Turing-Computability of Solution of Hirota Equation Dianchen Lu1, a and Liming Fu1, b

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1 Ieraioal Coferece o Comper Sciece a Elecroic Techology (ICCSET ) Trig-Compailiy of Solio of iroa Eqaio Diache L a a Limig F Facly of Sciece Jiag Uiveriy Zheiag Jiag 3P..Chia a cl@.e.c @qq.com Keywor: iroa eqaio Solio operaio Trig compailiy TTE heory Dhamel priciple Arac. I hi paper we maily ic he Trig-compale of he Solio operaio of iroa Eqaio. Firly we prove he exiece a iqee of he olio operaor of eqaio y priciple of coracio mappig a acclaim ha i local olio i Trig-Compale i e of TTE heory. By compale fcio corcig we exe he olio from he ieral o he eire pace. The he olio of hi eqaio i Trig-compale. The rel elarge he applicaio i compig iffereial eqaio o igial comper.. Irocio Differeial eqaio i a impora mahemaical moel of real prolem o all have goo properie of olio of iffereial eqaio. I realiy we are cocere ao how he olio of hi eqaio i calclae. A a rel he compailiy of olio for operaor ecome a ey ie i he y. K.Weihrach a N.Zhog[-] have ie he compailiy of he olio operaor of he wave eqaio a KDV eqaio. Diache L a oher have ie he compailiy of he oliear Kawahara eqaio[3]. I hi paper we will y he iiial prolem of iroa eqaio: + iα x + β 3x + µ x ( ) + iγ= ( x ) ( x) x () x where α β ( µ γ ) are real (complex) coa a αβ i complex vale fcio. iroa Eqaio () i a ypical moel i mahemaical phyic which ecompae he well-ow oliear Schröiger eqaio. aegawa a Koama [-5] propoe () a a moel for propagaio of ple i opical fier. The Cachy prolem of () chage a follow if µ = + iα x + β 3x + iγ = x = x. ( x ) ( x) () The rcre of he aricle i ha: I Par we maily iroce ome aic efiiio a lemma a which are releva o he proof of Par3; I Par 3 we prove he mai heorem of he paper maily.. Prelimiarie Lemma[] e i (ype coverio) fi : Le i : ω i e a repreeaio efie L ( x x )(= x ) : f ( x x ) he of he if f i ( ) -compale if a oly if L i ( [ ]) -compale. Lemma [] Le γ : Y M a γ : Y M are wo repreeaio vn i amiile repreeaio of N.The we have he followig propoiio: 5. The ahor - Plihe y Alai Pre 37

2 ) If f : M M i ( ) γγ -compale he f : N M M M compale. We efie a fcio g : N M M a followig: g x = f x g + x = f g x x ( ) ( ) ( ) ( ( ) ) Where x M Nhe g i ( v N γγ ) - compale. ) Ame ha h: M M i ( ) + : N M M : ( x) = x ( + x) = h ( x) = h ( x) So he fcio i ( v γγ) - compale. γγ - compale a efie a fcio. N - i ( v γ γγ ) N Defiiio [6] For he Borgai pace i he complee pace ao Schwarz pace S ( ) wih he followig orm: S 3 = ξ τ ξ + αξ where = ( + ) F LL ξ τ =. S 3 = ξ τ ξ αξ F or LL ξ τ Lemma 3 [6] If 7 < < ' >. The C (3) ψ 3 3 ' ' ' Le ψ C ( ) wih ψ = o () ψ ( ()) = for ome o-zero. Lemma [6] If Lemma 5 [6] If < < ' < <. The a p pψ [ ]. We eoe ψ () S () ϕ C ϕ S () ψ () F C F (5) ' ' (6) (7) L x ψ () S ( ) F( ) C F ψ () S ( ) F( ) C F ' < < < <. The ' ( + ). (8) 3. Mai el ( ) From he prolem () we ealih a oliear map K C ( ) iiial aa o he olio. The map Theorem ( ) i ( ) : ; which ralae he K i he olio operaor of he prolem (). whe > he olio operaor of he prolem () ρ -compale. K : C ; ( ) Proof We ca ge prolem () eqivale iegral eqaio y he Dhamel priciple: 38

3 where ( ) 3 i ( αξ + βξ ) ( ) = ( ) ( ) γ ( ) x S i S ' ' ' S = F e F F repree forier raform. For efie operaor ( ) ( ) = ( ) ( ) ( ) ( ) G ψ S i S ' ψ ' γ ( ') ' ( )( ) = ( ) G : G. Accorig he lemma 3. i [] i i eay way o prove he operaor G i ρ ρ -compale. By lemma he operaor G i ([ ] ) ([ ρ ] [ ρ ] ) -compale. Defie fcio v: S ( ) N CS ( ; ( )) v( ) = G( ) v( ) G( v( ) ) By Lemma v i ( vn [ ρ ] ) -compale. Le ω ( x ) = ( x + ) [ T]. If ( x ) = ϕ ( x) he 3 ω+ α xω+ β xω+ γω ω= i i x ω( x) = ϕ( x) x. We ame ha he iiial vale ( ) p=< p p > which i oaie y ( p ) = ϕ a exi compale We efie fcio aifyig ϕ + =. (9) ϕ i give y a c i i ϕ. ϕ + ( ) G( ) ω : = G ω : = ω. ϕ. For N -ame i.e. I i eay o prove ω ω ( ) here ω aifie he followig iegral eqaio: 3 ω ω ω + iα + β + iγω 3 ω = x x x ω ( x) = µ ( x) x. Sice limω = ω ( ) we ca elec iale ieger { ω } aifyig ω ω N.The { ω } I followig we prove { ω } From (3)-(8) we oai N N faly coverge o ω. i compale eqece. here o corac a eqece 39

4 ( ) ( ) ω ω = ψ () S ( ϕ ( x) ϕ( x)) ψ () i S ' ψ (') γ( ω ω ω ω ) ' ( ) ( ) ψ () S ( ϕ ( x) ϕ( x)) + ψ () i S ' ψ (') γ( ω ω ω ω ) ' ( ) ( ) C ϕ x ϕ x + C γ ω ω ω ω ' ' ϕ( ) ϕ( ) γ ( ω ω ) ω ω ' CCr γ ω ω C x x + C + C + C Chooig fficie mall ch ha < he ' CrC Th ω ω ω ω ω + ω ω The we have prove { ω } We have ow { ω } N N ω. +. faly coverge o ω a ω i compale. i compale eqece if ( q ) ω ( ) S = he. ece he olio ω of he ~ q q = ω() i.e. q q i he ~ -ame of ω ( ) iiial prolem (9) i compale o [ T ]. We efie a ( ρ ρ ) -compale map I : ( ) ( ) [ ] + T where ω ( ) = ω ( ) i he olio of he iiial prolem () o [ + T]. The we prove he olio ( T ) i compale. The fcio : ( φ ) = ( T ) efie y ( ) = ( ) I( T ( ) ( ) T) + = +. I i eay o prove he fcio i compale ice i erive y primiive recrio from compale fcio I. I he e we prove () i compale. Le T ( + ) T T we firly compe ( ) he compe I ( T ( T ) ) o ( ) = I ( T ( T ) ) i compale. I hi way we have ge he compale olio o. For he olio K : C ; ( ρ )-compale. ( ) operaor ( ) of iroa eqaio() i [ ] Similarly we ca prove he olio operaor of iroa eqaio i compale. Acowlegeme The wor wa ppore y he Naioal Nare Sciece Foaio of Chia (No. 673) a Naral Sciece Foaio of Jiag Provice (BK55). eferece [] Weihrach.K Zhog.N. Compig he olio of he Koreweg-e Vrie eqaio wih arirary preciio o Trig machie. Theore.Comp.SciV33 5 pp

5 [] Weihrach.K Zhog.N. I wave propagaio compale or ca wave comper ea he Trig machie?. Proc. Loo Mah. Soc. V85 () pp [3] Diache.LJiaxi.G. Compale aalyi of he olio of he Noliear Kawahara eqaio. IJCSETV Ie pp [] aegawa.a Koama.Y. Noliear ple propagaio i a moomoe ielecric gie. IEEE Joral of Qam ElecroicV3987 pp.5-5. [5] Koama.Y. Opical olio i a moomo fier. Joral of Saiical phyv pp [6] Zhaohi. Yelig.J. Well-poee for he cachy prolem o he iroa eqaio i oolev pace of egaive iice. Chi.A.Mah. V6B 5 pp [7] Caixia.S. Moifie KV eqaio a D-S poee a ill-poe prolem. Chiee Acaemy of Egieerig Phyic 6. 35