Journal of Mathematical Analysis and Applications

Transcription

1 J. Math. Anal. Appl. 394 (202) 2 28 Contents lists available at SciVerse ScienceDirect Journal of Mathematical Analsis and Applications journal homepage: Resonance of solitons in a coupled higher-order Ito equation Yi Zhang a Yun-Cheng You b Wen-Xiu Ma b Hai-Qiong Zhao c a Department of Mathematics Zhejiang Normal Universit Jinhua PR China b Department of Mathematics and Statistics Universit of South Florida Tampa FL USA c Business Information Management School Shanghai Institute of Foreign Trade Shanghai PR China a r t i c l e i n f o a b s t r a c t Article histor: Received 7 June 20 Available online 7 April 202 Submitted b Junping Shi In this Letter we propose a new coupled higher-order Ito equation and present its N-soliton solutions in Pfaffian form. Furthermore some interesting eamples of soliton resonance related to two solitons near the resonant state are pointed out. 202 Elsevier Inc. All rights reserved. Kewords: Soliton resonance Pfaffian Coupled higher-order Ito equation. Introduction It is important in mathematical phsics to look for eact solutions to soliton equations and to search for new soliton equations. In order to achieve these two objectives several approaches have been developed. One of them is the Hirota bilinear approach [ 3] which provides a direct powerful approach to nonlinear integrable equations and it is widel used in constructing N-soliton solutions. It is known that the Korteweg de Vries (KdV) equation u t + 6uu + u = 0 (.) can be transformed into the bilinear form D (D t + D 3 )f f = 0 (.2) b the dependent variable transformation u = 2(ln f ) where the bilinear operators D m Dn t are defined b (.3) D m Dn t a b = ( ) m ( t t ) n a( t)b( t ) =t =t. (.4) In [4] Ito investigated the following new tpe of bilinear equation: D t (D t + D 3 )f f = 0 and noted that Eq. (.5) and the KdV equation Eq. (.2) have the same -soliton solution but that their N-soliton solutions differ in the phase shift. Besides using the same dependent variable transformation as Eq. (.3) Eq. (.5) is transformed into the nonlinear form: u tt + u t + 6u u t + 3uu t + 3u u t = 0. (.6) (.5) Corresponding author. address: z2836@63.com (Y. Zhang) X/$ see front matter 202 Elsevier Inc. All rights reserved. doi:0.06/j.jmaa

2 22 Y. Zhang et al. / J. Math. Anal. Appl. 394 (202) 2 28 Moreover its Bäcklund transformation conservation laws and Hamiltonian structures are studied in [56]. In [47] the authors also presented a higher-order version of (.5): D (D + D 3 )f f = 0 (6D t D + D 5 D 5D 2 D2 )f f = 0 where is onl an auiliar variable. B the following transformations (.7a) (.7b) u = ln f p = (ln f ) w = (ln f ) (.8) we can derive a sstem with respect to u w p as follows: and w + p + 6u p = 0 6p t + p + 0p u + 60u 2 p + 20u p 5w 20p 2 0u w = 0 w t + w (w u + p p ) + 20(2u p p + u 2 w ) (.9a) (.9b) + 0(p p + u w ) + 0(p p + u w ) 20 3 p w = 0 (.9c) where the auiliar variable has vanished. The N-soliton solutions in Pfaffian form were obtained [8]. Meanwhile bilinear Bäcklund transformation and nonlinear superposition formulas for Eq. (.7) have also been presented [9]. Recentl the resonance of solitons has been studied theoreticall and eperimentall in man real phsical models [0 ] where the interactions between solitons ma be completel non-elastic. That is to sa the amplitude velocit and wave shape of a soliton ma change after the nonlinear interaction. For instance at a specific time one soliton can undergo fission into two or more solitons; or contraril two or more solitons ma fuse into one soliton [2 5]. These tpes of phenomena are also called soliton fission and soliton fusion respectivel [67]. Furthermore fission and fusion phenomena of the dromion peakon and compacton solutions have also be observed [89]. To describe the intermediate patterns of the resonant solitons of the shallow wave equation Kodama considered N-soliton solutions including all possible interactions and classified those N-soliton solutions b a chord diagram method [202]. In this paper we propose a new coupled higher-order Ito equation (D + D 3 )f g = 0 D (D + D 3 )f g = 0 (6D t + D 5 5D2 D )f g = 0 (6D t D + D 5 D 5D 2 D2 )f g = 0 (.0a) (.0b) (.0c) (.0d) where is an auiliar variable. It is obvious that Eq. (.0) ma lead to a higher-order Ito equation Eq. (.7) with g = f. Using the dependent variable transformation φ = g f u = 2(ln f ) (.) the coupled higher-order Ito equation Eq. (.0) can be rewritten in the following nonlinear form: φ + φ + 3uφ = 0 u + u + 3u u + 3uu = {[3(u + 6uu + 2u )φ + 6uφ (.2a) 3( u 6u 2 6u )φ + 8u φ + 2φ + 2uφ + 2φ ]/φ} (.2b) 6φ t + [φ + 0 u φ + 5( u + 3( u ) 2 )φ ] 5(φ + φ u + 2uφ ) = 0 u t 0 3 uu 5 6 u ( + 5 u ) 2 u 5 6 [ u u + u u u u u u ] [ u u + u u ] + 6 u u u u = 5 6 {[u φ 6 u u φ u φ + u φ (.2c) + 6 u uφ + 2u φ + uφ 2u φ 2uφ u φ ]/φ}. (.2d)

3 Y. Zhang et al. / J. Math. Anal. Appl. 394 (202) The purpose of this paper is to give the N-soliton solutions and to analze the resonant soliton phenomena for this coupled higher-order Ito equation. Using the perturbation and Pfaffian technique we present N-soliton solutions to Eq. (.0) and give a strict proof. In addition we discuss the resonance of solitons described b a 2-soliton solutions of the coupled higher-order Ito equation. 2. N-soliton solutions to the coupled higher-order Ito equation It is known that Eq. (.9) belongs to the DKP hierarch (dispersionless KP hierarch) and the solutions of the DKP hierarch can be written in Pfaffian form [22]. In the following in relation to the D-tpe Lie algebraic structure of the solutions we present the N-soliton solutions to Eq. (.0) b virtue of Pfaffians and give a strict proof b the Pfaffian identit. Using the perturbational method we obtain 2-soliton solutions and 3-soliton solutions to Eq. (.9) epressed as follows: and with f = + a e η + a 2 e η 2 + a 2 e η +η 2 g = + b e η + b 2 e η 2 + b 2 e η +η 2 f = + a e η + a 2 e η 2 + a 3 e η 3 + a 2 e η +η 2 + a 3 e η +η 3 + a 23 e η 2+η 3 + a 23 e η +η 2 +η 3 g = + b e η + b 2 e η 2 + b 3 e η 3 + b 2 e η +η 2 + b 3 e η +η 3 + b 23 e η 2+η 3 + b 23 e η +η 2 +η 3 η j = p j p 3 j p5 j t a ij = b ij = (p i p j ) (p i + p j )(p 3 i + p 3 j ) α ij (p i p j ) (p i + p j )(p 3 i + p 3 j ) β ij (2.a) (2.b) (2.2a) (2.2b) (2.3a) (2.3b) (2.3c) α ij = a i p 3 i b j a j p 3 j b i β ij = b i p 3 i a j b j p 3 j a i (i j = 2 3) (2.3d) a 23 = b 23 = (p p 2 )(p p 3 )(p 2 p 3 ) (p + p 2 )(p + p 3 )(p 2 + p 3 )(p 3 + p3 2 )(p3 + p3 3 )(p3 2 + p3 3 ) α 23 (2.3e) (p p 2 )(p p 3 )(p 2 p 3 ) (p + p 2 )(p + p 3 )(p 2 + p 3 )(p 3 + p3 2 )(p3 + p3 3 )(p3 2 + p3 3 ) β 23 (2.3f) α 23 = [a a 2 (p 6 p6 2 )b 3p 3 3 a a 3 (p 6 p6 3 )b 2p a 2a 3 (p 6 2 p6 3 )b p 3 ] β 23 = [b b 2 (p 6 p6 2 )a 3p 3 3 b b 3 (p 6 p6 3 )a 2p b 2b 3 (p 6 2 p6 3 )a p 3 ] where p j a j b j are free parameters. These epressions suggest that N-soliton solutions to Eq. (.0) are epressed b Pfaffians. In fact we find that f = pf (d 0 a r r 2... r N c N... c 2 c ) = pf (d 0 a ) g = pf (a b r r 2... r N c N... c 2 c ) = pf (a b ) where the entries of the Pfaffians are defined as where pf (d m r j ) = p m j e η j (m 0 j = 2... N) pf (d 0 a) = pf (a b) = pf (d m a) = 0 (m ) pf (a r j ) = e η j pf (a c j ) = a j pf (b c j ) = b j (j = 2... N) pf (r j r k ) = a jk e η j+η k pf (r j c k ) = δ jk pf (c j c k ) = c jk (j k = 2... N) pf (d m c j ) = pf (d m b) = pf (d m d n ) = pf (b r j ) = 0 (m n 0 j = 2... N) δ jk = j = k 0 j k η j = p j p 3 j p5 j t a jk = p j p k p j + p k c jk = a 3 jpj b k a k p 3 k b j. p 3 j + p 3 k (2.3g) (2.3h) (2.4a) (2.4b) (2.4c) (2.4d)

4 24 Y. Zhang et al. / J. Math. Anal. Appl. 394 (202) 2 28 The above Pfaffians have 3N parameters p j a j b j for j = 2... N. In what follows we show that f and g given b Eq. (2.4) are N-soliton solutions to Eq. (.0). B virtue of the above Pfaffians we come up with the following differential formulae for f and g: f = pf (d a ) f = pf (d 2 a ) (2.5a) f = pf (d 3 a ) + pf (d 0 d d 2 a ) f = pf (d 4 a ) + 2pf (d 0 d d 3 a ) f = pf (d 5 a ) + 3pf (d 0 d d 4 a ) + 2pf (d 0 d 2 d 3 a ) f = pf (d 3 a ) + 2pf (d 0 d d 2 a ) f = pf (d 4 a ) + pf (d 0 d d 3 a ) f = pf (d 5 a ) + pf (d 0 d 2 d 3 a ) f t = pf (d 5 a ) + 2pf (d 0 d d 4 a ) + 3pf (d 0 d 2 d 3 a ) (2.5b) (2.5c) (2.5d) (2.5e) (2.5f) (2.5g) (2.5h) g = pf (d 0 d a b ) g = pf (d 0 d 2 a b ) (2.5i) g = pf (d 0 d 3 a b ) + pf (d d 2 a b ) g = pf (d 0 d 4 a b ) + 2pf (d d 3 a b ) g = pf (d 0 d 5 a b ) + 3pf (d d 4 a b ) + 2pf (d 2 d 3 a b ) g = pf (d 0 d 3 a b ) + 2pf (d d 2 a b ) g = pf (d 0 d 4 a b ) + pf (d d 3 a b ) g = pf (d 0 d 5 a b ) + pf (d 2 d 3 a b ) g t = pf (d 0 d 5 a b ) + 2pf (d d 4 a b ) + 3pf (d 2 d 3 a b ). Substituting these relations into Eq. (.0a) Eq. (.0c) we find that Eq. (.0a) is reduced to the Pfaffian identit (2.5j) (2.5k) (2.5l) (2.5m) (2.5n) (2.5o) (2.5p) pf (d 0 d d 2 a )pf (a b ) pf (d 0 d a b )pf (d 2 a ) + pf (d 0 d 2 a b )pf (d a ) pf (d d 2 a b )pf (d 0 a ) 0 (2.6a) and Eq. (.0c) is reduced to Pfaffian identit 5[pf (a b )pf (d 0 d d 4 a ) pf (d 0 d a b )pf (d 4 a ) pf (d 0 a )pf (d d 4 a b ) + pf (d 0 d 4 a b )pf (d a )] 5[pf (a b )pf (d 0 d 2 d 3 a ) + pf (d 0 d 2 a b )pf (d 3 a ) pf (d 0 a )pf (d 2 d 3 a b ) pf (d 0 d 3 a b )pf (d 2 a )] 0. Furthermore in order to prove that (2.4) also satisfies Eq. (.0b) and Eq. (.0d) we have to use the second epression for g which is equal to (2.4b): g = pf (e 0 a r r 2... r N c N... c 2 c ) = pf (e 0 a ) where the new entries are defined b pf (e 0 a) = pf (e a) = 0 pf (e m c j ) = b j p 3m j (m = 0 ) pf (e m r j ) = pf (d m e ) = 0 (m 0 j = 2... N). Using the properties of the Pfaffian [] we obtain the following differential formulas: (2.6b) g = pf (e a ) g = pf (d 0 d e a ) g = pf (d 0 d 2 e a ) (2.7b) g = pf (d 0 d 3 e a ) + pf (d d 2 e a ) g = pf (d 0 d 4 e a ) + 2pf (d d 3 e a ) g = pf (d 0 d 5 e a ) + 3pf (d d 4 e a ) + 2pf (d 2 d 3 e a ) g = pf (d 0 d 3 e a ) + 2pf (d d 2 e a ) g = pf (d 0 d 4 e a ) + pf (d d 3 e a ) g = pf (d 0 d 5 e a ) + pf (d 2 d 3 e a ) g t = pf (d 0 d 5 e a ) + 2pf (d d 4 e a ) + 3pf (d 2 d 3 e a ). (2.7a) (2.7c) (2.7d) (2.7e) (2.7f) (2.7g) (2.7h) (2.7i)

5 Y. Zhang et al. / J. Math. Anal. Appl. 394 (202) Fig.. The plot of the regular interaction of two solitons to the new coupled higher-order Ito equation. The parameters used are a = b a 2 = b 2 p =.5 p 2 = 2 b = 3 b 2 = 4 = 0. The left figure shows a three-dimensional plot and the right figure shows a contour map. Moreover we have to rewrite Eq. (.9b) and Eq. (.9d) in the following form: (D + D 3 )f g = 0 (6D t + D 5 5D2 D )f g = 0 where we have used the -derivative of Eqs. (.9a) and (.9c). Substituting (2.4a) (2.5a) (2.5h) and (2.7) into (2.8) we find that Eq. (2.8a) is reduced to the Pfaffian identit (2.8a) (2.8b) pf (d 0 d d 2 a )pf (e a ) pf (d 0 d e a )pf (d 2 a ) + pf (d 0 d 2 e a )pf (d a ) pf (d d 2 e a )pf (d 0 a ) 0 (2.9a) and Eq. (2.8b) is reduced to the Pfaffian identit 5[pf (e a )pf (d 0 d d 4 a ) pf (d 0 d e a )pf (d 4 a ) pf (d 0 a )pf (d d 4 e a ) + pf (d 0 d 4 e a )pf (d a )] 5[pf (e a )pf (d 0 d 2 d 3 a ) + pf (d 0 d 2 e a )pf (d 3 a ) pf (d 0 a )pf (d 2 d 3 e a ) pf (d 0 d 3 e a )pf (d 2 a )] 0. Thus we have proved that the N-soliton solutions given in Eq. (2.4) actuall satisf the bilinear equations Eq. (.9). 3. Resonance phenomena of solitons In this section we discuss the details of interactions between two solitons using the 2-soliton solutions of the coupled higher-order Ito equation amongst which are resonant solitons and we compare their interaction properties with similar solutions for the higher-order Ito equation. The interactions are classified into four tpes depending on the value of the phase shift a 2. In the accompaning figures we plot onl u = 2(ln f ) of the coupled higher-order Ito equation Eq. (.). For the case of the a 2 being finite the resulting solution Eq. (2.a) represents regular interaction of two solitons in which the larger soliton takes over the smaller soliton (see Fig. ). For the case of the solution Eq. (2.a) under a resonant condition a 2 = 0 we show that two solitons fuse into one soliton after colliding with each other (see Fig. 2) or that one soliton splits into two solitons in the resonant state (see Fig. 3). For the case of the solution Eq. (2.a) under another resonant condition a 2 we show that two solitons fuse into one soliton after colliding with each other in the resonant state and then this splits into two solitons at the end of the resonant state (see Fig. 4). For the case of the solution Eq. (2.a) under a quasi-resonant condition a 2 0 the resulting solution Eq. (2.a) shows that a higher soliton splits into two solitons as it approaches a lower soliton then one of the two solitons moves and collides with the lower soliton and finall the lower soliton echanges energ with the higher soliton b fusing one of the two solitons (see Fig. 5). (2.9b)

6 26 Y. Zhang et al. / J. Math. Anal. Appl. 394 (202) 2 28 Fig. 2. The plot of two solitons fusing into a large soliton to the new coupled higher-order Ito equation. The parameters used are a = p 2 p b 2 = a 2p 2 2 b p p 2 =. p 2 =.9 a 2 = 2 = 0. The left figure shows a three-dimensional plot and the right figure shows a contour map. Fig. 3. The plot of one soliton splitting into two solitons to the new coupled higher-order Ito equation. The parameters used are a = p 2 p b 2 = a 2p 2 2 b p p 2 =.9 p 2 = a 2 = 2 = 0. The left figure shows a three-dimensional plot and the right figure shows a contour map. 4. Conclusions A new tpe of coupled higher-order Ito equation has been given and N-soliton solutions have been obtained in the form of Pfaffians. The result that the coupled higher-order Ito equation possesses N-soliton solutions suggests this sstem might be a candidate of integrable equations. In this aspect we will present a Bäcklund transformation for Eq. (.0) to confirm the integrabilit of the coupled higher-order Ito equation; details will be given elsewhere. For more parameters than the higher-order Ito equation Eq. (.7) the 2-soliton solutions of Eq. (.0) ehibit some special phenomena such as one soliton undergoing fission into two solitons or two solitons fusing into one soliton at the resonant state after colliding with each other. Owing to onl needing two solitons to analze fission fusion and mied

7 Y. Zhang et al. / J. Math. Anal. Appl. 394 (202) Fig. 4. The plot of two solitons fusing into one soliton and then splitting into two solitons to the new coupled higher-order Ito equation. The parameters used are a = a 2 = p = p 2 =.5 b = 0 7 b 2 = 0 9 = 0. The left figure shows a three-dimensional plot and the right figure shows a contour map. Fig. 5. The plot of two solitons splitting into three solitons and then fusing into two solitons to the new coupled higher-order Ito equation. The parameters used are a = p 2 p b 2 = a 2p 2 2 b p p 2 =.9 p 2 =.2 a 2 = 2 b = b 2 = 3 = 0. The left figure shows a three-dimensional plot and the right figure shows a contour map. collision phenomena we believe that three or more solitons must ehibit more special phenomena which have not observed before. We also have not considered the positive and negative nature of a a a 2. In fact for the case of a 2 being negative and finite the resulting solution Eq. (2.a) could represent one particular phenomenon in which two regular solitons transmute into two singular solitons after colliding with each other (see Fig. 6). Although we have obtained a novel coupled higher-order Ito sstem in bilinear form it is an interesting problem to derive a coupled higher-order Ito sstem without the auiliar variable. Detailed studies of these problems are left for the future.

8 28 Y. Zhang et al. / J. Math. Anal. Appl. 394 (202) () (2) (3) (4) (5) (6) (7) (8) Fig. 6. The plot of two solitons transmuting into two singular solitons after colliding with each other to the new coupled higher-order Ito equation. The parameters used are a = 2 a 2 = 4 p = p 2 =.5 b = 0 b 2 = 70 = 0 where ()t = 5 (2)t =.5 (3)t = 0.6 (4)t = 0.2 (5)t = 0.27 (2)t = 0. (3)t = 0 (4)t = Acknowledgments The authors would like to epress their sincere thanks to the referee for his valuable comments. This work is supported b the National Natural Science Foundation of China (No ) Zhejiang Innovation Project (No T200905). This work was also supported in part b an Established Researcher grant a CAS facult development grant and a CAS Dean research grant of the Universit of South Florida. References [] R. Hirota The Direct Methods in Soliton Theor Cambridge Universit Press 2004 (Edited and translated into English b A. Nagai J.J.C. Nimmo C.R. Gilson). [2] R. Hirota Eact soliton of the KdV equation for multiple collisions of solitons Phs. Rev. Lett. 27 (97) [3] R. Hirota X.B. Hu X.Y. Tang A vector potential KdV equation and vector Ito equation: soliton solutions bilinear Bäcklund transformations and La pairs J. Math. Anal. Appl. 288 (2003) [4] M. Ito An etension of nonlinear evolution equations of the KdV(mKdV) tpe to higher orders J. Phs. Soc. Japan 49 (980) [5] Y. Matsuno Properties of conservation laws of nonlinear evolution equations J. Phs. Soc. Japan 59 (990) [6] Q.P. Liu Hamiltonian structures for Ito s equation Phs. Lett. A 277 (2000) [7] J. Springael X.B. Hu I. Loris Bilinear characterization of higher order Ito equations J. Phs. Soc. Japan 65 (996) [8] C.X. Li Y.B. Zeng Soliton solutions to a higher order Ito equation: Pfaffian technique Phs. Lett. A 363 (2007) 4. [9] X.B. Hu A Bäcklund transformation and nonlinear superposition formula of a higher order Ito equation J. Phs. A: Math. Gen. 26 (993) [0] H. Ono I. Nakata Reflection and transmission of an ion-acoustic soliton at a step-like inhomogeneit J. Phs. Soc. Japan 63 (994) [] M. Hisakado Breather trapping mechanism in piecewise homogeneous DNA Phs. Lett. A 227 (997) [2] S. Isojima R. Willo J. Satsuma On various solutions of the coupled KP equation J. Phs. A: Math. Gen. 35 (2002) [3] Y. Ohta R. Hirota New tpe of soliton equation J. Phs. Soc. Japan 76 (2007) [4] M.J. Ablowitz Z.H. Musslimani Discrete vector spatial solitons in a nonlinear waveguide arra Phs. Rev. E 65 (2002) [5] Y. Zhang J.J. Yan Soliton resonance of the NI-BKP equation AIP Conf. Proc. 22 (200) [6] S. Wang X.Y. Tang S.Y. Lou Soliton fission and fusion: Burgers equation and Sharma Tasso Olver equation Chaos Solitons Fractals 2 (2004) [7] Z.J. Lian S.Y. Lou Smmetries and eact solutions of the Sharma Tass Olver equation Nonlinear Anal. 63 (2005) [8] L.M. Alonso E.M. Reus Eotic coherent structures in the Dave Stewartson equation Inverse Problems 8 (992) [9] X.Y. Tang J.M. Li S.Y. Lou Reflection and reconnection interactions of resonant dromions Phs. Scr. 75 (2007) [20] Y. Kodama Young diagrams and N-soliton solutions of the KP equation J. Phs. A: Math. Gen. 37 (2004) [2] Y. Kodama M. Oikawa H. Tsuji Soliton solutions of the KP equation with V-shape initial waves J. Phs. A 42 (2009) [22] C.R. Gilson Generalizing the KP hierarchies: Pfaffian hierarchies Theoret. Math. Phs. 33 (2002)