Conservation laws of a perturbed Kaup Newell equation

Transcription

1 Modern Physics Leers B Vol. 30, Nos. 32 & 33 (2016) (6 pages) c World Scienific Publishing Company DOI: /S Conservaion laws of a perurbed Kaup Newell equaion Jing-Yun Yang School of Mahemaical and Physical Sciences, Xuzhou Insiue of Technology, Xuzhou , Jiangsu, China @qq.com Wen-Xiu Ma Deparmen of Mahemaics and Saisics, Universiy of Souh Florida, Tampa, FL , USA maw@cas.usf.edu Received 9 Augus 2016 Acceped 20 Sepember 2016 Published 24 November 2016 A new La pair is inroduced for a perurbed Kaup Newell equaion and used o consruc wo series of conservaion laws hrough a Riccai equaion ha a raio of eigenfuncions saisfies. Boh series of conservaion laws are defined recursively, and he firs wo in each series are presened eplicily. Keywords: La pair; conservaion law; Riccai equaion. PACS Number(s): Ik 1. Inroducion Many nonlinear models are sudied and showed o possess infiniely many conservaion laws, and bi-hamilonian srucures play a crucial role in consrucing conserved densiies. 1 Associaed wih non-semisimple Lie algebras, he variaional ideniies have been developed o formulae generaing funcions for conserved densiies. 2,3 There are also oher approaches o conservaion laws including he mehod using adjoin symmeries 4 6 and he epansion echnique of raios of eigenfuncions of specral problems. 7,8 A perurbed Kaup Newell equaion was recenly inroduced from a generalized Kaup Newell specral problem wih linear perurbaion. 9 The generalized Corresponding auhors

2 J.-Y. Yang & W.-X. Ma Kaup Newell equaion is defined by p = 1 2 (p 2pp q p 2 q 4αpp ), q = 1 2 q 1 2 p q 2 pqq 2α(pq), (1.1) where α is an arbirary consan. The case of α = 0 reads o he sandard Kaup Newell equaion, 10 which possesses a ri-hamilonian srucure 11 and is he Euler Poincaré flow on he space of firs-order differenial operaors. 12 In his paper, we would like o presen a differen La pair for he perurbed Kaup Newell equaion (1.1) and consruc wo series of conservaion laws hrough a Riccai equaion which a raio of wo eigenfuncions needs o saisfy. 7 Such a scheme of consrucing conservaion laws using La pairs has been also applied o oher nonlinear inegrable equaions (see, e.g. Refs. 13 and 14). A few concluding remarks are given a he end of he paper. 2. A Riccai Equaion and Conservaion Laws Moivaed by a La pair for he Kaup Newell equaion (see, e.g. Ref. 15), we can derive a new La pair for he perurbed Kaup Newell equaion (1.1): λ + αp λp φ1 φ = Uφ = U(u, λ)φ, U = (U ij ) 2 2 =, φ = (2.1) q λ αp and where φ = V φ = V (u, λ)φ, V = (V ij ) 2 2, (2.2) V 11 = V 22 = λ λpq 1 2 αp2 q α 2 p αp, V 12 = 1 2 λ( 2λp p + p 2 q + 2αp 2 ), V 21 = λq 1 2 q 1 2 pq2 αpq. Tha is o say, he isospecral (λ = 0) zero curvaure equaion: U V + U, V = 0 φ 2 (2.3) eacly presens he perurbed Kaup Newell equaion (1.1). To apply he scheme of generaing conservaion laws based on La pairs, we consider he raio of he wo eigenfuncions φ 21 = φ 2 φ 1. (2.4) Obviously from he specral problem in (2.1), we see ha he raio φ 21 saisfies he following Riccai equaion: φ 21, = 2(λ + αp)φ 21 + q λpφ (2.5)

3 Conservaion laws of a perurbed Kaup Newell equaion Epanding φ 21 ino a Lauren series φ 21 = χ i λ i, (2.6) i=1 we obain from he above Riccai equaion a recursion relaion for defining χ i : χ 1 = q 2, Paricularly, his gives rise o χ 2 = 1 8 pq2 1 2 αpq 1 4 q, χ i+1 = 1 2 χ i, 1 i 2 p χ i j+1 χ j αpχ i, i 1. (2.7) j=1 χ 3 = 1 16 p2 q pqq p q α2 p 2 q αp2 q αpq αp q q. Epanding φ 21 ino anoher Lauren series φ 21 = κ i λ i, (2.8) i=0 we obain from he above Riccai equaion a recursion relaion for defining κ i : κ 0 = 2 p, κ 1 = 1 2 κ 0, 1 ( ) 1 2 q + αpκ 0 = 1 p 2 q 2α, κ i+1 = 1 2 κ i, + 1 i (2.9) 2 p κ i j+1 κ j + αpκ i, i 1. j=1 Noe ha here are he opposie signs in defining χ i+1 and κ i+1 due o he eisence of an addiional erm 2pκ 0 κ i λ i in he second recursive relaion. The recursion relaion (2.9) paricularly produces κ 2 = 1 8p 3 (p4 q 2 2p 3 q 4p 2 p q + 4pp 4p 2 + 4αp 4 q 8αp 2 p ), κ 3 = 1 16p 4 ( p6 q 3 + 4p 5 qq + 7p 4 p q 2 2p 4 q 8p 3 p q 8p 3 p q + 4p 2 p 8pp p + 4p 3 6αp 6 q 2 8α 2 p 6 q + 8αp 5 q + 16α 2 p 4 p + 28αp 4 p q 16αp 3 p ). Now o consruc conservaion laws, we inroduce I = V 11 + V 12 φ 21, J = U 11 + U 12 φ 21 (2.10) and hen, from he presened La pair (2.1) and (2.2), we obain ( ) ( ) φ1, φ1, I = = = J. (2.11) φ 1 φ

4 J.-Y. Yang & W.-X. Ma Subsiuing he wo Lauren series epansions of φ 21 ino his basic formula and equaing he same powers of λ will produce conservaion laws for he perurbed Kaup Newell equaion (1.1). In wha follows, we compue hese wo series of conservaion laws. Firs, using (2.6) and comparing differen powers of λ from zeroh in (2.11) direcly yields infiniely many conservaion laws ( (pχ 1 + αp) = 1 2 p2 qχ 1 + pχ p χ 1 αp 2 χ αp2 q α 2 p ) 2 αp, (pχ i+1 ) = ( 12 p2 qχ i+1 + pχ i ) p χ i+1 αp 2 χ i+1, i 1, (2.12) where χ i, i 1, are defined recursively by (2.7). The firs wo conservaion laws in (2.12) can be worked ou as follows: ( ) ( 1 pq + αp = p2 q pq p q 3 2 αp2 q α 2 p ) 2 αp (2.13) and ( 1 8 p2 q αp2 q 1 ) 4 pq ( 1 = 8 p3 q p2 qq pq 1 8 p q αp3 q 2 + α 2 p 3 q αp2 q ). (2.14) Second, by use of (2.8) in (2.11), he coefficiens of λ presens a rivial conservaion law, and comparing differen powers of λ from zeroh eacly yields infiniely many conservaion laws ( (pκ 1 + αp) = 1 2 p2 qκ 1 + pκ p κ 1 αp 2 κ αp2 q α 2 p ) 2 αp, (pκ i+1 ) = ( 12 p2 qκ i+1 + pκ i ) p κ i+1 αp 2 κ i+1, i 1, (2.15) where κ i, i 1, are defined recursively by (2.9). The firs wo conservaion laws in (2.15) can be compued as follows: 1 2p (p2 q 2p + 2αp 2 ) = 1 8p ( 3p3 q 2 + 2p 2 q + 10pp q 4p 8α 2 p 3 12αp 3 q + 20αpp ) (2.16)

5 Conservaion laws of a perurbed Kaup Newell equaion and 1 8p 2 ( p4 q 2 + 2p 3 q + 4p 2 p q 4pp + 4p 2 4αp 4 q + 8αp 2 p ) = 1 8p 2 (p5 q 3 6p 3 p q 2 3p 4 qq + 5p 2 p q + 6p 2 p q + p 3 q + 2p p 2pp + 6αp 5 q 2 + 8α 2 p 5 q 24αp 3 p q 16α 2 p 3 p 6αp 4 q + 12αp 2 p ). (2.17) Observing he wo recursive formulas (2.7) and (2.9), we see ha he firs series of conservaion laws in (2.12) is of differenial polynomial ype and he second series of conservaion laws in (2.15) is of raional differenial funcion ype. We poin ou ha he oher Riccai equaion, φ 12, = 2(λ + αp)φ 12 + λp qφ 2 12, (2.18) which he oher raio φ 12 = φ1 φ 2 of he wo eigenfunions saisfies, generaes wo equivalen classes of conservaion laws, by use of similar Lauren epansions of φ Concluding Remarks Based on a La pair, we consruced wo series of conservaion laws for he discussed perurbed Kaup Newell equaion. Boh series of conservaion laws were generaed from aking wo Lauren epansions of a raio of eigenfuncions, which saisfy a Riccai equaion, and he firs wo conservaion laws in each series were eplicily presened. Such Riccai equaions are also used o presen algebro-geomeric soluions by epanding raios of eigenfuncions, called he Baker Akhiezer funcions, ino Lauren series. In 2 2 mari specral problems, hyperellipical curves are adoped, and in 3 3 mari specral problems, rigonal curves are inroduced o consruc algebro-geomeric soluions o solion equaions, by aking characerisic polynomials of La marices Moreover, by symbolic compuaions, raional and lump soluions are discussed recenly for many ineresing nonlinear equaions, and Hiroa bilinear forms 23 or generalized bilinear forms have been used o presen raional and lump soluions by links of logarihmic derivaives. Acknowledgmens The work was suppored in par by he Naional Naural Science Foundaion of China under he Gran Nos , , , and , he Jiangsu Qing Lan Projec (2014) and XZIT (XKY ), and he disinguished professorships of he Shanghai Universiy of Elecric Power and he Shanghai Second Polyechnic Universiy

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