Two Kinds Of Methods For Finding The Exact Solutions Of Nonlinear Partial Differential Equations

A lot of nonlinear phenomena of modern science andtechnology are mostly described by nonlinear partial differentialequations (NPDEs), it is great significance for finding the exactsolutions of NPDEs in theory and application.In recent years, the research of exact solutions to the NPDEs inmathematical physics has attracted much attention by many authors,and up to now a number of powerful methods to construct the exactsolutions of NPDEs have been found and developed: for example,the inverse scattering theory, Hirota's bilinear method,Painlevé analysis method, Backlund transformations,Darboux transformations, the homogeneous balancemethod, the hyperbolic function expansion method[,Jacobi elliptic function expansion method, F-expansionmethod, etc. From the way of solving problems, these methods can bedivided into two kinds. One is based on nonlinear transformation,such as Hirota's bilinear methods, Painlevé analysis method,B?cklund transformations, Darboux transformations, etc. Byperforming nonlinear transformation to the equation, the originalequation is turned into an equation which is simple and easy tosolve.The other kind is on the basis of the priori assumption of thesolution. This kind of methods is often used to find the travelingwaves solution of the mathematical physical equation. The typicalmethods are the hyperbolic function expansion method, Jacobielliptic function expansion method, F-expansion method, which isproduced by the application of homogeneous balanced principledirectly. In these methods, the solution of NPDEs is supposed to beexpressed by combination of some special functions, these specialfunctions satisfied some classical auxiliary ordinary differentialequations, the exact solutions of NPDEs can be obtained by solvingthe auxiliary ordinary differential equations. Suppose choosingappropriate auxiliary equation, different kinds of periodic wavesolutions and solitary wave solutions of NPDEs can be obtained.This article mainly studies two kinds of methods to constructthe exact solutions of NPDEs based on the homogeneous balanceprinciple proposed by professor Wang Mingliang.1.Homogeneous Balance MethodGiven a partial differential equation (PDE), say, in two variables,P (u , u x , ut , u xx , u xt , ut t, ) = 0, (1)in which both nonlinear terms and higher order derivatives ofu ( x , t ) are all involved. In general, the left-hand side of Eq. (1) is apolynomial in u and its various derivatives.A function ? ( x , t) is called a quasi-solution of Eq. (1), ifthere exists a function f = f (? ) of one variable only so that asuitable linear combination,[ ] [ ]u ( x , t ) = ??m +x n m f ?(t ? n )+Φ f (? ) = f ( m +n )(? )? xm ? tn+Ψ?( x , t), (2)where Φ [ f (? )] is all partial derivative terms with lower than(m + n) order of f (? ), Ψ [? ( x , t)] is all terms with lower than(m + n) degree in various partial derivatives of ? ( x , t), is actually asolution of Eq. (1), where m ≥ 0, n ≥ 0, f (? ) and ? ( x , t) areintegers to be determined. The homogeneous balance method forsolving Eq. (1) proceeds in the following four steps:First step: according to the idea of homogeneous balanceprinciple, considering the balance in part between the highest orderpartial derivative term and the nonlinear term in Eq.(1), so m andn are determined.Second step: substituting the linear combination chosen in thefirst step into Eq.(1), collecting all terms with the highest degree ofderivatives of ? ( x , t) and setting its coefficient to zero, we obtainan ordinary differential equation for f (? ) and then solve it, inmost cases f (? ) is a logarithm function.Third step: starting from the ODE and its solution obtainedabove, the nonlinear terms of various derivatives of f (? ) in theexpression obtained in the second step can be replaced by thecorresponding higher order derivatives of f (? ). After doing this,collecting all terms with the same order derivatives of f (? ) andsetting the coefficient of each order derivative of f (? ) to zerorespectively, we obtain a set of PDEs for ? ( x , t). We can choosecoefficients of the linear combination properly so that the solutionsof the PDEs can be found.Fourth step: substituting f (? ) and ? ( x , t) as well as someconstants obtained in the second and third steps into thecombination chosen in the first step, after doing some calculations,we then obtain an exact solution of Eq.(1).Consider a Generalized Fisher equation with variablecoefficientsu t ? α (t )u xx? β( t )(u 3? u) = 0. (3)An auto-B?cklund Transformation (auto-BT) to the Eq. (3) isderived if the coefficients of the equation are linearly dependent,u ( x , t ) = ?2 a ??x+ V ( x , t), (4)? ?2 a? t + 3 ?2 aa β? xx + 6 a β V?x= 0, (5)? xt ? a β? xxx ? 3 β V2? x + β?x= 0. (6)The meaning is: If V = V ( x , t) is one solution of Eq. (3), thenu ( x , t ) is the solution of Eq. (3) when ? = ?( x , t) satisfy Eqs. (5)and (6).In terms of the auto-BT, several solitary solutions to Eq. (3)are obtained in the condition of V = 0, V = ±1 .2.Extended F-expansion MethodF-expansion method has been proposed recently, which can bethought of as any kind of Jacobi elliptic function expansion methodand hence an over-all generalization of it. The extendedF-expansion method has been proposed and researched in thisarticle.The extended F-expansion method for solving Eq. (1)proceeds in the following four steps:Step 1. Seek traveling wave solution of Eq. (1) by takingu ( x , t ) = u (ξ ), ξ = kx + ω t+ ξ0, (7)where k ,ω are constants to be determined, ξ 0 is an arbitrary realconstant. Substituting (7) into Eq. (1) yields an ordinarydifferential equation (ODE) for u (ξ ),P (u , ku ′, ku ′, k 2 u ′′, ω 2 u ′′, kω u ′′, k 3u′′′, ) = 0. (8)11( ) N n n ( ) Nnn( ) ( )n N nu ξ a F ξ b F ?ξ Gξ=? == ∑ +∑ , a N≠ 0, (9)where an , bn are constants to be determined, F (ξ ) and G (ξ )satisfy first order ODEF ′ 2 (ξ ) = P1 F 4 (ξ ) + Q1 F 2(ξ )+ R1, ( F ′′= 2P1 F 3+ Q1F), (10)2 4 2G ′ (ξ ) = P2 G (ξ ) + Q2 G (ξ )+ R2, ( G ′′ = 2P2 G 3+ Q2 G), (11)respectively. Both ODE (10) and (11) are all included in AppendixA provided Pi , Qi , Ri (i = 1,2) are selected such that the solutions ofODE (10) and (11) are different Jacobi elliptic functions satisfyingthe relation (see Appendix B)G 2 = μ F2+ ν, (12)μ ,ν are constants, Integer N can be determined by consideringthe homogeneous balance between governing nonlinear terms andthe highest order derivatives of u in Eq. (8). Substituting (9) intoODE (8), and using ODE (10), (11) as well as (12), the left-handside of ODE (8) can be converted into a polynomial inF ? 1(ξ ), F (ξ ) and G (ξ ). Setting the coefficients of the polynomialto zero yields a set of algebraic equations for ai ,b j( i = ? N , ? N + 1, ,0,1, ,N, j = 1,2, ,N), and ω (possiblyk ).Step 3. With the aid of Mathematica or Maple, solving thealgebraic equations obtained in Step 2, then ai ,b j ( i = ? N , ? N+ 1,,0,1, , N , j = 1,2, ,N) and ω (possibly k ) can be expressedby Pi , Qi , Ri (i = 1,2), μ and ν . Substituting these results intoF-expansion (9), we can obtain a general form of traveling wavesolution to Eq. (1).Step 4. With the aid of Appendices A and B, from the generalform of traveling wave solutions, many periodic wave solutionsexpressed by various Jacobi elliptic functions can simultaneouslybe obtained. Under the limit cases when the modulus m → 1 andm → 0, the solitary wave solutions and trigonometric functionsolutions are also obtained.We studied the following NPDEs by applying the extendedF-expansion method.Sinh-Laplace equation[27]u xx + u yy= sinhu,Variant Boussinesq equations[29]( ) 00t x xxxt x xH Hu uu H uu??? ++ + + ==BBM equation[35]ut + u x + uu x ? uxxt= 0,(2+1) dimension BBM equationut + u x + u y + uu x + uu y ? u xxt ? uyyt= 0,Nonlinear Klein-Gordon equation[40]-[41]ut t ? u xx+ f ′(u ) = 0,more precisely,set f ′( u )= au + bu3,Nonlinear Klein-Gordonequation becomesu tt ? u xx+ au + bu3 = 0,(it contains Φ 4 equation,Approximate Sine-Gordon equation,Landau-Ginzburg-Higgs equation).The coupled KdV equations[47]0( ) 0t x x xxxt x xu pvv quu ruv s uv hvv??? + + + + + ==.An abundance of exact traveling wave solutions of theseNPDEs which includes doubly periodic wave solutions expressedby Jacobi elliptic functions, single periodic wave solutions andsolitary wave solutions are obtained by using the extendedF-expansion method.