KdV equation obtained by Lie groups and Sturm-Liouville problems

Transcription

1 KdV equation obtained by Lie groups and Sturm-Liouville problems M. Bektas Abstract In this study, we solve the Sturm-Liouville differential equation, which is obtained by using solutions of the KdV equation. M.S.C. : 53C, 34B4. Key words: KdV equation, Sturm-Liouville equation.. Introduction In [3], Gardner C.S. et al, announced a method for exact solution of the Kortewegde Vries equation (KdV for short) u t + uu x + u xxx =. This is a nonlinear partial differential equation and the equation arises in different physical applications. The problems in long waves, in water of relatively shallow depth, for very small amplitudes, would drop the nonlinear term uu x [6]. Kortewegde Vries [] first derived it in the study of long water waves in a channel of finite depth. Other fluid dynamical applications have been studied in [5]. On the other hand, this equation was obtained from a special case of the structure equation of the Lie group SL(, R) (the special linear group of all ( )-real unimodular matrices) by S.S. Chern and C.K. Peng []. Also, in [4], Gardner C.S. et al, showed how the initial value problem for the (nonlinear) Korteweg-de Vries equation can be reduced to a sequence of nonlinear problems. It is convenient to raplace u by 6u and thereby transform the KdV equation into u t 6uu x + u xxx =. Various distinct sets of boundary conditions can be specified, each posing a KdV equation with initial data u(x, ) = u (x), where u (x) is assumed to be bounded and three times continuously differentable. Certain special conditions must also be given to completely specify the problem. Differential Geometry - Dynamical Systems, Vol.5, No., 3, pp c Balkan Society of Geometers, Geometry Balkan Press 3.

2 M. Bektas This paper is organized in the following way. In the first section, it is obtained from SL(, R) the Korteweg-de Vries equation u t = 4 u xxx 3 uu x, which was obtained by Chern and Leng, and in the second section the Sturm-Liouville differential equation is solved and obtained from the solutions of the KdV equation.. Lie groups and the KdV equation In this section we will recall the notions and terminology used in []. Let SL(, R) be the group of all ( )-real unimodular matrices. If we choose X SL(, R), then we have (.) det X =, and therefore det X =. Its right-invariant Maurer-Cartan form is ( ) w = dx.x w = w w w, where w + w =. The structure (Maurer Cartan) equation of SL(, R), is (.) dw = w w. Let U be a neighbourhood in the (x, t)-plane and consider a smooth mapping f : U SL(, R). The pull-backs of the Maurer-Cartan forms can be written (.3) w = dx + Adt w = qdx + Bdt w = rdx + Cdt, where the coefficients are functions of x and t. The forms in (.3) satisfy the equations (.). This gives t + A x qc + rb = (.4) q t + B x B + qa = r t + C x ra + C =. We consider the special case when r = ±, and is a parameter indepent of x and t. Writing q = u(x, t), we get from (.4) { A = +C + C x B = uc C x C xx. Substitution into the second equation of (.4) we obtain (.5) u t = K(u) = u x C + uc x + C x C xxx. As an example we take C = u. Then, (.5) becomes u t = 4 u xxx 3 uu x,

3 KdV equation obtained by Lie groups which is a well-known KdV equation mentioned in [].. Spectral theory of Sturm-Liouville problems vs. KdV equation We consider the problem (.) Ly = y + u (x)y = λy, t, x (.) (.3) where y () =, y (, λ) + Hy(, λ) =, H, u(x, t) = x 3( t) is a solution of the KdV equation and u (x, t) t= = u (x). Since u is, the operator L is self-adjoint, has a discret spectrum, which consists of simple eigenvalues. As known, a solution of the problem (.)-(.3) is ϕ (x, λ) = sin sx s + s s sin {s (x τ)} u (τ, λ) dτ. On the other hand, if we solve (.3) equation for λ, we find s n = n + + H ( ) (.4) n + + O n, where s = λ and H = H + u (τ, λ) dτ. s n are called the eigenvalues of the problem (.)-(.3). Using the formula (.4), we will obtain an asymptotic formula for the eigenfunctions x, ( ϕ n (x) = n + ) ( sin n + ) ( ) x + O n, and the normalized eigenfunctions are V n (x) = ϕ n (x) ϕ n (x). Now, we obtain a solution of (.)-(.3) using the translation operator. A solution of the problem { y + u (x) y = λy y () = is y(x, λ) and the solution of the problem, where u (x, t) = x 3 ( t)

4 M. Bektas is a solution of the KdV equation and u (x, t) t= = u (x). A solution for { y = λy y () = is cos λx. These solutions are connected by χ(cos λx) = cos λx + K(x, s) cos λsds, where χ is a translation operator. Now, we consider the two operators A = x + u (x), Since Aχ = χb, we obtain a hyperbolic-type PDE, (.5) and the boundary conditions K(x, ) = K(x, x) = K x B = x. u (x)k = K s u (r, t) dr = r dr = 3 ( t) 6 x ( t) for K(x, s). If we consider two characteristics, ξ = x + s and = x s, the equation (.5) is reduced to the canonical form K ξ = 4 Ku. The function yields the equation (.6) and the conditions ( ξ + m A(ξ, ) = K, m ) A ξ = 4 ua. A (, ) = A (ξ, ) = ξ 4( t). Integration of the equation (.6) with respect to the variable from to gives On the other hand A ξ A ξ = = A ξ = A (ξ, α) udα. 4 ξ = ( t),

5 KdV equation obtained by Lie groups 3 and therefore, A (.7) ξ = 4 A (ξ, α) udα ξ ( t). Integration of the equation (.7) with respect to ξ from to ξ gives A (ξ, ) A (, ) = Since A (, ) =, it follows that On the other hand, with (.8) A (ξ, ) = K(x, s) = A (β, α) udα 4 β ( t). ξ β A (β, α) udα 4 ( t). ( ξ + m A(ξ, ) = K, m ) = K(x, s), +s x s x s 4 K (β, α) udα xs 6 ( t). The equation (.8) is a Volterra-type integral equation. Therefore, its solution is unique and the solution can be provided by successive approximations. References [] S.S. Chern and C. K. Peng, Lie groups and KdV Equations, Manuscripta Mathematika 8 (979), 7, 7-7. [] D.J. Korteweg and G. de Vries, On the change of form of long waves advancing a rectangular canal and on a new type of long stationary waves, Phil. Mag. 39 (985), [3] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Method for solving the Korteweg-De Vries equation, Phys. Rev. Letters, 9 (967), [4] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Korteweg-De Vries equation and generalizations VI, Methods for Exact Solutions Communications on Pure and Applied Mathematics, V.XXVII (974), [5] M.C. Shen, Asymptotic theory of unsteady three-dimensional waves in a channel of arbitary cross section, SIAM. J. Appl. Math, 7 (969), 6-7. [6] G.B. Whitham, Linear and Nonlinear Waves, John Willey & Sons, California, 974. Author s address: Mehmet BEKTAS Department of Mathematics, Firat University, 39 Elazig, Turkey mbektas@firat.edu.tr