Case 2: a0ð/Þ ¼ A2ð/� cÞ; pð/Þ ¼ B2ð/� 1Þ, using the same arguments as the above, we have A2B2 ¼ bn; ð1þ 1nÞA2 þ a� B2 ¼ 0; �c ¼ 1n A2c� B2: 8>< >: ð21Þ Solving Eq. (21), we obtain A2 ¼ nð�a�qÞ2ðnþ1Þ ; B2 ¼ a�q2 ; c ¼ ða�qÞðnþ1Þþða�qÞc2ðnþ1Þ : 8>>< >>: ð22Þ In this case, (9) becomes y ¼ nða� qÞ 2ðnþ 1Þ ð/� cÞ: ð23Þ So, the second-order ordinary differential equation (8) reduces to d/ dn ¼ nða� qÞ 2ðnþ 1Þ /ð/� cÞ: ð24Þ Solving Eq. (24) and changing to the original variables, we obtain travelling solitary wave solutions to Eq. (1) uðx; tÞ ¼ c 2 � c 2 tanh ncðq� aÞ 4ðnþ 1Þ x� ða� qÞcþ ða� qÞðnþ 1Þ 2ðnþ 1Þ t þ x0 � �� �� �1 n ; ð25Þ uðx; tÞ ¼ c 2 � c 2 coth ncðq� aÞ 4ðnþ 1Þ x� ða� qÞcþ ða� qÞðnþ 1Þ 2ðnþ 1Þ t þ x0 � �� �� �1 n ; ð26Þ where x0 is an arbitrary constant. It is easy to see that the exact solution (3) described in [29] by using nonlinear transformations is merely identical to our (25). However, there is a minor error in (3), that is, x� ð ca1þn � ð1þn�cÞðq�aÞ2ð1þnÞ Þt should be x� ða�qÞcþðaþqÞðnþ1Þ2ðnþ1Þ t. 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