Computable Analysis of the Solution of the Nonlinear Kawahara Equation

In this paper, we study the computability of the solution operator of the initial problem for the nonlinear Kawahara equation, which is based on the Type-2 Turing machines. We will prove that in Sobolev space 2 ( ) s H R , for 0 s ≥ , the solution operator: 2 : ( ) s R K H R → ( ; C R 2 ( )) s H R is ( ,[ s H δ ρ → ]) s H δ -computable. The conclusion enriches the theory of computability. Key words— the nonlinear Kawahara equation, initial problem, computability, Type-2 theory of effectivity (TTE), Sobolev space I.INTRODUCTION At present, the computability of solutions of the nonlinear evolution equations have become an important topic to the workers of physics, mechanics, life science, applied mathematics, engineering and theoretical computer. Researching boundedness and computability of the solutions of the nonlinear equations, will offer effective tools for the application of equations, enrich theoretical foundation of computer science and promote the development of computer software. In 1985, K.Weihrauch and others established a computational model, called Type-2 theory of effectivity (TTE for short). K.Weihrauch and N. Zhong have studied the computability of the generalized functions, the KdV equation and the Schrödinger equation [3]–[5], Dianchen Lu and others have studied the computability of the mKdV equation [1]. The nonlinear Kawahara equation was first proposed by Kawahara in 1972, this equation has wide applications in physics such as in the theory of magneto–acoustic waves in plasma, in the theory of long waves in shallow liquid under ice cover and so on. In this paper, we will discuss the nonlinear Kawahara equation as follows: 3 5 2 0 3 5 2 1 ( ) 2 u u u u u au c x t x x x y β γ ∂ ∂ ∂ ∂ ∂ ∂ + + + = − ∂ ∂ ∂ ∂ ∂ ∂ (1) ( , ,0) ( , ), , , 0 u x y x y x y R t φ = ∈ ≥ (2) where , R β γ ∈ are dispersion coefficients, a is nonlinear perturbance coefficient, and 0 c >0 is sound velocity. The paper is organized as follows. In Section2, we mainly review some basic definitions, lemmas and conclusions of TTE, which are relevant to the proof of section3. Section3 is devoted to the proof of the main theorem. II. PRELIMINARIES This section we will give a brief introduction of TTE. For details the reader can refer to [2]. Lemma 2.1 1) In Schwarz space ( ) S R , the function ψ ψ a a  ) , ( is ( , , ) s s ρ δ δ − computable; ) ( ) , ( t t ψ ψ  is ( , , s δ ρ ) ρ − computable; ψ φ ψ φ +  ) , ( and ψ φ ψ φ ⋅  ) , ( are ( , s δ , ) s s δ δ − computable. Dianchen Lu et al IJCSET |April 2012| Vol 2, Issue 4,1059-1064