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whereΔisn-dimensional Laplace operator:Δ≡∑ n i=1 2 x2i,γ(>0)is diffusion coefficient. In this paper, we consider (n+1)-dimensional Chaffee-Infante equation with variable coefficients ut(t,x)-α(t)Δu(t,x)+β(t)u3(t,x)-γ(t)u(t,x) =0, x∈Rn, (1) whereα(t)β(t)≠0.Furthermore, if the problem is considered in random environment, we can get (n+1)- dimensional random Chaffee-Infante equation,which can better reflect objective phenomena, so studying exact solutions of this stochastic equation has more practical significance. In order to give the exact solu- tions for random Chaffee-Infante equation, we only consider this problem in white noise environment. We shall investigate the following (n+1)-dimensional Wick-type stochastic Chaffee-Infante equation Ut(t,x)-A(t) ΔU(t,x)+B(t) U 3(t,x)-Γ(t) U(t,x) =0, x∈Rn, (2) and give white noise functional solutions. Where is Wick product on the Kondratiev distribution space (S)-1, A(t), B(t),Γ(t)are(S)-1-valued functions withA(t) B(t)≠0. As is well known, random wave equation is an important branch of the modern probability theory. There are many authors studying them, such as Konotop and Vázquez in their bookstudied nonlinear random wave equations systematically,Wadati first introduced and studied the stochastic KdV equation and gave the diffusion of solutions of the KdV equation under Gaussian noise in ,Chen and Xie[4-5]have al-ready done a lot of work using white noise analysis method in the field of random wave equation and got some important results.As for Chaffee-Infante equations, some authors studied them through nonlinear partial differential methods,such as the homogeneous balance method, the first integral method, the direct hypothetical method, etc., and got some periodic solitary wave solutions and elliptic function solu- tionsWe shall use white noise analysis, Hermite transform and truncated expansion methodto get exact solutions and Wick versions of white noise functional solutions for Eq.(1) and (2), respectively.
Solutions of Eq.(1) and (2) Taking the Hermite transform for Eq.(2), we get the following definite equation: Ut(t,x,z)- A(t,z)∑ n i=1 2 x2i U(t,x,z)+~B(t,z) U(t,x,z)-~Γ(t,z) U(t,x,z) =0, (3) wherex=(x1,x2,…,xn)∈Rn, z=(z1,z2,…)∈CN+is a parameter. Let Ut,x,z)=u(t,x,z), A(t,z)=α(t,z),~B(t,z)=β(t,z),~Γ(t,z)=γ(t,z), thus Eq.(3) can be expressed as ut(t,x,z)-α(t,z)∑ n i=1 2 x2iu(t,x,z)+β(t,z)u3(t,x,z)-γ(t,z)u(t,x,z) =0. (4) In order to obtain the solutions of Eq.(4), we assume that the solutions have the form u(t,x,z) = u(ξ),ξ=∑ n i=1 ki(t,z)xi+l(t,z)+c0, (5) whereki(t,z)(i=1,…,n), l(t,z)are nonzero functions with respect totandz, c0is an arbitrary con- stant. Substituting Eq.(5) into Eq.(4) yields ∑ n i=1 xikit(t,z)+lt(t,z) u′(ξ)-α(t,z)∑ n i=1 k2i(t,z) u″(ξ)+β(t,z)u3(ξ)-γ(t,z)u(ξ) =0, (6) where′denotesddξ, kit(t,z)denotes ki(t,z) t. We suppose that Eq.(6) has the following formal solution u(ξ) =∑ m j=0 aj(t,z)Fj(ξ), (7) whereaj(t,z)(j=0,1,…,m)are functions to be determined later,F(ξ)satisfies F(ξ) =11+eξ. (8) Substituting (7) into Eq.(6) and balancing the highest order derivative -α(t,z)∑ n i=1 k2i(t,z) u″(ξ)and the highest nonlinear termβ(t,z)u3(ξ), we can easily findm=1,so Eq.(7) can be simplified as u(ξ) = a0(t,z)+a1(t,z)F(ξ). (9) Note that u′(ξ) = a1(t,z)F2(ξ)-a1(t,z)F(ξ), (10) u″(ξ) =2a1(t,z)F3(ξ)-3a1(t,z)F2(ξ)+a1(t,z)F(ξ). (11) Substituting Eq.(9)-(11) into Eq.(6), sincexiFj(ξ), Fk(ξ)(i=1,…,n; j,k=0,1,2,3)are linear inde- pendent, setting their coefficients to be zero, we have -γ(t,z)a0(t,z)+β(t,z)a30(t,z) =0, β(t,z)a31(t,z)-2α(t,z)a1(t,z)∑ n i=1 k2i(t,z) =0, -γ(t,z)a1(t,z)+3β(t,z)a20(t,z)a1(t,z)-a1(t,z)lt(t,z)-α(t,z)a1(t,z)∑ n i=1 k2i(t,z) =0, 3β(t,z)a0(t,z)a21(t,z)+a1(t,z)lt(t,z)+3α(t,z)a1(t,z)∑ n i=1 k2i(t,z) =0, -a1(t,z)kit(t,z) =0(i =1,…,n).
