The Bäcklund Transformations and Abundant Exact Explicit Solutions for a General Nonintegrable Nonlinear Convection-Diffusion Equation

and Applied Analysis 3 From 2.1 , we have ut f ′′ ( φ ) φxφt f ′ ( φ ) φxt u1t, 2.2 ux f ′′ ( φ ) φ2 x f ′φ ) φxx u1x, 2.3 uxx f ′′′ ( φ ) φ3 x 3f ′′φ ) φxφxx f ′ ( φ ) φxxx u1xx, 2.4 u2 ( f ′ )2( φ ) φ2 x 2f ′φxu1 x, t u1 x, t , u3 ( f ′ )3( φ ) φ3 x 3 ( f ′ )2 φ2 xu1 x, t 3f φxu1 x, t u 3 1 x, t . 2.5 Substituting 2.1 – 2.5 into the left side of 1.1 and collecting all terms with φ3 x, we obtain ut − uxx αuux βu γu2 δu3 ( αf ′′f ′ − f ′′′ δf ′)3 ) φ3 x [ f ′′φxφt − 3f ′′φxφxx αf ′′φ2 xu1 α ( f ′ )2 φxφxx γ ( f ′ )2 φ2 x 3δ ( f ′ )2 φ2 xu1 x, t ] f ′ [ φxt − φxxx αφxxu1 αφxu1x βφx 2γφxu1 3δφxu1 ] [ u1t − u1xx αu1u1x βu1 γu1 δu1 ] 0. 2.6 Setting the coefficient of φ3 x in 2.6 to be zero, we obtain an ordinary differential equation for f αf ′′f ′ − f ′′′ δf ′)3 0, 2.7 which has a solution f ( φ ) λ ln ( φ ) , 2.8 where λ α ± √ α2 8δ /2δ. And then ( f ′ )2 −λ f ′′. 2.9 By virtue of 2.7 – 2.9 , 2.6 becomes ut − uxx αuux βu γu2 δu3 f ′′ [ φxφt − 3φxφxx αφ2 xu1 − αλφxφxx − γλφ2 x − 3δλφ2 xu1 x, t ] f ′ [ φxt − φxxx αφxxu1 αφxu1x βφx 2γφxu1 3δφxu1 ] [ u1t − u1xx αu1u1x βu1 γu1 δu1 ] 0. 2.10 4 Abstract and Applied Analysis Setting the coefficients of f ′′, f ′, f0 to be zero, respectively, it is easy to see from 2.10 that φt ( αu1 − γλ − 3δλu1 ) φx − 3 αλ φxx 0, 2.11 φxt − φxxx αφxxu1 αφxu1x βφx 2γφxu1 3δφxu1 0, 2.12 u1t − u1xx αu1u1x βu1 γu1 δu1 0. 2.13 Substituting 2.8 into 2.1 , we obtain a Bäcklund transformation u x, t λ φx φ u1 x, t , 2.14 where λ α± √ α2 8δ /2δ, φ, u1 satisfy 2.11 – 2.13 . Substituting a seed solution u1 x, t of 1.1 into linear equations 2.11 and 2.12 , then solving 2.11 and 2.12 , we can get a new solution of 1.1 from 2.14 . Thus we can obtain infinite solutions of 1.1 by the Bäcklund transformation 2.14 and 2.11 2.12 from a seed solution of 1.1 . Taking u1 0, by 2.11 – 2.14 , we obtain a transformation u x, t λ φx φ , 2.15 that transforms 1.1 into linear equations φt − γλφx − 3 αλ φxx 0, φt − φxx βφ E, 2.16 where λ α ± √ α2 8δ /2δ, E is an arbitrary constant. Taking u1 −γ ± √ Δ /2δ, from 2.11 – 2.14 we obtain another transformation u x, t −γ ± √ Δ 2δ λ φx φ . 2.17 Equation 1.1 can be solved by solving two linear equations φt ( αu1 − γλ − 3δλu1 ) φx − 3 αλ φxx 0, φxt − φxxx αφxx βφx 2γφxu1 3δφxu1 0, 2.18 where u1 −γ ± √ Δ /2δ, λ α ± √ α2 8δ /2δ, Δ γ2 − 4βδ. 3. Exact Explicit Solutions to 1.1 In this section we want to obtain abundant exact explicit particular solutions of 1.1 from the Bäcklund transformation 2.14 and a trivial solution of 1.1 . Abstract and Applied Analysis 5and Applied Analysis 5 Noting the homogeneous property of 2.16 we can expect that φ in 2.16 is of the form φ x, t A sinh kx ωt ξ0 B cosh kx ωt ξ0 C 3.1 with A, B, C, k, ω, and ξ0 constants to be determined. Substituting 3.1 into 2.16 , one gets a set of nonlinear algebraic equation Aω − γλAk − 3 αλ Bk2 0, Bω − γλBk − 3 αλ Ak2 0, Aω − Bk2 βB 0, Bω −Ak2 βA 0,