New Types of Interactions between Solitons of the (2+1)-Dimensional Broer-Kaup-Kupershmidt Equation

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1 ew Types of Interactons between Soltons of the (2+1)-Dmensonal Broer-Kaup-Kupershmdt Equaton Chao-Qng Da, Jan-Png Meng, and Je-Fang Zhang Insttute of onlnear Physcs, Zhejang ormal Unversty, Jnhua , P. R. Chna Reprnt requests to C.-Q. D.; E-mal: Z. aturforsch. 60a, (2005); receved August 5, 2005 By means of the extended homogeneous balance method and the varable separaton approach, more general varable separaton solutons of the (2+1)-dmensonal Broer-Kaup-Kupershmdt equaton are obtaned. Based on the varable separaton soluton and by selectng approprate functons, new types of nteractons between the mult-valued and the sngle-valued soltons, such as compactonlke sem-foldon and compacton, peakon-lke sem-foldon and peakon, and bell-lke sem-foldon and dromon, are nvestgated. Meanwhle, we also dscuss the phase shft of these nteractons. PACS: Jr, Ik Key words: ew Types of Interactons between Soltons; (2+1)-Dmensonal BKK System; Extended Homogeneous Balance Method. 1. Introducton Snce Lou and Lu successfully appled a knd of varable separaton method, now called the multlnear varable separaton approach (MLVSA), frstly n 1995 for the Davey-Stewartsen (DS) equaton [1], the MLVSA has been a nearly systematc process to solve (2+1)-dmensonal nonlnear evoluton systems. Recently, a qute unversal formula has been found [1 4] λ (a 0 a 3 a 1 a 2 )P x Q y u (a 0 + a 1 P + a 2 Q + a 3 PQ) 2, (1) whch s vald for the sutable physcal felds or potentals of varous (2+1)-dmensonal physcally nterestng nonlnear models, such as the DS equaton, the asymmetrc DS equaton, the long dspersve wave (LDW) equaton, the dspersve long wave (DLW) equaton, the Broer-Kaup-Kupershmdt (BKK) system, the zhnk-ovkov-veselov (V) equaton, the asymmetrc V system, the generalzed V equaton, the long wave-short wave nteracton model, the Maccar system, the (2+1)-dmensonal Ablowtz- Kaup-ewell-Segur (AKS) system, a nonntegrable (2+1)-dmensonal Korteweg-de Vres (KdV) equaton, the general ( + M)-component AKS system and the (2+1)-dmensonal sne-gordon equaton. In expresson (1), P P(x,t) s an arbtrary functon of {x,t}, Q Q(y,t) may be ether an arbtrary func- ton of {y,t} or an arbtrary soluton of a Rccat equaton, whle λ, a 0, a 1, a 2 and a 3 are taken as constants. In usual cases, the constant λ = ±2 or λ = ±1. owadays the MLVSA s stll n progress amng at two aspects. One s to extend ths method to other nonlnear systems, such as the dfferental-dfference equatons [5] and the (1+1)-dmensonal nonlnear systems [6]. Another s to obtan more general solutons n the sense that t admts more arbtrary separaton functons nto the solutons, and to search more new solton structures. Apart from the tradtonal localzed exctatons ncludng lumps, dromons, peakons, compactons, foldons, rng soltons, fractal soltons, chaotc soltons and so on [1 4], the propertes of dromon-dromon, peakon-peakon, compactoncompacton, foldon-foldon nteractons were studed [2 4]. More recently, the propertes of dromoncompacton, peakon-compacton, dromon-peakon nteractons [7] were dscussed. From mentoned above, we know that the nteractons were dscussed ether between sngle-valued soltons, or between mult-valued soltons (foldons). To our best knowledge, the nteractons between mult-valued and sngle-valued soltons, such as compacton-lke sem-foldon and compacton, peakon-lke sem-foldon and peakon, and belllke sem-foldon and dromon, whch are focused n the present paper, were lttle reported n prevous lterature / 05 / $ c 2005 Verlag der Zetschrft für aturforschung, Tübngen

2 688 C.-Q. Da et al. ew Types of Interactons between Soltons of the (2+1)-Dmensonal BKK Equaton In order to study these new types of nteractons, we take the (2+1)-dmensonal Broer-Kaup-Kupershmdt equaton (BKKE) H ty H xxy + 2(HH x ) y + 2G xx = 0, (2) G t + G xx + 2(HG) x = 0 (3) as a concrete example. The (2+1)-dmensonal BKKE may be derved from the nner parameter-dependent symmetry constrant of the Kadomtsev-Petvashvl (KP) equaton [8]. Usng some sutable dependent and ndependent varable transformatons, Chen and L have proved that the (2+1)-dmensonal BKKE can be transformed to the (2+1)-dmensonal ntegrable dspersve long wave equaton (DLWE) [8] u ty = η xx 1 2 (u2 ) xy, (4) reduced to the usual (1+1)-dmensonal BKKE, whch can be used to descrbe the propagaton of long wave n shallow water [9]. In [11], usng the extended homogeneous balance method [12], we have proved that the (2+1)- dmensonal BKKE possesses a general varable separaton soluton (1). Zhu et al. [13] presented a more general soluton for the (2+1)-dmensonal BKKE. In ths paper, we extend these results n [11, 13] to a more general one wth more arbtrary functons. Meanwhle, based on the unversal formula (1), some new types of nteractons between compacton-lke sem-foldon and compacton, peakon-lke sem-foldon and peakon, and bell-lke sem-foldon and dromon wll be dscussed. 2. Varable Separaton Solutons for the (2+1)-Dmensonal BKKE η t = (uη + u + u xy ) x, (5) and the (2+1)-dmensonal ntegrable Ablowtz-Kaup- ewell-segur equaton (AKSE) ψ t = ψ xx + ψu, (6) φ t = φ xx φu, (7) u y = ψφ. (8) When we take y = x, the (2+1)-dmensonal BKKE s Accordng to the extended homogeneous balance method, we assume (2) and (3) have the solutons H = f (w)w x + u 0, (9) G = g (w)w x w y + g (w)w xy + v 0, (10) where f (w), g(w), w(x, y,t) are to be determned later, and u 0, v 0 are two seed solutons. It s evdent that (2) and (3) possess trval seed solutons u 0 = u 0 (x,t), v 0 = 0. ow substtutng (9) and (10) together wth the seed solutons nto (2) and (3) yelds H ty H xxy + 2(HH x ) y + 2G xx = ( 2 f f f (3) f (4) + 2g (4)) w 3 xw y + f (3)( w x w y w t 3w 3 xw xy 3w x w y w xx + 2u 0 w 2 ) xw y + f ( ) w xy w t + w x w ty + w y w xt 3w xy w xx 3w x w xxy w y w xxx + 4u 0 w x w xy + 2u 0 w xx w y + 2u 0x w x w y + f ( ) w xyt w xxxy + 2u 0 w xxy + 2u 0x w xy + 2g w xxxy + g (3)( 6w x w xx w y + 6w 2 ) xw xy + g ( ) 2w xxx w y + 6w xx w xy + 6w x w xxy + f f ( 6w x w xx w y + 4w 2 ) xw xy + f 2 ( ) 2w xx w xy + 2w x w xxy = 0, (11) G t + G xx + 2(HG) x = ( g (4) + 2g (3) f + 2g f ) w 3 x w y + g (3)( 3w x w xx w y + 3w 2 x w xy + w x w y w t + 2u 0 w 2 x w y) + g ( ) w xxx w y + 3w xx w xxy + w xy w yt + w y w xt + 2u 0 w xx w y + 4u 0 w x w xy + 2u 0x w x w y (12) + g ( w xyt + w xxxy + 2u 0 w xxy + 2u 0x w xy ) + f g ( 4w x w xx w y + 4w 2 x w xy) + f g w 2 x w xy + f g ( w xx w xy + 2w x w xxy ) = 0. Settng the coeffcents of the term w 3 xw y n (11) and (12) to zero, we obtan two ordnary dfferental equatons for functons f (w) and g(w): g (4) + 2g (3) f + 2g f = 0. (14) The followng specal solutons exst for (13) and (14): 2 f f f (3) f (4) + 2g (4) = 0, (13) f (w)=g(w)=ln(w). (15)

3 C.-Q. Da et al. ew Types of Interactons between Soltons of the (2+1)-Dmensonal BKK Equaton 689 Therefore f f = 1 2 f, f 2 = f. (16) Usng these results, (11) and (12) can be smplfed as f (w t + w xx + 2u 0 w x ) xy + f [ w xy (w t + w xx + 2u 0 w x ) + w x (w t + w xx + 2u 0 w x ) y ] + w y (w t + w xx + 2u 0 w x ) x + f (3)[ w x w y (w t + w xx + 2u 0 w x ) ] = 0, g (w t + w xx + 2u 0 w x ) xy + g [ w xy (w t + w xx + 2u 0 w x ) + w x (w t + w xx + 2u 0 w x ) y ] + w y (w t + w xx + 2u 0 w x ) x + g (3)[ w x w y (w t + w xx + 2u 0 w x ) ] = 0. (17) (18) Settng the coeffcents of f (3), f, f and g (3), g, g n (17) and (18) to zero yelds a set of partal dfferental equatons for w(x, y,t): (w t + w xx + 2u 0 w x ) xy = 0, (19) w xy (w t + w xx + 2u 0 w x ) + w x (w t + w xx + 2u 0 w x ) y + w y (w t + w xx + 2u 0 w x ) x = 0, (20) w x w y (w t + w xx + 2u 0 w x )=0. (21) By analyss, we fnd that (19) (21) are satsfed automatcally under the condton w t + w xx + 2u 0 w x = 0. (22) From mentoned above, (2) and (3) are lnearzed by the transformatons (9) and (10). Snce the (22) s only a lnear equaton, one can certanly utlze the lnear superposton theorem. For nstance w = k=1 w k (x,y,t)=q 0 (y)+ k=1 P k (x,t)q k (y,t), (23) where the varable separated functons P k (x,t) P k and Q k (x,t) Q k (k = 1,2,,) are only the functons of {x,t} and {y,t}, respectvely, and Q 0 (y) Q 0. Insertng the ansatz (23) nto (22) yelds the followng smple varable separated equatons: P kt + P kxx + 2u 0 P kx + b(t)p k = 0, (24) Q kt b(t)q k = 0, (25) where b(t) s an arbtrary functon of ndcated varable. Then the general varable separaton soluton for the BKKE yelds G = H = k=1 P kxq k Q 0 + k=1 P kq k + u 0 (x,t), (26) k=1 P kxq ky Q 0 + k=1 P k=1 P kxq k [Q 0y + k=1 P kq ky ] kq k [Q 0 + k=1 P, kq k ] 2 (27) where u 0,P k and Q k satsfy (24) and (25). In order to dscuss new types of nteractons between soltons, we have to leave the complex forms (26) and (27) and need to make some smplfcatons further. If = 3, Q 0 = a 0, Q 1 = a 1, Q 2 = a 2 Q, Q 3 = a 3 Q,(a = constants, = 0,,3), P 1 = P 3 = P, P 2 = 1 wth P P(x,t) and Q Q(y,t), then (23) (25) change nto w = a 0 + a 1 P + a 2 Q + a 3 PQ, (28) P t + P xx + 2u 0 P x + b(t)p = 0, (29) Q t b(t)q = 0. (30) From (29) and (30), u 0 and Q(y,t) can be solved, that s u 0 = 1 [P t + P xx + b(t)p], (31) 2P x Q(y,t)=ϕ(y)exp[ t b(τ)dτ], (32) where ϕ(y) ϕ s an arbtrary functon of {y}. Therefore, a specal varable separaton exctaton of the (2+1)-dmensonal BKKE s H = (a 1 + a 3 Q)P x a 0 + a 1 P + a 2 Q + a 3 PQ 1 2P x [P t +P xx +b(t)p], G = (33) (a 0 a 3 a 1 a 2 )P x Q y [a 0 + a 1 P + a 2 Q + a 3 PQ] 2, (34) where P s an arbtrary functon of {x,t} and Q s gven n (32). From (34), we can see that t possesses the same form of the unversal formula (1) wth λ = 1.

4 690 C.-Q. Da et al. ew Types of Interactons between Soltons of the (2+1)-Dmensonal BKK Equaton 3. ew Types of Interactons between Soltons of the (2+1)-Dmensonal BKKE Here we do not dscuss the general form (27) of G, and only dscuss some localzed structures for G expressed by (34). Actually, even n ths specal stuaton, one stll fnds abundant localzed exctatons for the (2+1)-dmensonal BKKE because of the arbtrarness of the functons P, ϕ and b ncluded n (32) (34). In [11], we have gven out some coherent solutons for the feld G such as dromons, conc soltons, lumps, oscllatng dromons, breathers, and nstantons n detals. Yng and Lou have obtaned some coherent structures of the (2+1)-dmensonal BKKE by means of the truncated Panlevé expanson and the related Bäcklund [14]. Lou has also constructed (2+1)- dmensonal compacton solutons and dscussed ther nteracton behavors [15]. Here we are nterested n revealng some new types of nteractons between soltons n the (2+1)-dmensonal BKKE such as the nteracton behavors between compacton-lke sem-foldon and compacton, peakon-lke sem-foldon and peakon, and bell-lke sem-foldon and dromon. Based on the physcal quantty (34), sem-folded localzed structures can be dscussed, when functon Q s a sngle-valued functon and P s selected va the relatons P x = =1 P = ζ P x x ζ dζ, κ (ζ d t), x = ζ + =1 χ (ζ d t), (35) where d ( = 1,2,...,) are arbtrary constants, κ and χ are localzed exctatons wth the propertes κ (± )=0, χ (± )=constants. From (35), one can know that ζ may be a mult-valued functon n some sutable regons of x by choosng the functons χ approprately. Therefore, the functon P x, whch s obvously an nteracton soluton of M localzed exctatons due to the property ζ x, may be a mult-valued functon of x n these areas; though t s a sngle-valued functon of ζ. Actually, most of the known mult-loop solutons are specal cases of (35). More propertes of the mult-valued functon under change of the degree of multvaluedness can be found n [14]. In order to dscuss the nteracton property of localzed exctatons related to the physcal quantty (1) or (34), we frst study the asymptotc behavors of the localzed exctatons produced from the formula (1) when t Asymptotc Behavors of the Localzed Exctatons Produced from (1) In general, f functons P [consderng (35)] and Q are selected as mult-localzed soltonc exctatons wth (z ζ d t) P t = Q t = =1 M j=1 P,P (z ) P (ζ d t) κ dx z, (36) Q j,q j Q j (y c j t + j ), (37) where {P,Q j } and j are localzed functons, then the physcal quantty expressed by (1) delvers M (2+1)-dmensonal localzed exctatons wth the asymptotc behavor wth u t =1 M j=1 M =1 j=1 (a 3 a 0 a 1 a 2 )Pz Q jy (1 + χz )[a 0 + a 1 (P (z )+ P )+a 2 (Q j + Q j )+a 3(P (z )+ P )(Q j + Q j )]2 u j, x t ζ + δ + χ (z ), (39) P Q = P j ( )+P j (± ), (40) j< j> = Q j ( )+Q j (± ), (41) j< j> δ (38) = χ j ( )+χ j (± ). (42) j< j> In the above dscusson, t has been assumed, wthout loss of generalty, that c > c j, d > d j f > j. From the asymptotc result (38), we dscover some mportant and nterestng facts: () The j-th localzed exctaton u j s a travellng wave movng wth the velocty d along the postve

5 C.-Q. Da et al. ew Types of Interactons between Soltons of the (2+1)-Dmensonal BKK Equaton 691 Fg. 1. Compacton-lke sem-foldon structure for G wth condtons (49) and (50) (a) at tme t = 2; (b) a sectonal vew related to (a) at x = 0.5; (c) a sectonal vew related to (a) at y = 0. (d > 0) or negatve (d < 0) x drecton, and c j along the postve (c j > 0) or negatve (c j < 0) y drecton. () The propertes of the j-th localzed exctaton u j s only determned by P of (36) and Q j of (37). () The shape of the j-th localzed exctaton u j wll be changed (non-completely elastc or completely nelastc nteracton) f P + P, (43) and (or) Q + j Q j, (44) followng the nteracton. On the contrary, t wll preserve ts shape (completely elastc nteracton) durng the nteracton f P + = P, (45) Q + j = Q j. (46) (v) The phase shft of the j-th localzed exctaton u j reads δ + δ, (47) n the x drecton and + j j. (48) n the y drecton Compacton-lke Sem-foldon Structure Due to the arbtrares of constants a 0 a 3 n (34), we can dscuss varous localzed exctatons based on (34) by choosng specal values of these constants a 0 a 3 (see [2] n detals) and the arbtrary functons P, ϕ and b. For dscussng the followng new types of sem-foldons and nteracton behavors, we take a 0 = 8, a 1 = a 2 = 1, a 3 = 0.3 for convenence. ow we dscuss some new coherent structures for the physcal quantty G, and focus our attenton on (2+1)-dmensonal sem-folded localzed structures, whch may exst n certan ntrcate stuatons. When the functon Q s a sngle-valued functon and P s selected as (35) wth = 1, namely P x = 0.5sech(ζ 0.3t) 2, x = ζ 1.5tanh(ζ 0.3t),

6 692 C.-Q. Da et al. ew Types of Interactons between Soltons of the (2+1)-Dmensonal BKK Equaton Fg. 2. Evoluton profles of the nteracton between sem-foldon and compacton determned by (34) wth (50) and (51) at (a) t = 25, (b) t = 1, (c) t = 25. (d) The correspondng sectonal vew at {t = 25,y = 0} (dotted lne before collson), {t = 1, y = 0} (sold lne n collson), {t = 25, y = 0} (dashed lne after collson), respectvely. ζ P = Px x ζ dζ, (49) 0, y π 2, (sn(y) +1) exp(sech(t)), Q = 2 π < y 2 π, (50) 2exp(sech(t)), y > 2 π, then we can obtan a new type of sem-folded localzed exctaton lookng lke a compacton-lke loop solton (compacton-lke sem-foldon). The correspondng coherent structure s depcted n Fg. 1, from whch one can fnd that the sem-folded coherent structure possesses novel propertes, namely, t folds as loop solton n the x drecton and localzes as compacton n the y drecton Interacton between Compacton-lke Sem-foldon and Compacton Due to the arbtrarness of the functons n soluton (34), we can dscuss the nteracton behavor between compacton-lke sem-foldon and compacton by selectng Q as a sngle-valued pecewse smooth functon (50) and P as (35) wth = 2, χ 2 = 0, d 1 = 0.3, d 2 = 0,.e. P x = 0.5sech(ζ 0.3t) 2 + sech(ζ) 2, ζ x = ζ 1.5tanh(ζ 0.3t), P = Px x ζ dζ. (51) From Fg. 2, one can see the nteracton between mult-valued compacton-lke sem-foldon and snglevalued compacton s non-completely elastc snce the ampltude of the movng sem-foldon decreases. Ths property s dfferent from the completely elastc nteracton between ether sngle-valued and sngle-valued soltons [7, 15], or mult-valued and mult-valued soltons (foldons) [4]. Meanwhle, the phase shft can be observed. In order to reveal the phase shft more clearly and vsually, t has proved convenent to fx the compacton possessng zero velocty. Pror to nteracton, the compacton has set to be {v 0x = d 2 = 0,v 0y = 0}, however, the poston located by compacton s stll altered about from x = 1.5 tox = 1.5, then stops at x = 1.5 and preserves ts ntal veloctes {v x = v 0x,v y = v 0y } after nteracton. Therefore the phase shft of the statc compacton s δ 2 + δ 2 = χ 1( ) χ 1 (+ )=3. The fnal veloctes V x and V y of the mov-

7 C.-Q. Da et al. ew Types of Interactons between Soltons of the (2+1)-Dmensonal BKK Equaton 693 Fg. 3. Evoluton profles of the nteracton between sem-foldon and peakon determned by (34) wth (51) and (52) at (a) t = 20, (b) t = 1, (c) t = 20. (d) The correspondng sectonal vew at {t = 20,y = 0.01} (dotted lne before collson), {t = 1, y = 0.01} (sold lne n collson), {t = 20, y = 0.01} (dashed lne after collson), respectvely. ng compacton-lke sem-foldon also completely preserved ts ntal veloctes {V x = V 0x = d 1 = 0.3,V y = V 0y = 0}. The phase shft of the movng sem-foldon s δ 1 + δ 1 = χ 2(+ ) χ 2 ( ) =0, that s, the for movng sem-foldon exsts no phase shft Interacton between Peakon-lke Sem-foldon and Peakon Smlarly, one of the smplest selectons for dscussng the nteracton between peakon-lke semfoldon and peakon s to take an arbtrary mult-valued functon and an arbtrary sngle-valued functon,.e. P s selected as (51) and Q has the followng form { exp(y)exp(sech(t)), y < 0, Q = ( exp( y) +2) exp(sech(t)), y 0. (52) From Fg. 3, one can see the nteracton behavor between peakon-lke sem-foldon and peakon s also non-completely elastc. Ths sem-folded localzed exctaton folds as loop solton n the x drecton and localzes as peakon n the y drecton; so we call t a peakon-lke sem-foldon. The shape and ampltude of the statc peakon hardly change after collson, whle the ampltude of the movng sem-foldon wth the velocty {V x = 0.3,V y = 0} decreases. Before the nteracton, the statc peakon s located at x = 1.5 and after the nteracton, t s shfted to x = 1.5. By careful analyss smlar to Secton 3.3, we know that the phase shft of the statc peakon s 3, and for the movng sem-foldon exsts no phase shft. These propertes are analogous to the case n Secton Interacton between Bell-lke Sem-foldon and Dromon Fnally, we dscuss the nteracton between bell-lke sem-foldon and dromon. When P possesses the form of (51) and Q s taken as Q = tanh(y)exp(sech(t)), (53) then the nteracton between bell-lke sem-foldon and dromon can be observed. Ths sem-folded localzed exctaton folds as loop solton n the x drecton and

8 694 C.-Q. Da et al. ew Types of Interactons between Soltons of the (2+1)-Dmensonal BKK Equaton Fg. 4. Evoluton profles of the nteracton between sem-foldon and dromon determned by (34) wth (51) and (53) at (a) t = 20, (b) t = 1, (c) t = 20. (d) The correspondng sectonal vew at {t = 20,y = 0} (dotted lne before collson), {t = 1, y = 0} (sold lne n collson), {t = 20, y = 0} (dashed lne after collson), respectvely. localzes lke a bell n the y drecton; so we call t a bell-lke sem-foldon. From Fg. 4, one can see the nteracton between mult-valued sem-foldon and sngle-valued dromon s also non-completely elastc whch s analogous to the above two cases. The shape and ampltude of the statc dromon hardly change after collson, whle the ampltude of the movng sem-foldon wth the velocty {V x = 0.3,V y = 0} decreases. Furthermore, the phase shft of the statc dromon can be observed. Before the collson, the statc dromon s located at x = 1.5 and after the collson, t s shfted to x = 1.5. Smlarly to analyss n Secton 3.3, we know that the phase shft of the statc dromon s 3, and for the movng sem-foldon exsts no phase shft. 4. Summary and Dscusson In summary, usng the extended homogeneous balance method and the varable separaton approach, more general varable separaton solutons of the (2+1)-dmensonal Broer-Kaup-Kupershmdt equaton are obtaned. From the specal varable separaton soluton (34) and by selectng approprate functons, new types of nteractons between the mult-valued and the sngle-valued soltons, such as compacton-lke sem-foldon and compacton, peakon-lke sem-foldon and peakon, and bell-lke sem-foldon and dromon, are found. By means of analyss of asymptotc behavors and the profles n Fgs. 2 4, one can fnd that ther nteractons are non-completely elastc whch s dfferent from the completely elastc nteracton between ether sngle-valued and sngle-valued soltons or mult-valued and mult-valued soltons (foldons). Moreover, the statc sngle-valued soltons all exst phase shft, whle the ampltudes of sem-foldons all decrease. What we further verfed that the extended homogeneous balance method and the varable separaton approach are qute useful to generate abundant localzed exctatons for many models. In our future work, on the one hand, we devote to extend ths method to other nonlnear systems, such as the dfferental-dfference equatons and (1+1)-dmensonal nonlnear systems. On the other hand, we wll look for more nterestng localzed exctatons.

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