nlin.SI 2021-04-14 0

New hierarchy of multiple soliton solutions for the (2+1)-dimensional Sawada-Kotera equation

A new transformation $u=4 ({\rm ln}f)_x$ that can formulate a quintic linear equation and a pair of Hirota's bilinear equations for the (2+1)-dimensional Sawada-Kotera (2DSK) equation is reported firstly, which enables one to obtain a new hierarchy o…

Authors: Ruoxia Yao, Yan Li, Senyue Lou

New hierar c h y of m ulti ple solit on solut ions o f the (2+1 )-dimensi onal Sa wada-Kotera eq uatio n Ruoxia Y ao 1 Scho ol of Computer Scienc e, Shaanxi Normal University, Xi’an, 710119, China Y an Li 2 Scho ol of Computer Scienc e, Shaanxi Normal University, Xi’an, 710119, China Senyue Lou 3 Scho ol of Physic al Scienc e and T e chno lo gy, Ni ngb o University, Ningb o, 315211 , China Abstract A new transfor mation u = 4(ln f ) x that can formulate a quint ic linear e quation and a pa ir of Hirota’s bilinear equations for the (2+ 1)-dimensional Sa wada-Kotera (2DSK) equation is repo r ted firstly , which enables one to obtain a new hierar ch y of multiple solito n so lutions of the 2DSK equation. It tells a crucial fact that a nonlinear par tia l different ial equation could p os sess tw o hierarchies of m ultiple soliton so lutions and the 2DSK equation is the firs t a nd only one found in this pap er. The quintic linear equation is solved by a pair of Hiro ta’s bilinea r equatio ns, of which one is the (2+1 )-dimensional bilinear SK equation o btained b y u = 2(ln f ) x , and the other is the bilinear KdV equation. The (1+1)-dimensio na l SK eq uation do es not p o ssess this prop erty . As another exa mple, a (3+1)-dimensiona l nonlinea r par tial differential equation p os s essing a pair o f Hirota’s bilinear equations, how ev er only b ear ing one hiera r ch y of multiple so liton s olutions is studied. Keywor d s: Multiple soliton solutio ns ; Dep endent v ariable transfo rmation; Q uin tic linear equation; Hiorta’s bilinear equation; (2+1 )-dimensional Sa wada-Kotera equatio n. 1. In tro duction Solitons of nonlinear partial differential equatio ns play imp ortant r oles in b oth fundamental theory and applications of nonlinea r s cience. As shown in [1], the soliton and solitar y wav e solutions that reflect a common nonlinea r phenomenon in nature provide ph ysical infor mation a nd more insights into the physical a nd mathematical asp ects of the problem thus lea ding to further applications. In the last tw o decades, more and more resear chers have taken their attent ions to the s tudy o f so litary wav e and soliton s olutions of co mpletely in tegra ble nonlinear evolution 1 Corresp onding author: rxy ao2@hotmail.com; r xy ao@snn u.edu.cn 2 Email: y anli@snn u.edu.cn 3 Email: louse nyue @nbu.edu.cn Pr eprint submitte d to Journal of L A T E X T emp lates Mar ch 9, 2020 equations. Up to now, we know that an integrable nonlinear par tial differen tial equatio n (PDE) usually po s sesses one hierarch y of multiple so liton solutions, ho wev er we find that the 2DSK equation p ossesses tw o hiera rchies o f multiple soliton solutions. This very sp ecial prop erty is nov el and first prop os ed in this pap er. The 2 DSK equation v t + v 5 x + 15 v x v xx + 15 v v 3 x + 45 v 2 v x + 5 v xxy + 15 v v y + 15 v x Z v y d x − 5 Z v y y d x = 0 , (1) where v = v ( x, y , t ), was fir s t pr op osed by Ko nop elchenk o and Dubrovsky [2], and has b een re- garded as a (2+ 1)-dimensional int egra ble genera lization o f the (1+1)-dimensiona l Saw ada-Ko tera equation [3]. Now it is k nown a ls o a s a member of the so- c alled CK P hierarchy [4]. The 2DSK hierarch y with almost the sa me form and only having different c o efficients of se veral terms of Eq. (1) are studied extensively , suc h that abundant results have b een reported in the Refs. [5, 6, 4, 7, 8 , 9, 10, 11, 1 2] and there in. Recent ly , man y ne w impo rtant t yp es o f solutions such as lump solutions, molecule soliton solutions and resona n t so liton solutions are r ep o rted constantly in Refs. [13, 14, 15, 16, 17]. 2. New transformation and a pair of Hirota’s biline ar equations of the 2DSK equation Eq. (1) reduces to u xt + u 6 x + 5 u 3 xy − 5 u y y + 15 u xx u 3 x + 15 u x u 4 x + 15 u x u xy + 15 u xx u y + 45 u 2 x u xx = 0 , (2) after introducing a p o tent ial v ariable transfor ma tion v = u x , u = u ( x, y , t ) , and u xt = ∂ 2 ∂ x∂ t u ( x, y , t ). Eq. (2) is the tw o (space)-dimensional extension of the famous KP equation first rep orted by Kadomtsev and Petviash vili [18] . In this pap er, we mainly consider the reduced Eq. (2 ). The 2DSK equation usually can b e written in Hiro ta ’s bilinear form b y expressing solutions in terms of a τ -function. If u = 2(ln τ ) x then τ ( x, y , t ) sa tisfies the Hirota’s bilinear equation ( D 6 x + D x D t − 5 D 3 x D y + 5 D 2 y ) τ · τ = 0 , (3) where the Hirota’s D -o per ator is de fined in Ref. [19]. In the following, we will show that we obtain different bilinear expres sions o f E q. (2). In Refs. [2 0, 2 1, 22, 2 3], Hietarinta provided a complete classification for KdV-type, mKdV- t yp e, sine- Gordon-type and complex bilinea r equations passing Hirota’s three-solito n condition. In Ref. [24], Cheng studied some decomp os itio ns including (1+1)-dimensiona l KdV+5 th -order KdV and Ito+K dV equa tions. The r esults presented in this pap er hav e not yet b een r epo rted therein or elsewhere. Starting with a dependent v ariable tra nsformation u = p (ln f ) x (4) 2 where f = f ( x, y , t ) and co mbin ing with the assumption f ≡ f 1 = 1 + exp( ξ 1 ) , ξ 1 = k 1 x + l 1 y + w 1 t, (5) instead of using the Hirota’s bilinear metho d, we get (see Ref. [25 ] for more details) p = 2 , w 1 = − ( k 6 1 + 5 k 3 1 l 1 − 5 l 2 1 ) /k 1 , (6) p = 4 , l 1 = − k 3 1 , w 1 = 9 k 5 1 , (7) and then a quint ic linear equation for Eq. (2) later cor resp onding to (7). The adv an tage is that the direct metho d c o uld av oid se e k ing for bilinear fo r m at the b eginning of constructing multiple soliton solutions. W e pr ov e that even a nonlinear PDE do es no t p ossess the so- c alled Hiro ta ’s bilinear fo rm appar ently [26, 27, 19], maybe we co uld obtain a new hierar ch y of r elated so lito n solutions. In this pap er, w e adopt the determined dep endent v ariable transforma tion u = 4(ln f ) x (8) instead of u = 2(ln f ) x to find soliton so lutio ns first of Eq. (2), a nd then use v → u x , namely , v = 4(ln f ) xx to obtain the multiple solito n solutions o f E q. (1 ). Substituting E q. (8 ) into Eq. (2) yields the following quintic linear equa tion P ≡ f 4 ( f 7 x + 5 f 4 xy + f 2 xt − 5 f x 2 y ) + f 3 (25 f 4 x f 3 x + 39 f 5 x f 2 x − 5 f 4 x f y − f 2 x f t − 7 f 6 x f x − 20 f 3 xy f x + 40 f 3 x f xy + 5 f x f 2 y + 30 f 2 x f 2 xy + 10 f xy f y − 2 f xt f x ) +2 f 2  − 10 f 3 x f x f y + f 2 x f t − 50 f 2 3 x f x − 135 f 4 x f x f 2 x − 9 f 5 x f 2 x − 90 f 2 x f x f xy + 75 f 3 x f 2 2 x − 5 f x f 2 y − 15 f 2 2 x f y  + 2 f (90 f 2 x f 2 x f y + 270 f 3 x f 2 x f 2 x + 60 f 3 x f xy − 225 f 3 2 x f x + 105 f 4 x f 3 x ) − 40(12 f 3 x f 4 x + 9 f 2 2 x f 3 x − 3 f 4 x f y ) = 0 . (9) Therefor, if f = f ( x, y , t ) solves Eq. (9), u = 4(ln f ) x solves E q . (2), and v obta ined by computing once deriv ativ e of the o btained solution u with resp ect to x solves Eq. (1). Theorem 1 If f = f ( x, y , t ) solves a c ouple of biline ar form e quations ( D 6 x + D x D t + 5 D 3 x D y − 5 D 2 y ) f · f = 0 , (10) ( D 4 x + D x D y ) f · f = 0 , (11) wher e Eq. (10) is of the standar d Hir ota’s biline ar 2DSK e quatio n, and Eq. (11) is of the standar d Hir ota’s biline ar KdV e quation, then u = 4(ln f ) x solves Eq. (2 ) with f also solve d by the quintic line ar e quation (9) . Pro of F r om the definition of the Hirota’s bilinea r der iv ativ e, the q uintic line a r e q uation (9) can be w r itten as the following form P ≡ 2 (∆ sk ) x /f 2 − 4 [∆ sk f x − 15∆ kdv ( f xx + f 3 x )] /f 3 − 60∆ kdv  f 2 x + 5 f xx f x  /f 4 − 4 f 3 x /f 5  , (12) 3 where ∆ kdv ≡ ( D 4 x + D x D y ) f · f , ∆ sk ≡ ( D 6 x + D x D t + 5 D 3 x D y − 5 D 2 y ) f · f . It shows that if ∆ sk = 0 and ∆ kdv = 0, then P = 0. O b viously , E q. (10), na mely ∆ sk , is the standar d Hiro ta ’s bilinear 2DSK equation, and Eq. (11), ∆ kdv , is the Hirota’s bilinear KdV equation. T o show that f = f ( x, y , t ) solving E qs. (10) and (11) is a solution of Eq. (9 ), one s ho uld prov e the consistency of E qs. (10 ) and (11), say , o f which the solution set is no t empty . F o r simplicity and without lo ss of gener ality , setting f = exp( ˜ v ) , ˜ v = ˜ v ( x, y , t ) in Eq s. (10), (11) and solving ˜ v xt , ˜ v y x resp ectively yields ˜ v xt = 120 ˜ v 3 2 x − 5 ˜ v 3 xy + 5 ˜ v 2 y − ˜ v 6 x , ˜ v xy = − ˜ v 4 x − 6 ˜ v 2 2 x . (13) Next, one should prov e ˜ v xt,xy = ˜ v xy ,xt . E asily , we get ˜ v xy ,xt = − ˜ v 5 xt − 12 ˜ v 3 x ˜ v 2 xt − 12 ˜ v 2 x ˜ v 3 xt , ˜ v xt,xy = 720 ˜ v 2 x ˜ v 2 xy ˜ v 3 x + 360 ˜ v 2 x ˜ v 3 xy − 5 ˜ v 4 x 2 y + 5 v x 3 y − ˜ v 7 xy . (14) Computing ˜ v xy ,xt − ˜ v xt,xy and substituting ˜ v xt and ˜ v xy with the for ms (13) into the o btained expression, one can obtain ˜ v xy ,xt − ˜ v xt,xy ≡ 0 .  Theorem 1 shows that the obtained q uintic linear equation is solved by the couple of Hiro ta’s bilinear equations whic h usually serves as a role to construct the new hiera r ch y of s oliton so lutions. It is well known that E q. (10) ensures the N -soliton s olutions of the 2DSK equation starting from u = 2(ln f ) x , a nd Eq. (11) ensures the N -solito n s olutions o f the KdV equa tio n. Ther efore, an explicit connection b etw een the KdV eq ua tion and the 2DSK equation is established. 3. Tw o hi erarc hies of multiple sol ition sol utions of the 2DSK equation T o c o nstruct multiple soliton solutions of the 2DSK Eq. (2 ), usually one can start from Eq. (8) with [19] f 1 = 1 + exp( ξ 1 ) , (15) f 2 = 1 + exp( ξ 1 ) + exp( ξ 2 ) + h 1 , 2 exp( ξ 1 + ξ 2 ) , (16) f 3 = 1 + exp( ξ 1 ) + exp( ξ 2 ) + exp( ξ 3 ) + h 1 , 2 exp( ξ 1 + ξ 2 ) + h 1 , 3 exp( ξ 1 + ξ 3 ) + h 2 , 3 exp( ξ 2 + ξ 3 ) + h 1 , 2 h 1 , 3 h 2 , 3 exp( ξ 1 + ξ 2 + ξ 3 ) , (17) . . . . . . , f n = X µ ∈ 0 , 1 exp n X i =1 µ i ξ i + X µ i µ j H ij ! , (18) where ξ i = ξ i ( x, y , t ) = k i x + l i y + w i t, ( i = 1 , . . . , n ), and k i , l i , w i are arbitra r y constants for all i . It is actually a simplified direct Hirota metho d. Usually , r esearchers pro ceed with the studies such as cons tructing multiple soliton solutions by taking p = 2 in Eq. (4). The tra nsformation u = 2(ln f ) x enables one to o btain a Hiro ta ’s 4 bilinear form, and then soliton solutions. It is deemed that one could not o btain high-o rder m ultiple so liton solutions if u = 4(ln f ) x do es not make the 2DSK to transfo rm into a Hiro ta’s bilinear form directly . It is crucial that we do o btain a new hierarchy of multiple soliton solutions of the 2DSK equation, which have not b een given in prev ious references. The 2 DSK equation (2) is the first and only one that b earing t wo hiera rchies of m ultiple solito n solutions. F or completeness, tw o one-solito n s olutions o f E q. (2 ) are listed b ellow u 1 = 4(ln f 1 ) x with f 1 = 1 + exp( k 1 x − k 3 1 y + 9 k 5 1 t ) , ˜ u 1 = 2(ln ˜ f 1 ) x with ˜ f 1 = 1 + exp  k 1 x + l 1 y + 5 l 2 1 − k 6 1 +5 k 3 1 l 1 k 1 t  , (19) where k 1 6 = 0 , l 1 6 = 0 are free parameters. They can b e written as the following forms [28]. u 1 = 4ln  cosh( k 1 2 x − k 3 1 2 y + 9 k 5 1 2 t )  x , ˜ u 1 = 2ln  cosh( p 1 + q 1 2 x − p 3 1 + q 3 1 2 y + 9( p 5 1 + q 5 1 ) 2 t )  x , (20) where p 1 , q 1 are arbitra r y constants. 3.1. Two-soliton solutions of the 2DSK e quation As for tw o -soliton solutions, starting fro m u = 4(ln f 2new ) x of Eq. (2 ) we obtain a new tw o - soliton solution with f 2new having the form f 2new = a 1 , 2 cosh( ξ 1 2 + ξ 2 2 ) + b 1 , 2 cosh( ξ 1 2 − ξ 2 2 ) , ξ i = k i x + l i y + w i t, (21) and l i = − k 3 i , w i = 9 k 5 i , ( i = 1 , 2) , a 1 , 2 = k 1 − k 2 , b 1 , 2 = k 1 + k 2 , k 1 6 = ± k 2 . As a comparison, we g ive the t wo-soliton so lution in terms of c osh function obtained by u = 2(ln f 2old ) x with f 2old having the same form with that of Eq. (21), wherea s k i = p i + q i , l i = − ( p 3 i + q 3 i ) , w i = 9( p 5 i + q 5 i ) , ( i = 1 , 2) , a 1 , 2 = p ( q 1 − q 2 )( q 1 − p 2 )( p 1 − q 2 )( p 1 − p 2 ) , b 1 , 2 = p ( q 1 + q 2 )( q 1 + p 2 )( p 1 + q 2 )( p 1 + p 2 ) , (22) where, and from now on, p i , q i , ( p i 6 = ± p j , q i 6 = ± q j , p i 6 = ± q j ) are constants for all i . 3.2. Thr e e-soliton solutions of the 2DSK e quation Starting from u = 4(ln f ) x we obtain a new three-s oliton so lution with f 3new = K 0 cosh( ξ 1 2 + ξ 2 2 + ξ 3 2 ) + K 1 cosh( ξ 1 2 − ξ 2 2 − ξ 3 2 ) + K 2 cosh( ξ 1 2 − ξ 2 2 + ξ 3 2 ) + K 3 cosh( ξ 1 2 + ξ 2 2 − ξ 3 2 ) , (23) where ξ i = k i x − k 3 i y + 9 k 5 i t, ( i = 1 , . . . , 3) , a nd K 0 = a 1 , 2 a 1 , 3 a 2 , 3 , K 1 = b 1 , 2 b 1 , 3 a 2 , 3 , K 2 = b 1 , 2 a 1 , 3 b 2 , 3 , K 3 = a 1 , 2 b 1 , 3 b 2 , 3 , a i,j = k i − k j , b i,j = k i + k j , k i 6 = ± k j , (1 ≤ i < j ≤ 3) . (24) 5 F or the known three-solito n so lution in ter ms of c osh function, f 3old has the s ame form with that of Eq. (23), where ξ i = k i x + l i y + w i t, k i = p i + q i , l i = − ( p 3 i + q 3 i ) , w i = 9( p 5 i + q 5 i ) , ( i = 1 , . . . , 3) , and K i ( i = 0 , . . . , 3) is the same with (24), how ever a i,j , b i,j hav e the following different forms a 1 , 2 = p ( q 1 − q 2 )( q 1 − p 2 )( p 1 − q 2 )( p 1 − p 2 ) , b 1 , 2 = p ( q 1 + q 2 )( q 1 + p 2 )( p 1 + q 2 )( p 1 + p 2 ) , a 1 , 3 = p ( q 1 − q 3 )( q 1 − p 3 )( p 1 − q 3 )( p 1 − p 3 ) , b 1 , 3 = p ( q 1 + q 3 )( q 1 + p 3 )( p 1 + q 3 )( p 1 + p 3 ) , a 2 , 3 = p ( q 3 − q 2 )( q 3 − p 2 )( p 3 − q 2 )( p 3 − p 2 ) , b 2 , 3 = p ( q 2 + q 3 )( p 2 + q 3 )( q 2 + p 3 )( p 2 + p 3 ) . 3.3. N -soliton solutions of the 2DSK e quation Starting from u = 4(ln f N new ) x we obtain an N -soliton solution with f N new = X ν K ν cosh N X i =1 ν i ξ i 2 ! , K ν = Y i

Original Paper

Loading high-quality paper...

Comments & Academic Discussion

Loading comments...

Leave a Comment