Journal of Mathematical Analysis and Applications
|
|
- Terence Wheeler
- 5 years ago
- Views:
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)
Generalized bilinear differential equations
Generalized bilinear differential equations Wen-Xiu Ma Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA Abstract We introduce a kind of bilinear differential
More informationLump solutions to dimensionally reduced p-gkp and p-gbkp equations
Nonlinear Dyn DOI 10.1007/s11071-015-2539- ORIGINAL PAPER Lump solutions to dimensionally reduced p-gkp and p-gbkp equations Wen Xiu Ma Zhenyun Qin Xing Lü Received: 2 September 2015 / Accepted: 28 November
More informationGrammian and Pfaffian solutions as well as Pfaffianization for a (3+1)-dimensional generalized shallow water equation
Grammian and Pfaffian solutions as well as Pfaffianization for a (3+1)-dimensional generalized shallow water equation Tang Ya-Ning( 唐亚宁 ) a), Ma Wen-Xiu( 马文秀 ) b), and Xu Wei( 徐伟 ) a) a) Department of
More informationSolitary Wave Solutions of KP equation, Cylindrical KP Equation and Spherical KP Equation
Commun. Theor. Phs. 67 (017) 07 11 Vol. 67 No. Februar 1 017 Solitar Wave Solutions of KP equation Clindrical KP Equation and Spherical KP Equation Xiang-Zheng Li ( 李向正 ) 1 Jin-Liang Zhang ( 张金良 ) 1 and
More informationLUMP AND INTERACTION SOLUTIONS TO LINEAR (4+1)-DIMENSIONAL PDES
Acta Mathematica Scientia 2019 39B(2): 498 508 https://doi.org/10.1007/s10473-019-0214-6 c Wuhan Institute Physics and Mathematics Chinese Academy of Sciences 2019 http://actams.wipm.ac.cn LUMP AND INTERACTION
More informationEXACT BREATHER-TYPE SOLUTIONS AND RESONANCE-TYPE SOLUTIONS OF THE (2+1)-DIMENSIONAL POTENTIAL BURGERS SYSTEM
EXACT BREATHER-TYPE SOLUTIONS AND RESONANCE-TYPE SOLUTIONS OF THE (+)-DIMENSIONAL POTENTIAL BURGERS SYSTEM YEQIONG SHI College of Science Guangxi University of Science Technology Liuzhou 545006 China E-mail:
More informationIMA Preprint Series # 2050
FISSION, FUSION AND ANNIHILATION IN THE INTERACTION OF LOCALIZED STRUCTURES FOR THE (+)-DIMENSIONAL ENERALIZED BROER-KAUP SYSTEM B Emmanuel Yomba and Yan-ze Peng IMA Preprint Series # ( Ma ) INSTITUTE
More informationMultiple-Soliton Solutions for Extended Shallow Water Wave Equations
Studies in Mathematical Sciences Vol. 1, No. 1, 2010, pp. 21-29 www.cscanada.org ISSN 1923-8444 [Print] ISSN 1923-8452 [Online] www.cscanada.net Multiple-Soliton Solutions for Extended Shallow Water Wave
More informationPainlevé Property and Exact Solutions to a (2 + 1) Dimensional KdV-mKdV Equation
Journal of Applied Mathematics and Phsics, 5, 3, 697-76 Published Online June 5 in SciRes http://wwwscirporg/ournal/amp http://ddoiorg/36/amp53683 Painlevé Propert and Eact Solutions to a ( + ) Dimensional
More informationA MULTI-COMPONENT LAX INTEGRABLE HIERARCHY WITH HAMILTONIAN STRUCTURE
Pacific Journal of Applied Mathematics Volume 1, Number 2, pp. 69 75 ISSN PJAM c 2008 Nova Science Publishers, Inc. A MULTI-COMPONENT LAX INTEGRABLE HIERARCHY WITH HAMILTONIAN STRUCTURE Wen-Xiu Ma Department
More informationNEW PERIODIC WAVE SOLUTIONS OF (3+1)-DIMENSIONAL SOLITON EQUATION
Liu, J., et al.: New Periodic Wave Solutions of (+)-Dimensional Soliton Equation THERMAL SCIENCE: Year 7, Vol., Suppl., pp. S69-S76 S69 NEW PERIODIC WAVE SOLUTIONS OF (+)-DIMENSIONAL SOLITON EQUATION by
More informationExact Periodic Solitary Wave and Double Periodic Wave Solutions for the (2+1)-Dimensional Korteweg-de Vries Equation*
Exact Periodic Solitary Wave Double Periodic Wave Solutions for the (+)-Dimensional Korteweg-de Vries Equation* Changfu Liu a Zhengde Dai b a Department of Mathematics Physics Wenshan University Wenshan
More informationKdV extensions with Painlevé property
JOURNAL OF MATHEMATICAL PHYSICS VOLUME 39, NUMBER 4 APRIL 1998 KdV etensions with Painlevé propert Sen-ue Lou a) CCAST (World Laborator), P. O. Bo 8730, Beijing 100080, People s Republic of China and Institute
More informationFrom bell-shaped solitary wave to W/M-shaped solitary wave solutions in an integrable nonlinear wave equation
PRAMANA c Indian Academ of Sciences Vol. 74, No. journal of Januar 00 phsics pp. 9 6 From bell-shaped solitar wave to W/M-shaped solitar wave solutions in an integrable nonlinear wave equation AIYONG CHEN,,,
More informationExotic Localized Structures of the (2+1)-Dimensional Nizhnik-Novikov- Veselov System Obtained via the Extended Homogeneous Balance Method
Exotic Localized Structures of the (2+1)-Dimensional Nizhnik-Novikov- Veselov System Obtained via the Extended Homogeneous Balance Method Chao-Qing Dai a, Guo-quan Zhou a, and Jie-Fang Zhang b a Department
More informationMultisoliton solutions, completely elastic collisions and non-elastic fusion phenomena of two PDEs
Pramana J. Phys. (2017) 88:86 DOI 10.1007/s12043-017-1390-3 Indian Academy of Sciences Multisoliton solutions completely elastic collisions and non-elastic fusion phenomena of two PDEs MST SHEKHA KHATUN
More informationLump solutions to a generalized Bogoyavlensky-Konopelchenko equation
Front. Math. China 2018, 13(3): 525 534 https://doi.org/10.1007/s11464-018-0694-z Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation Shou-Ting CHEN 1, Wen-Xiu MA 2,3,4 1 School of Mathematics
More informationarxiv:nlin/ v1 [nlin.si] 7 Sep 2005
NONSINGULAR POSITON AND COMPLEXITON SOLUTIONS FOR THE COUPLED KDV SYSTEM arxiv:nlin/5918v1 [nlin.si] 7 Sep 25 H. C. HU 1,2, BIN TONG 1 AND S. Y. LOU 1,3 1 Department of Physics, Shanghai Jiao Tong University,
More informationDeformation rogue wave to the (2+1)-dimensional KdV equation
Nonlinear Dyn DOI 10.1007/s11071-017-3757-x ORIGINAL PAPER Deformation rogue wave to the +1-dimensional KdV equation Xiaoen Zhang Yong Chen Received: 9 November 01 / Accepted: 4 May 017 Springer Science+Business
More informationToward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach
Commun. Theor. Phys. 57 (2012) 5 9 Vol. 57, No. 1, January 15, 2012 Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach G. Darmani, 1, S. Setayeshi,
More informationNew approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations
Volume 28, N. 1, pp. 1 14, 2009 Copyright 2009 SBMAC ISSN 0101-8205 www.scielo.br/cam New approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations HASSAN A. ZEDAN Mathematics
More informationA SEARCH FOR LUMP SOLUTIONS TO A COMBINED FOURTH-ORDER NONLINEAR PDE IN (2+1)-DIMENSIONS
Journal of Applied Analysis and Computation Volume *, Number *, * *, 1 15 Website:http://jaac.ijournal.cn DOI:10.11948/*.1 A SEARCH FOR LUMP SOLUTIONS TO A COMBINED FOURTH-ORDER NONLINEAR PDE IN (2+1)-DIMENSIONS
More informationSolving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm
Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm Wen-Xiu Ma and Zuonong Zhu Department of Mathematics and Statistics, University of South Florida, Tampa,
More informationModified Simple Equation Method and its Applications for some Nonlinear Evolution Equations in Mathematical Physics
Modified Simple Equation Method and its Applications for some Nonlinear Evolution Equations in Mathematical Physics Elsayed M. E. Zayed Mathematics department, Faculty of Science Zagazig University, Zagazig,
More informationNew Approach of ( Ǵ/G ) Expansion Method. Applications to KdV Equation
Journal of Mathematics Research; Vol. 6, No. ; ISSN 96-9795 E-ISSN 96-989 Published by Canadian Center of Science and Education New Approach of Ǵ/G Expansion Method. Applications to KdV Equation Mohammad
More informationBreaking soliton equations and negative-order breaking soliton equations of typical and higher orders
Pramana J. Phys. (2016) 87: 68 DOI 10.1007/s12043-016-1273-z c Indian Academy of Sciences Breaking soliton equations and negative-order breaking soliton equations of typical and higher orders ABDUL-MAJID
More informationNew Analytical Solutions For (3+1) Dimensional Kaup-Kupershmidt Equation
International Conference on Computer Technology and Science (ICCTS ) IPCSIT vol. 47 () () IACSIT Press, Singapore DOI:.776/IPCSIT..V47.59 New Analytical Solutions For () Dimensional Kaup-Kupershmidt Equation
More informationA New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent Sources
Commun. Theor. Phys. Beijing, China 54 21 pp. 1 6 c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 1, July 15, 21 A New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent
More informationGroup analysis, nonlinear self-adjointness, conservation laws, and soliton solutions for the mkdv systems
ISSN 139-5113 Nonlinear Analysis: Modelling Control, 017, Vol., No. 3, 334 346 https://doi.org/10.15388/na.017.3.4 Group analysis, nonlinear self-adjointness, conservation laws, soliton solutions for the
More informationHOMOTOPY ANALYSIS METHOD FOR SOLVING KDV EQUATIONS
Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (21), 89 98 HOMOTOPY ANALYSIS METHOD FOR SOLVING KDV EQUATIONS Hossein Jafari and M. A. Firoozjaee Abstract.
More informationBäcklund transformation and soliton solutions in terms of the Wronskian for the Kadomtsev Petviashvili-based system in fluid dynamics
Pramana J. Phys. (08) 90:45 https://doi.org/0.007/s043-08-53- Indian Academy of Sciences Bäcklund transformation and soliton solutions in terms of the Wronskian for the Kadomtsev Petviashvili-based system
More informationA NEW VARIABLE-COEFFICIENT BERNOULLI EQUATION-BASED SUB-EQUATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS
U.P.B. Sci. Bull., Series A, Vol. 76, Iss., 014 ISSN 1-707 A NEW VARIABLE-COEFFICIENT BERNOULLI EQUATION-BASED SUB-EQUATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS Bin Zheng 1 In this paper,
More informationExact Interaction Solutions of an Extended (2+1)-Dimensional Shallow Water Wave Equation
Commun. Theor. Phys. 68 (017) 165 169 Vol. 68, No., August 1, 017 Exact Interaction Solutions of an Extended (+1)-Dimensional Shallow Water Wave Equation Yun-Hu Wang ( 王云虎 ), 1, Hui Wang ( 王惠 ), 1, Hong-Sheng
More information2. The generalized Benjamin- Bona-Mahony (BBM) equation with variable coefficients [30]
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.12(2011) No.1,pp.95-99 The Modified Sine-Cosine Method and Its Applications to the Generalized K(n,n) and BBM Equations
More informationResearch Article Two Different Classes of Wronskian Conditions to a (3 + 1)-Dimensional Generalized Shallow Water Equation
International Scholarly Research Network ISRN Mathematical Analysis Volume 2012 Article ID 384906 10 pages doi:10.5402/2012/384906 Research Article Two Different Classes of Wronskian Conditions to a 3
More informationTwo-Componet Coupled KdV Equations and its Connection with the Generalized Harry Dym Equations
Two-Componet Coupled KdV Equations and its Connection with the Generalized Harr Dm Equations Ziemowit Popowicz arxiv:1210.5822v1 [nlin.si] 22 Oct 2012 Ma 5, 2014 Institute of Theoretical Phsics, Universit
More informationScattering of Solitons of Modified KdV Equation with Self-consistent Sources
Commun. Theor. Phys. Beijing, China 49 8 pp. 89 84 c Chinese Physical Society Vol. 49, No. 4, April 5, 8 Scattering o Solitons o Modiied KdV Equation with Sel-consistent Sources ZHANG Da-Jun and WU Hua
More informationWronskians, Generalized Wronskians and Solutions to the Korteweg-de Vries Equation
Wronskians, Generalized Wronskians and Solutions to the Korteweg-de Vries Equation Wen-Xiu Ma Department of Mathematics, University of South Florida, Tampa, FL 33620-5700, USA arxiv:nlin/0303068v1 [nlin.si]
More informationNew explicit solitary wave solutions for (2 + 1)-dimensional Boussinesq equation and (3 + 1)-dimensional KP equation
Physics Letters A 07 (00) 107 11 www.elsevier.com/locate/pla New explicit solitary wave solutions for ( + 1)-dimensional Boussinesq equation and ( + 1)-dimensional KP equation Yong Chen, Zhenya Yan, Honging
More informationA NEW APPROACH FOR SOLITON SOLUTIONS OF RLW EQUATION AND (1+2)-DIMENSIONAL NONLINEAR SCHRÖDINGER S EQUATION
A NEW APPROACH FOR SOLITON SOLUTIONS OF RLW EQUATION AND (+2-DIMENSIONAL NONLINEAR SCHRÖDINGER S EQUATION ALI FILIZ ABDULLAH SONMEZOGLU MEHMET EKICI and DURGUN DURAN Communicated by Horia Cornean In this
More informationExact Solutions for the Nonlinear (2+1)-Dimensional Davey-Stewartson Equation Using the Generalized ( G. )-Expansion Method
Journal of Mathematics Research; Vol. 6, No. ; 04 ISSN 96-9795 E-ISSN 96-9809 Published by Canadian Center of Science and Education Exact Solutions for the Nonlinear +-Dimensional Davey-Stewartson Equation
More informationLump-type solutions to nonlinear differential equations derived from generalized bilinear equations
International Journal of Modern Physics B Vol. 30 2016) 1640018 16 pages) c World Scientific Publishing Company DOI: 10.1142/S021797921640018X Lump-type solutions to nonlinear differential equations derived
More informationApplication of Laplace Adomian Decomposition Method for the soliton solutions of Boussinesq-Burger equations
Int. J. Adv. Appl. Math. and Mech. 3( (05 50 58 (ISSN: 347-59 IJAAMM Journal homepage: www.ijaamm.com International Journal of Advances in Applied Mathematics and Mechanics Application of Laplace Adomian
More informationAn Improved F-Expansion Method and Its Application to Coupled Drinfel d Sokolov Wilson Equation
Commun. Theor. Phys. (Beijing, China) 50 (008) pp. 309 314 c Chinese Physical Society Vol. 50, No., August 15, 008 An Improved F-Expansion Method and Its Application to Coupled Drinfel d Sokolov Wilson
More informationPeriodic and Solitary Wave Solutions of the Davey-Stewartson Equation
Applied Mathematics & Information Sciences 4(2) (2010), 253 260 An International Journal c 2010 Dixie W Publishing Corporation, U. S. A. Periodic and Solitary Wave Solutions of the Davey-Stewartson Equation
More informationarxiv: v1 [nlin.ps] 3 Sep 2009
Soliton, kink and antikink solutions o a 2-component o the Degasperis-Procesi equation arxiv:0909.0659v1 [nlin.ps] 3 Sep 2009 Jiangbo Zhou, Liin Tian, Xinghua Fan Nonlinear Scientiic Research Center, Facult
More informationNew Jacobi Elliptic Function Solutions for Coupled KdV-mKdV Equation
New Jacobi Elliptic Function Solutions for Coupled KdV-mKdV Equation Yunjie Yang Yan He Aifang Feng Abstract A generalized G /G-expansion method is used to search for the exact traveling wave solutions
More informationThe (2+1) and (3+1)-Dimensional CBS Equations: Multiple Soliton Solutions and Multiple Singular Soliton Solutions
The (+1) and (3+1)-Dimensional CBS Equations: Multiple Soliton Solutions and Multiple Singular Soliton Solutions Abdul-Majid Wazwaz Department of Mathematics, Saint Xavier University, Chicago, IL 60655,
More informationA Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning
Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 22, 1097-1106 A Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning M. T. Darvishi a,, S.
More informationSoliton solutions of Hirota equation and Hirota-Maccari system
NTMSCI 4, No. 3, 231-238 (2016) 231 New Trends in Mathematical Sciences http://dx.doi.org/10.20852/ntmsci.2016115853 Soliton solutions of Hirota equation and Hirota-Maccari system M. M. El-Borai 1, H.
More informationInvariant Sets and Exact Solutions to Higher-Dimensional Wave Equations
Commun. Theor. Phys. Beijing, China) 49 2008) pp. 9 24 c Chinese Physical Society Vol. 49, No. 5, May 5, 2008 Invariant Sets and Exact Solutions to Higher-Dimensional Wave Equations QU Gai-Zhu, ZHANG Shun-Li,,2,
More informationAuto-Bäcklund transformation and exact solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear terms of any order
Physics Letters A 305 (00) 377 38 www.elsevier.com/locate/pla Auto-Bäcklund transformation and exact solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear terms of any
More informationThe Solitary Wave Solutions of Zoomeron Equation
Applied Mathematical Sciences, Vol. 5, 011, no. 59, 943-949 The Solitary Wave Solutions of Zoomeron Equation Reza Abazari Deparment of Mathematics, Ardabil Branch Islamic Azad University, Ardabil, Iran
More informationSolutions of the nonlocal nonlinear Schrödinger hierarchy via reduction
Solutions of the nonlocal nonlinear Schrödinger hierarchy via reduction Kui Chen, Da-jun Zhang Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China June 25, 208 arxiv:704.0764v [nlin.si]
More informationRelation between Periodic Soliton Resonance and Instability
Proceedings of Institute of Mathematics of NAS of Ukraine 004 Vol. 50 Part 1 486 49 Relation between Periodic Soliton Resonance and Instability Masayoshi TAJIRI Graduate School of Engineering Osaka Prefecture
More informationNew Formal Solutions of Davey Stewartson Equation via Combined tanh Function Method with Symmetry Method
Commun. Theor. Phys. Beijing China 7 007 pp. 587 593 c International Academic Publishers Vol. 7 No. April 5 007 New Formal Solutions of Davey Stewartson Equation via Combined tanh Function Method with
More informationNew Homoclinic and Heteroclinic Solutions for Zakharov System
Commun. Theor. Phys. 58 (2012) 749 753 Vol. 58, No. 5, November 15, 2012 New Homoclinic and Heteroclinic Solutions for Zakharov System WANG Chuan-Jian ( ), 1 DAI Zheng-De (à ), 2, and MU Gui (½ ) 3 1 Department
More informationNew Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four and Drinfeld-Sokolov Equations
ISSN 1749-3889 print), 1749-3897 online) International Journal of Nonlinear Science Vol.008) No.1,pp.4-5 New Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four and Drinfeld-Sokolov Euations
More informationExact Solutions of The Regularized Long-Wave Equation: The Hirota Direct Method Approach to Partially Integrable Equations
Thai Journal of Mathematics Volume 5(2007) Number 2 : 273 279 www.math.science.cmu.ac.th/thaijournal Exact Solutions of The Regularized Long-Wave Equation: The Hirota Direct Method Approach to Partially
More informationSOLITON SOLUTIONS OF SHALLOW WATER WAVE EQUATIONS BY MEANS OF G /G EXPANSION METHOD
Journal of Applied Analysis and Computation Website:http://jaac-online.com/ Volume 4, Number 3, August 014 pp. 1 9 SOLITON SOLUTIONS OF SHALLOW WATER WAVE EQUATIONS BY MEANS OF G /G EXPANSION METHOD Marwan
More informationNew Integrable Decomposition of Super AKNS Equation
Commun. Theor. Phys. (Beijing, China) 54 (2010) pp. 803 808 c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 5, November 15, 2010 New Integrable Decomposition of Super AKNS Equation JI Jie
More informationMulti-Soliton Solutions to Nonlinear Hirota-Ramani Equation
Appl. Math. Inf. Sci. 11, No. 3, 723-727 (2017) 723 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/110311 Multi-Soliton Solutions to Nonlinear Hirota-Ramani
More informationYimin Wei a,b,,1, Xiezhang Li c,2, Fanbin Bu d, Fuzhen Zhang e. Abstract
Linear Algebra and its Applications 49 (006) 765 77 wwwelseviercom/locate/laa Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices Application of perturbation theory
More informationarxiv:nlin/ v1 [nlin.ps] 12 Jul 2001
Higher dimensional Lax pairs of lower dimensional chaos and turbulence systems arxiv:nlin/0107028v1 [nlin.ps] 12 Jul 2001 Sen-yue Lou CCAST (World Laboratory), PO Box 8730, Beijing 100080, P. R. China
More informationA note on the G /G - expansion method
A note on the G /G - expansion method Nikolai A. Kudryashov Department of Applied Mathematics, National Research Nuclear University MEPHI, Kashirskoe Shosse, 115409 Moscow, Russian Federation Abstract
More informationThe (G'/G) - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics
Vol.3, Issue., Jan-Feb. 3 pp-369-376 ISSN: 49-6645 The ('/) - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics J.F.Alzaidy Mathematics Department, Faculty
More informationThe Traveling Wave Solutions for Nonlinear Partial Differential Equations Using the ( G. )-expansion Method
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.8(009) No.4,pp.435-447 The Traveling Wave Solutions for Nonlinear Partial Differential Equations Using the ( )-expansion
More informationSupersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and. Applications
Supersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and arxiv:1802.04922v1 [nlin.si] 14 Feb 2018 Applications Hui Mao, Q. P. Liu and Lingling Xue Department of Mathematics, China University
More informationExotic localized structures based on the symmetrical lucas function of the (2+1)-dimensional generalized Nizhnik-Novikov-Veselov system
Turk J Phs 35 (11), 41 56. c TÜBİTAK doi:1.396/fiz-19-1 Eotic localized structures based on the smmetrical lucas function of the (+1)-dimensional generalized Nizhnik-Novikov-Veselov sstem Emad A-B. ABDEL-SALAM
More informationA Block Matrix Loop Algebra and Bi-Integrable Couplings of the Dirac Equations
East Asian Journal on Applied Mathematics Vol. 3, No. 3, pp. 171-189 doi: 10.4208/eajam.250613.260713a August 2013 A Block Matrix Loop Algebra and Bi-Integrable Couplings of the Dirac Equations Wen-Xiu
More informationNew Variable Separation Excitations of a (2+1)-dimensional Broer-Kaup- Kupershmidt System Obtained by an Extended Mapping Approach
New ariable Separation Ecitations of a (+1)-dimensional Broer-Kaup- Kupershmidt System Obtained by an Etended Mapping Approach Chun-Long Zheng a,b, Jian-Ping Fang a, and Li-Qun Chen b a Department of Physics,
More informationPainlevé Test for the Certain (2+1)-Dimensional Nonlinear Evolution Equations. Abstract
Painlevé Test for the Certain (2+1)-Dimensional Nonlinear Evolution Equations T. Alagesan and Y. Chung Department of Information and Communications, Kwangju Institute of Science and Technology, 1 Oryong-dong,
More informationKeywords: Exp-function method; solitary wave solutions; modified Camassa-Holm
International Journal of Modern Mathematical Sciences, 2012, 4(3): 146-155 International Journal of Modern Mathematical Sciences Journal homepage:www.modernscientificpress.com/journals/ijmms.aspx ISSN:
More informationDynamics of multiple pendula without gravity
Chaotic Modeling and Simulation (CMSIM) 1: 57 67, 014 Dnamics of multiple pendula without gravit Wojciech Szumiński Institute of Phsics, Universit of Zielona Góra, Poland (E-mail: uz88szuminski@gmail.com)
More informationExact Solutions for Generalized Klein-Gordon Equation
Journal of Informatics and Mathematical Sciences Volume 4 (0), Number 3, pp. 35 358 RGN Publications http://www.rgnpublications.com Exact Solutions for Generalized Klein-Gordon Equation Libo Yang, Daoming
More informationA multiple Riccati equations rational expansion method and novel solutions of the Broer Kaup Kupershmidt system
Chaos, Solitons and Fractals 30 (006) 197 03 www.elsevier.com/locate/chaos A multiple Riccati equations rational expansion method and novel solutions of the Broer Kaup Kupershmidt system Qi Wang a,c, *,
More informationarxiv:nlin/ v1 [nlin.si] 25 Sep 2006
Remarks on the conserved densities of the Camassa-Holm equation Amitava Choudhuri 1, B. Talukdar 1a and S. Ghosh 1 Department of Physics, Visva-Bharati University, Santiniketan 73135, India Patha Bhavana,
More informationApplication of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics
PRMN c Indian cademy of Sciences Vol. 77, No. 6 journal of December 011 physics pp. 103 109 pplication of the trial equation method for solving some nonlinear evolution equations arising in mathematical
More informationExact traveling wave solutions of nonlinear variable coefficients evolution equations with forced terms using the generalized.
Exact traveling wave solutions of nonlinear variable coefficients evolution equations with forced terms using the generalized expansion method ELSAYED ZAYED Zagazig University Department of Mathematics
More informationNew Exact Solutions for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli Equation
Applied Mathematical Sciences, Vol. 6, 2012, no. 12, 579-587 New Exact Solutions for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli Equation Ying Li and Desheng Li School of Mathematics and System Science
More informationResearch Article Chaos and Control of Game Model Based on Heterogeneous Expectations in Electric Power Triopoly
Discrete Dnamics in Nature and Societ Volume 29, Article ID 469564, 8 pages doi:.55/29/469564 Research Article Chaos and Control of Game Model Based on Heterogeneous Epectations in Electric Power Triopol
More informationTravelling Wave Solutions for the Gilson-Pickering Equation by Using the Simplified G /G-expansion Method
ISSN 1749-3889 (print, 1749-3897 (online International Journal of Nonlinear Science Vol8(009 No3,pp368-373 Travelling Wave Solutions for the ilson-pickering Equation by Using the Simplified /-expansion
More informationBuilding Generalized Lax Integrable KdV and MKdV Equations with Spatiotemporally Varying Coefficients
Journal of Physics: Conference Series OPEN ACCESS Building Generalized Lax Integrable KdV and MKdV Equations with Spatiotemporally Varying Coefficients To cite this article: M Russo and S R Choudhury 2014
More informationSimilarity Reductions of (2+1)-Dimensional Multi-component Broer Kaup System
Commun. Theor. Phys. Beijing China 50 008 pp. 803 808 c Chinese Physical Society Vol. 50 No. 4 October 15 008 Similarity Reductions of +1-Dimensional Multi-component Broer Kaup System DONG Zhong-Zhou 1
More informationSUB-MANIFOLD AND TRAVELING WAVE SOLUTIONS OF ITO S 5TH-ORDER MKDV EQUATION
Journal of Applied Analysis and Computation Volume 7, Number 4, November 07, 47 430 Website:http://jaac-online.com/ DOI:0.94/0706 SUB-MANIFOLD AND TRAVELING WAVE SOLUTIONS OF ITO S 5TH-ORDER MKDV EQUATION
More informationBäcklund transformation and special solutions for Drinfeld Sokolov Satsuma Hirota system of coupled equations
arxiv:nlin/0102001v1 [nlin.si] 1 Feb 2001 Bäcklund transformation and special solutions for Drinfeld Sokolov Satsuma Hirota system of coupled equations Ayşe Karasu (Kalkanli) and S Yu Sakovich Department
More information(2.5) 1. Solve the following compound inequality and graph the solution set.
Intermediate Algebra Practice Final Math 0 (7 th ed.) (Ch. -) (.5). Solve the following compound inequalit and graph the solution set. 0 and and > or or (.7). Solve the following absolute value inequalities.
More informationarxiv:nlin/ v1 [nlin.si] 25 Jun 2004
arxiv:nlin/0406059v1 [nlinsi] 25 Jun 2004 Resonance and web structure in discrete soliton systems: the two-dimensional Toda lattice and its fully discrete and ultra-discrete versions Ken-ichi Maruno 1
More informationNew Exact Solutions of the Modified Benjamin-Bona-Mahony Equation Yun-jie YANG and Li YAO
06 International Conference on Artificial Intelligence and Computer Science (AICS 06) ISBN: 978--60595-4-0 New Exact Solutions of the Modified Benamin-Bona-Mahony Equation Yun-ie YANG and Li YAO Department
More informationA new integrable system: The interacting soliton of the BO
Phys. Lett., A 204, p.336-342, 1995 A new integrable system: The interacting soliton of the BO Benno Fuchssteiner and Thorsten Schulze Automath Institute University of Paderborn Paderborn & Germany Abstract
More informationKINK DEGENERACY AND ROGUE WAVE FOR POTENTIAL KADOMTSEV-PETVIASHVILI EQUATION
THERMAL SCIENCE, Year 05, Vol. 9, No. 4, pp. 49-435 49 KINK DEGENERACY AND ROGUE WAVE FOR POTENTIAL KADOMTSEV-PETVIASHVILI EQUATION by Hong-Ying LUO a*, Wei TAN b, Zheng-De DAI b, and Jun LIU a a College
More informationComputers and Mathematics with Applications
Computers and Mathematics with Applications 60 (00) 3088 3097 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Symmetry
More informationSimultaneous Orthogonal Rotations Angle
ELEKTROTEHNIŠKI VESTNIK 8(1-2): -11, 2011 ENGLISH EDITION Simultaneous Orthogonal Rotations Angle Sašo Tomažič 1, Sara Stančin 2 Facult of Electrical Engineering, Universit of Ljubljana 1 E-mail: saso.tomaic@fe.uni-lj.si
More informationIntroduction to the Hirota bilinear method
Introduction to the Hirota bilinear method arxiv:solv-int/9708006v1 14 Aug 1997 J. Hietarinta Department of Physics, University of Turku FIN-20014 Turku, Finland e-mail: hietarin@utu.fi Abstract We give
More informationNew Application of the (G /G)-Expansion Method to Excite Soliton Structures for Nonlinear Equation
New Application of the /)-Expansion Method to Excite Soliton Structures for Nonlinear Equation Bang-Qing Li ac and Yu-Lan Ma b a Department of Computer Science and Technology Beijing Technology and Business
More informationThe General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method
Applied and Computational Mathematics 015; 4(5): 335-341 Published online August 16 015 (http://www.sciencepublishinggroup.com/j/acm) doi: 10.11648/j.acm.0150405.11 ISSN: 38-5605 (Print); ISSN: 38-5613
More informationTraveling Wave Solutions For Two Non-linear Equations By ( G G. )-expansion method
Traveling Wave Solutions For Two Non-linear Equations By ( )-expansion method Qinghua Feng Shandong University of Technology School of Science Zhangzhou Road 1, Zibo, 55049 China fqhua@sina.com Bin Zheng
More informationProlongation structure for nonlinear integrable couplings of a KdV soliton hierarchy
Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy Yu Fa-Jun School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang 110034, China Received
More informationMultisolitonic solutions from a Bäcklund transformation for a parametric coupled Korteweg-de Vries system
arxiv:407.7743v3 [math-ph] 3 Jan 205 Multisolitonic solutions from a Bäcklund transformation for a parametric coupled Korteweg-de Vries system L. Cortés Vega*, A. Restuccia**, A. Sotomayor* January 5,
More informationSolution of the Hirota Equation Using Lattice-Boltzmann and the Exponential Function Methods
Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 7, 307-315 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.7418 Solution of the Hirota Equation Using Lattice-Boltzmann and the
More information