ANALYSIS OF A NONLINEAR SURFACE WIND WAVES MODEL VIA LIE GROUP METHOD

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1 Electronic Journal of Differential Equations, Vol. 206 (206), No. 228, pp. 8. ISSN: URL: or ANALYSIS OF A NONLINEAR SURFACE WIND WAVES MODEL VIA LIE GROUP METHOD BEN GAO Abstract. This article focuses on two aspects. Firstly, symmetry analysis is performed for a nonlinear equation which can model surface wind wave patterns in nature. As a byproduct, the similarity reductions and exact solutions of the equation are constructed based on the optimal systems. Secondly, the explicit solutions are considered by the power series method. Moreover, the convergence of the power series solutions are shown.. Introduction Manna [6] studied the surface wind waves qualitative behavior, represented by the nonlinear equation u xt = u uu xx + (u x ) 2, (.) where u(x, t) is considered as an unidirectional surface wave propagating in the x-direction on a fluid medium involved in a large scale flow. h is the unperturbed initial depth, g is the acceleration of gravity, c 0 is the wind velocity and subscripts denote partial derivatives. For the sake of providing more information to understand (.), some works have been devoted to study (.) [6, 2]. The author in [6] provided some peakon solutions with amplitude, velocity, and width in interrelation and static compacton solutions with amplitude and width in interrelation for (.). The exact explicit traveling wave solutions of (.) are given by using the method of dynamical systems in [2]. However, as the authors known, the Lie group analysis and explicit power series solutions of (.) are left as open problems. The application of Lie transformation group theory for the construction of solutions of nonlinear partial differential equations (PDEs) is one of the most active fields of research in the theory of nonlinear PDEs and applications [3, 4, 5, 7, ]. The main idea of Lie group method is to transform solutions of a system of differential equations to other solutions. Once the symmetry group of a system of differential equations has be determined, one can directly use the defining property of such a group and construct new solutions to the system from known ones. The rest of article is arranged as follows: Section 2 concentrates on symmetries of (.); in Section 3, the similarity reductions for (.) are dealt with and exact solutions are provided by using Lie group method; in Section 4, the explicit solutions 200 Mathematics Subject Classification. 35J05, 35E05, 43A80, 26A8. Key words and phrases. Nonlinear surface wind waves; symmetry analysis; similarity reduction; power series method. c 206 Texas State University. Submitted July 2, 206. Published August 23, 206.

2 2 B. GAO EJDE-206/228 for the reduced equations are obtained by using the power series method; the last section contains a conclusion of our work. 2. Lie point symmetry In this section, we apply Lie point symmetry method for (.), and obtain its infinitesimal generators, commutation table of Lie algebra. First of all, let us consider a one-parameter Lie symmetry group admitted by (.) with an infinitesimal operator of the form X = ξ x + τ t + φ u. (2.) where ξ, τ, φ are functions of x, t, u respectively and are called infinitesimals of the symmetry group. The classical infinitesimal Lie invariance criterion for (.) with respect to the operator (2.) reads as in [2, 7], pr X (2)[ u xt + u + uu xx (u x ) 2] = 0 for any u solves (.). Here, the symbol pr X (2) is the usual 2 th-order prolongation of the operator [2, 7], in this situation, where pr X (2) = X + φ () x + φ (2) xx + φ (2) xt, u x u xx u xt φ () x = D x φ u x D x ξ u t D x τ, φ (2) xx = D 2 x(φ ξu x τu t ) + ξu xxx + τu xxt, φ (2) xt = D x D t (φ ξu x τu t ) + ξu xxt + τu xtt, and D x, D t stand for the total derivative operators, for example, D t = t + u t u + u tx + u tt u x u t Substituting pr X (2) into (.) and splitting it with respect to the different order derivatives of u, one obtains a system of linear over-determining equations for the unknown functions ξ, τ and φ. We can find the following equations for the symmetry group of (.) ξ x = τ t, ξ t = ξ u = 0, Solving above (2.2), we obtain τ x = τ u = 0, τ tt = 0, φ = 2uτ t. ξ = c x + c 3, τ = c t + c 2, φ = 2c u, (2.2) where c, c 2, c 3 and c 4 are arbitrary constants, we find that (.) admits the operators X = x x + t t 2u u, X 2 = t, X 3 = x. It is easy to check that {X, X 2, X 3 } is closed under the Lie bracket. In fact, we have [X, X ] = [X 2, X 2 ] = [X 3, X 3 ] = 0, [X, X 2 ] = [X 2, X ] = X 2, [X, X 3 ] = [X 3, X ] = X 3,

3 EJDE-206/228 NONLINEAR SURFACE WIND WAVES MODEL 3 [X 2, X 3 ] = [X 3, X 2 ] = 0. Furthermore, to obtain their adjoint representation, employing the following Lie series Ad(exp(ɛX i ))X j = X j ɛ[x i, X j ] + 2 ɛ2 [X i, [X i, X j ]]..., we can compute the following results Ad(exp(ɛX i ))X i = X i, i =, 2, 3, Ad(exp(ɛX ))X 2 = e ɛ X 2, Ad(exp(ɛX 2 ))X = X ɛx 2, Ad(exp(ɛX ))X 3 = e ɛ X 3, Ad(exp(ɛX 3 ))X = X + ɛx 3, Ad(exp(ɛX 2 ))X 3 = X 3, Ad(exp(ɛX 3 ))X 2 = X 2, where ɛ is an arbitrary constant. Based on the adjoint representation of the infinitesimal operators, we obtain the optimal systems of (.) as follows, where a is an arbitrary constant. {X, X 2, X 3, X 3 + ax 2 }, Remark 2.. The optimal systems of (.) can also be obtained using the results of the paper [9]. 3. Similarity reductions and exact solutions The symmetry group properties are very useful for the construction of invariant solutions of the differential equation under study. Next, We will consider the following similarity reductions and group-invariant solutions for (.) based on the optimal system. Case : Reduction by X. Integrating the characteristic equation for X, we get similarity variables z = tx, p = u x 2, and the group-invariant solution is p = f(z), that is, Substituting this expression into (.), we obtain u = x 2 f(tx). (3.) f 2f 2 + 3f z 2 f 2 + zf + z 2 ff = 0 (3.2) where f = df dz. Case 2: Reduction by X 2. Similarly, we have u = f(z) in which z = x. Substituting it into (.), we obtain f f 2 + ff = 0 (3.3) where f = df dz. Case 3: For generator X 3, we have z = t, u = f(z). The corresponding reduction equation is f = 0. (3.4) Therefore, (.) has a solution u = 0. Obviously, the solution is not meaningful.

4 4 B. GAO EJDE-206/228 Case 4: For generator X 3 +ax 2, we have z = ax+t, u = f(z). The corresponding reduction equation is where f = df dz. f a 2 f 2 af + a 2 ff = 0 (3.5) 4. Explicit power series solutions In Section 4, we obtained the reduced equations by using symmetry analysis. The power series can be used to treat differential equations, including many complicated differential equations with nonconstant coefficients []. In this section, we solve the nonlinear ODEs (3.2), (3.3), and (3.5) by the power series method. 4.. Explicit solutions to (3.2). Now, we seek a solution of (3.2) in a power series of the form f(z) = p n z n, (4.) n=0 where the coefficients p n (n = 0,, 2,... ) are constants to be determined. Substituting (4.) into (3.2), we have n p n z n 2 p k p n k z n + 3 (n + )p n+ z n n=0 n=0 n=0 n z 2 (k + )(n + k)p k+ p n+ k z n + z (n + )(n + 2)p n+2 z n (4.2) + z 2 n=0 n=0 n (k + )(k + 2)p n k p k+2 z n = 0. From this equality, comparing coefficients, we have p = 3 p 0(2p 0 ), p 2 = 8 p (4p 0 ). n=0 Generally, for n 2, we have { p n+ = 2p n p ( ) 2p 0 pn (n + )(n + 3) [2p k p n k + (k + )(n k)p k+ p n k + } (k + )(k + 2)p k p k+2 ]. (4.3) (4.4) In view of this equality, we can obtain all the coefficients p i (i 3) of the power series (4.), e.g., p 3 = 5 (3p2 p 2 + 2p 0 p 2 ). (4.5) Therefore, for arbitrary chosen constant number p 0, the other terms of the sequence {p n } n=0

5 EJDE-206/228 NONLINEAR SURFACE WIND WAVES MODEL 5 can be determined by (4.3) and (4.4). This implies that for (3.2), there is a power series solution (4.) with the coefficients constructed by (4.3) and (4.4). Now we show that the convergence of the power series solution (4.) of (3.2). In fact, from (4.4), we have p n+ M[ p n + p n + ( p k p n k + p k+ p n k + p k p k+2 )], for n = 2, 3,..., where M = max{ p, 2p 0, }. Now, we define a power series R = R(z) = n=0 r nz n by and r i = p i, i = 0,, 2, r n+ = M[r n + r n + (r k r n k + r k+ r n k + r k r k+2 )] where n = 2, 3,.... Then, it is easily seen that p n r n, n = 0,, 2,.... Thus, the series R = R(z) = n=0 r nz n is a majorant series of (4.). Next, we show that the series R = R(z) has a positive radius of convergence. R(z) = r 0 + r z + r 2 z 2 + r n+ z n+ ( = r 0 + r z + r 2 z 2 + M r n z n+ + r n z n+ + r k r n k z n+ + r k+ r n k z n+ + r k r k+2 z n+) = r 0 + r z + r 2 z 2 + M[z 2 (R r 0 ) + z(r r 0 r z) + zr(r r 0 r z) + z(r r 0 ) 2 + zr(r r 0 r z)]. Consider now the implicit functional equation with respect to the independent variable z, F (z, R) = R r 0 r z r 2 z 2 M[z 2 (R r 0 ) + z(r r 0 r z) + zr(r r 0 r z) + z(r r 0 ) 2 + zr(r r 0 r z)]. Since F is analytic in the neighborhood of (0, r 0 ) and F (0, r 0 ) = 0, F R (0, r 0 ) = 0. if we choose the parameter r 0 = p 0 properly. By the implicit function theorem [0], we see that R = R(z) is analytic in a neighborhood of the point (0, r 0 ) and with the positive radius. This implies that the power series (4.) converges in a neighborhood of the point (0, r 0 ). This completes the proof. Hence, the power series solution (4.) for (3.2) is an analytic solution. The power series solution of (3.2) can be written as f(z) = p 0 + p z + p 2 z 2 + p n+ z n+

6 6 B. GAO EJDE-206/228 = p p 0(2p 0 )z + 8 p (4p 0 )z 2 + (n + )(n + 3) {2p n p ( 2p 0 )p n + [2p k p n k + (k + )(n k)p k+ p n k (k + )(k + 2)p k p k+2 ]}z n+. Furthermore, the explicit power series solution of (.) is u(x, t) = p 0 x 2 + p tx 3 + p 2 t 2 x 4 + p n+ t n+ x n+3 = p 0 x p 0(2p 0 )tx p (4p 0 )t 2 x 4 + (n + )(n + 3) {2p n p ( 2p 0 )p n + [2p k p n k + (k + )(n k)p k+ p n k (k + )(k + 2)p k p k+2 ]}t n+ x n+3, where p 0 is an arbitrary constant, the other coefficients p n (n ) depend on (4.3) and (4.4) completely Explicit solutions to (3.3). Similarly, we seek a solution of (3.3) in a power series of the form (4.). Substituting it into (3.3), and comparing coefficients, we have p 2 = (p 2 p 0 ). (4.6) 2p 0 Generally, for n, we have { n p n+2 = (k + )[(n + k)p k+ p n+ k (n + )(n + 2)p 0 (k + 2)p n k p k+2 ] + (n + )p p n+ p n }. (4.7) In view of (4.6) and (4.7), we can obtain all the coefficients p i (i 3) of the power series (4.), e.g., p 3 = p 6p 0 (2p 2 ), p 4 = p 2 2p 0 (2p 2 ). Therefore, for arbitrary chosen constant numbers p 0 0 and p, the other terms of the sequence {p n } n=0 can be determined by (4.6) and (4.7). This implies that for (3.3), there is a power series solution (4.) with the coefficients constructed by (4.6) and (4.7) Explicit solutions to (3.5). Now, we seek a solution of (3.5) in a power series of the form (4.). Substituting (4.) into (3.5), and comparing coefficients, we have p 2 = 2a(ap 0 ) (a2 p 2 p 0 ). (4.8)

7 EJDE-206/228 NONLINEAR SURFACE WIND WAVES MODEL 7 Generally, for n, we have p n+2 = (n + )(n + 2)a(ap 0 ) {a 2 n (k + )[(n + k)p k+ p n+ k (k + 2)p n k p k+2 ] + a 2 (n + )p p n+ p n }. (4.9) In view of (4.8) and (4.9), we can obtain all the coefficients p n (n 3) of the power series (4.), e.g., p 3 = p 6a(ap 0 ) (2a2 p 2 p 2 ), p 4 = 2a(ap 0 ) (2a2 p 2 ). Thus, for arbitrary chosen constant numbers a 0, p 0 a and p, the other terms of the sequence {p n } n=0 can be constructed by (4.8) and (4.9). This implies that for (3.5), there is a power series solution (4.) with the coefficients determined by (4.8) and (4.9). Furthermore, the explicit power series solution of (.) is u(x, t) = p 0 + p ( ax + t) + p 2 ( ax + t) 2 + = p 0 + p ( ax + t) + + n= (n + )(n + 2)a(ap 0 ) p n+2 ( ax + t) n+2 n= 2a(ap 0 ) (a2 p 2 p 0 )( ax + t) 2 {a 2 n (k + )[(n + k)p k+ p n+ k (k + 2)p n k p k+2 ] + a 2 (n + )p p n+ p n } ( ax + t) n+2, where a 0, p 0 a and p are arbitrary constants, the other coefficients p n (n 2) depend on (4.8) and (4.9) completely. Remark 4.. The proofs of convergence of the power series solutions to (3.3), (3.5) are similar to the one of (3.2). The details are omitted here. Furthermore, such power series solutions can greatly enrich the solutions of the (.) and converge quickly, so it is convenient for computations in both theory and applications. Remark 4.2. The solutions of equations (3.3) and (3.5) (obviously, it is sufficiently to consider only the case a = ) in the explicit form can also be found without using the power series method (see the handbook [8]) and read as and f(z) = f(x) = be cx d c 2 + f(z) = be cz ( d c 2 ) + where d =, b 0, c 0. d2 4bc 4 e cx, d 4bc 2 ( d c 2 )e cz (fora = ),

8 8 B. GAO EJDE-206/228 Conclusions. In this article, We present Lie approach for a nonlinear surface wind waves model. As a byproduct, new invariant solutions are constructed based on the optimal systems which may exert potential applications about the problems of physical phenomena. Moreover, we apply the power series method to obtain the explicit solutions of (.). It can be seen that the Lie symmetry analysis and the power series method are very efficient to research the explicit solutions of PDEs. Acknowledgments. This research was supported by the Natural Science Foundation of Shanxi (No ). References [] N. H. Asmar; Partial Differential Equations with Fourier Series and Boundary Value Problems, second edn., China Machine Press, Beijing, [2] G. W. Bluman, S. C. Anco; Symmetry and Integration Methods for Differential Equations, Springer, New York, [3] G. W. Bluman, S. Kumei; Symmetries and Differential Equations, Springer-Verlag, World Publishing Corp., 989. [4] B. J. Cantwell; Introduction to Symmetry Analysis, Cambridge University Press, [5] M. Craddock, K. Lennox; Lie group symmetries as integral transforms of fundamental solutions, J. Differential Equations. 232 (2007), [6] M. A. Manna; Asymptotic dynamics of monochromatic short surface wind waves, Physica D. 49 (200), [7] P. J. Olver; Applications of Lie Groups to Differential Equations, in: Grauate Texts in Mathematics, vol. 07, Springer, New York, 993. [8] A. D. Polyanin, V. F. Zaitsev; Handbook of exact solutions for ordinary differential equations, Moscow, Fizmatlit, 200. [9] J. Patera, P. Winternitz; Subalgebras of real three-and four-dimensional Lie algebras, J. Math. Phys. 8 (977), [0] W. Rudin; Principles of Mathematical Analysis, third ed., China Machine Press, Beijing, [] W. Sinkala, P. Leach, J. O Hara; Invariance properties of a general-pricing equation, J. Differential Equations. 244 (2008), [2] Y. Xie, S. Tang, L. Zhong, A. Y. Chen; Bifurcations and exact solutions for a nonlinear surface wind waves equation, Mathematical Methods in the Applied Sciences. 39 (205), Ben Gao College of Mathematics, Taiyuan University of Technology, Taiyuan , China address: benzi06@63.com