Turing computability of the Solution Operator of a ...

2014 International Conference on Computer Science and Electronic Technology (ICCSET 2014)

Turing computability of the Solution Operator of a higher modified Camassa-Holm equation

order

Dianchen Lu1, a * and Qiaoqiao Chen1, b 1

Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang Jiangsu, 212013, P.R.China a

b

Email: [email protected],

Email: [email protected]

Keywords: Modified Camassa-Holm equation, Solution operator, Turing computability, TTE theory, Duhamel principle

Abstract. In this paper, we mainly discuss the Turing computability of the solution operator of a higher order modified Camassa-Holm equation. Firstly, we transform the equation to its integral equation by Duhamel principle. Then applying the TTE theory, we prove that the solution operator of the integral equation is computable in a short interval. Finally, by constructing computable function, we extend the solution from partial internal to the entire space. The result enlarge the application in computing differential equations on digital computers. Introduction Camassa-Holm equation is an important equation. In the shallow water wave theory, different nonlinear completely integrable partial differential equation can be obtained by different levels of approximation. These equations with soliton solution, that is, they tend to be the determine constant in the infinity of space. Not disappear when they collide with each other and waveform, and speed also have no or only a small change. In addition, they show similar particle scattering behavior. In this paper, for m ≥ 3 odd, we will prove that the solution operator of the initial problem(1) is computable by the TTE theory [1-3].  1 2  2 1 m 2 2 −1 ∂ t u + ∂ x u + ∂ x u + 1 − ∂ x ∂ x u + ( ∂ x u )  = 0 2 2   (1)  u ( x, 0= ) ϕ ( x ) , t ∈ R, x ∈ R.  Erika A.Olson show in [4] that the periodic initial value problem for Eq.(1) is well posed. At present, the computability of solutions of the nonlinear evolution equations have become an important topic to the workers of physics and mathematics [5-6]. 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. The structure of the article is that: In part 2, we introduce some basic definitions and lemmas, which are relevant to the proof of part3; In part 3, we prove the main theorem of the paper mainly.

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Preliminaries Definition 1[4] For any ∀s,b ∈ R , X

s ,b

denotes the completion of the Schwartz space S ( R 2 ) with

respect to the norm  ' u=  ∫ ∫ (1 + ξ s ,b RR

1

) (1 + τ − ξ ) 2s

© 2015. The authors - Published by Atlantis Press

m

2b

u (ξ ,τ )

2

2 d ξ dτ  . 

374

Definition 2[4] In the sequel, let

ψ = ψ ( t ) ∈ C0∞ ( −1,1) be a cut-off function with 0 ≤ ψ ≤ 1 and

1 2

ψ ( t ) ≡ 1 for t < . t δ 

Definition 3[4] For 0 < δ < 1 , define = ψ δ ( t ) ψ=   , Tδ u ψ δ T (ψ δ u ) .

Lemma 1[1] (type conversion) Let δ i :⊆ ∑ω → X i be a representation of the set X i ( 0 ≤ i ≤ k ) , fi :⊆ X 1 ×⋅⋅⋅× X k → X 0

,

and

L ( x1 , ⋅⋅⋅, xk −1 )(= xk ) : f ( x1 , ⋅⋅⋅, xk ) ,

define

then

if

f

is (δ1 , ⋅⋅⋅, δ k , δ 0 ) -computable if and only if L is (δ1 , ⋅⋅⋅, δ k −1 , [δ k → δ 0 ]) -computable. Lemma 2[1] Let γ :⊆ Y → M and γ ′ :⊆ Y → M ′ are two representations, vN is admissible representation of N . Then we have the following propositions: 1) If f :⊆ M → M ′ is ( γ , γ ′ ) - computable, then f ′ :⊆ N × M ′ × M → M ′ is

( vN , γ ′, γ , γ ′ ) - computable. We define a function g ′ :⊆ N × M → M ′ as follow: g ′ (= 0, x ) f ( x ) , g ′ ( n += 1, x ) f ′ ( n, g ′ ( n, x ) , x ) , where x ∈ M ， n ∈ N , then g ′ is 2 ) Assuming that h :⊆ M → M

( vN , γ , γ ′) - computable. is ( γ , γ ) - computable, Define a function

H :⊆ N × M → M H ( 0, x= ) x, H ( n + 1, x=) h  H ( n, x=) h n+1 ( x ) .

( vN , γ , γ ) - computable.

So, the function H is [4]

Lemma

3

1 1  if b ∈  , + b  , 2 2  f , g ∈ X s ,b , w fg '

 1 m 2 − 3m + 1 1  Given s ∈  ⋅ 2 there exists β > 0 such that ,  ,  2 m − 3m + 2 2  1 1 b ' ∈  − b ,  , and b + b ' ≤ 1 , then there exists c > 0 such that for any 2 2 ' s ,b −1

≤c f

' s ,b '

g

' s ,b '

, where w fg satisfies

� � 2xx � fg (x ,τ ) �  xx w f * g ( ,τ ) + ∂�x f * ∂�x g (x ,τ ) .  + 2  2 1 + xx 1 +   Main Result

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)

From the problem (1), we establish a nonlinear map K R : H s → C R; H s (R ) , which translate the initial data ϕ to the solution u . The map K R is the solution operator of the initial problem. 1 m 2 − 3m + 1 1 Theorem 3.1 For any t ∈ R , when . 2 < s < , the solution operator 2 m − 3m + 2 2 s s K R : H → C R; H (R ) is δ H s ,  ρ → δ H s  -computable , where m ≥ 3 odd.

(

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−1 1 1 2  ∂ x ( u 2 ) + (1 − ∂ 2x ) ∂ x u 2 + ( ∂ x u )  , We can get Eq.(1) equivalent 2 2   integral equation by Duhamel principle:

Proof

Defining w=

u ( x,= t ) W ( t ) ϕ ( x ) − ∫ W ( t − ttt )w ( x, ) d . t

0

375

(2)

i xx x + mt ) where W ( t ) ϕ ( x ) := ∫ ϕ (xx d . )e ( R

 1 m 2 − 3m + 1 1 < s <  , define operator 2 2  2 m − 3m + 2

ϕ∈Hs  .

For

S ( u , ϕ , t= ) ψ ( t )W ( t ) ϕ ( x ) −ψ d ( t ) ∫ W ( t − ttt )w( x, )d , t

0

S ( u , ϕ )( t ) := S ( u , ϕ , t ) .

According the Lemma 3.2 in [1], it is easy to prove S is ([ ρ → δ s ] , δ s , ρ , δ s ) -computable. By lemma 1, S is

([ ρ → δ ] , δ , [ ρ → δ ]) -computable. s

s

s

Define function v : S ( R ) × N → C ( R; S ( R ))

v (ϕ , 0 ) = S ( 0, ϕ ) , v (jjj , j + 1) = S (v ( , j ) , ) . The function v is defined by primitive recursion from computable functions. By Lemma 2 v is (δ s , vN , [ ρ → δ s ]) -computable. Let ω= ( x, t ) u ( x, t0 + t ) ,where t ∈ [0, T ] , t0 ≥ 0 . If u ( x, t0 ) = µ ( x ) , then

 1 2  2 1 m 2 2 −1 ∂ tω + ∂ x ω + ∂ x (ω ) + (1 − ∂ x ) ∂ x ω + ( ∂ xω )  = 0 2 2   (3)  ω ( x, 0= ) m ( x ) , t ∈ R, x ∈ R.  ~ We assume that the initial value µ ∈ H s ( R ) is given by a δ H -name , i.e., p = p0 , p1 , which is s

obtained by

µn − µ s ≤ 2− n − 2 . For ∀k ∈ N , there exist appropriate computable

δ sc ( pi ) = µi and

nk satisfying µnk − µ

Hs

≤ 2− nk − 2 ≤ 2− k − 2 .

Define

= ωn0 : S= ( 0, µn ) , ωnj +1 : S (ωnj , µn ) . It is easy to prove

ωnj → ωn

( j → ∞) , there ωn satisfies the following integral equation:

ww ψ ( t ) W ( t ) µn ( x ) −ψ d ( t ) ∫ W ( t − ttt S ( n , µn ) = )w( x, )d , n (t ) = t

0

2 1  2 1 2 2 −1 where w= ∂ x (www ∂x n )  . ( n ) + (1 − ∂ x ) ∂ x  n + 2 2   j Since ωn → ωn ( j → ∞) , we can select suitable integer nk ， jk to constrct a sequence

{ω } jk nk

k∈N

，satisfying ωnjkk − ωnk

s

≤ 2− k −1 .

{ }

In the following , we prove ωnjkk

fastly converges to ω .

k∈N

By the Lemma 3,we obtain

(

)

= ww ψ ( t ) W ( t ) µnk ( x ) − µ ( x ) −ψ dww ( t ) ∫0 W ( t − tttt )  w nk ( x, ) − w ( x, ) d nk − s ,b '

(

≤ ψ ( t ) W ( t ) µnk ( x ) − µ ( x )

(

≤ ψ ( t ) W ( t ) µnk ( x ) − µ ( x )

≤ C1 µnk ( x ) − µ ( x )

Hs

) )

' s ,b ' s ,b

t

+ ψ dww ( t ) ∫0 W ( t − tttt )  w nk ( x, ) − w ( x, ) d t

t

'

0

s ,b

+ ψ d ( t ) ∫ W ( t − ttt ) w fg ( x, ) d

+ Cδ

1 −b 2

ωn + ω k

' s ,b

ωn − ω k

376

' s ,b

' s ,b

' s ,b

1

≤ C1 2− k − 2 + 2rCδ 2

−b

ωn − ω k

' s ,b

, where f =ωnk + ω , g =ωnk − ω .

C1

Choosing sufficient small δ such that

1 − 2rCδ Thus

ωnj − ω

'

k

k

s ,b

≤ ωnjkk − ωnk

{ }

Then we have proved ωnjkk

s

{ω }

s ,b

k∈N

+ ωnk − ω

' s ,b

' s ,b

≤ 2− k −1 .

≤ 2− k −1 + 2− k −1 ≤ 2− k .

fastly converges to ω and ω is computable.

computable sequence , if δ S ( qk ) = ωnjkk ( t ) , then ~ q0 , q1 , = ω ( t ) ，i.e., q0 , q1 , is the δ H s -name of ω ( t ) . Hence the solution ω of the

We

δH

'

1 −b 2

< 2 ，then ωnk − ω

jk nk

known

k∈N

is

initial problem(3) is computable on t ∈ [ 0, T ] , that is solution operator map S is computable. Define a

( ρ , δ

Hs

)

, ρ , δH s -computable map P : ( t0 , µ , t ) → u ( t ) , t ∈ [t0 , t0 + T ] ，

where ω ( t0 ) = µ , ω ( t ) is the solution of the initial problem(1)on t ∈ [t0 , t0 + T ] . Then we prove the solution u (n ⋅ T ) is computable. Define function H : H (ϕ , n ) = u ( nT )

H= , n + 1) P ( nT , H (ϕ , n ) , ( n + 1) T ) . (ϕ , 0 ) ϕ , H (ϕ=

Then H is computable since H is derived by primitive recursion from computable function P . In the end, we prove u (t ) is computable. let n ⋅ T ≤ t ≤ (n + 1) ⋅ T ，we first compute u (n ⋅ T ) ， then compute P ( nT , u ( nT ) , t ) ，so u ( t ) = P ( nT , u ( nT ) , t ) is computable. this way, we have get the computable solution on t ∈ R . For 1 m 2 − 3m + 1 1 < s < ，the solution operator K R : H s → C R; H s (R ) of the higher ∀t ∈ R , . 2 2 m − 3m + 2 2 order modified Camassa-Holm equation (1) is (δ s , [ ρ → δ s ]) -computable. In

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Summary and Outlook The paper study computability of the solution operator of the higher order modified Camassa-Holm equation. On the basis of computablity theory, whether problem can be implemented on computer is an important problem. Computational complexity theory just can be used to solve the problem. The topic we will study in the future. Acknowledgements The work was supported by the National Nature Science Foundation of China (61070231) and the Natural Science Foundation of Jiangsu Province (BK20140525). References [1] K.Weihrauch, N.Zhong. Computing the solution of the Korteweg-de Vries equation with arbitrary precision on Turing machines. Theoret.Comput.Sci.. 2005.332 (1-3): 337-366. [2] Ning Zhong, Klaus Weihrauch, Computability theory of generalized functions [J]. Assoc. for Computing Mathinery 50 (4) (2003)469-505. [3] N. Zhong. Computability structure of the Sobolev spaces and its applications. Theoretical Computer Science. 1999 (219): 487-510. [4] Erika A.Olson. Well posedness for a higher order modified Camassa-Holm equation.J. Differential Equations 2009 (246) :4154–4172. 377

[5] Dianchen Lu，Qingyan Wang. Computing the solution of the m-Korteweg-deVries equation on Turing machines, Electronic Notes in Theoretical Computer Science, 2008(202):219-236. [6] Dianchen Lu，Jiaxin Guo. Computable analysis of the solution of the Nonlinear Kawahara equation.IJCSET.Vol 2(2012),Issue 4,1059-1064.

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