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2 Available online a Chaos, Solions and Fracals 4 (9) A new inegrable equaion wih no sooh solions Zhijun Qiao *, Liping Liu Deparen of Maheaics, The Universiy of Teas Pan-Aerican, Wes Universiy Drive, Edinburg, TX 7854, USA Acceped Noveber 7 Counicaed by Prof. M. Wadai Absrac In his paper, we propose a new copleely inegrable equaion: ¼ ; which has no sooh solions. This equaion is shown o have bi-hailonian srucure and La pair, which iply inegrabiliy of he equaion. Sudying his new equaion, we develop wo new kinds of solion soluions under he inhoogeneous boundary condiion li jj! ¼ B where B is nonzero consan. One is coninuous and piecewise sooh W/M -shape-peaks soliary soluion and he oher one-single-peak solion. The wo new kinds of peaked solions can no be wrien as he regular ype peakon: ce j cj, where c is a consan. We will provide graphs o show hose new kinds of peaked solions. Ó 8 Elsevier Ld. All righs reserved.. Inroducion Recenly, he sudy of peaked and cusped solion equaions has arisen lo of aracive aenion. The ypical represenaive of such equaions is he well-known Harry Dy (HD) equaion [8] u ¼ pffiffi u : Wadai e al. [8] generalized he HD equaion o an inegrable hierarchy. In heir paper [9,], Wadai e al. firs ie proposed he cusp solion, which is a kind of peaked solion whose lef and righ derivaives equal infiniies, for he HD equaion. Laer, here are several auhors sudying he cusp and peaked solion soluions for he inegrable equaions [ 5,9,,3,5 7]. In his paper, we propose a new peaked solion equaion: ¼ ; ðþ * Corresponding auhor. E-ail address: qiao@upa.edu (Z. Qiao) /$ - see fron aer Ó 8 Elsevier Ld. All righs reserved. doi:.6/j.chaos.7..34

3 588 Z. Qiao, L. Liu / Chaos, Solions and Fracals 4 (9) where is a scalar funcion and subscrips denoe he parial derivaives. This equaion is shown o have bi-hailonian srucure, and La pair ha iplies is inegrabiliy. Through sudying equaion (), we develop wo new kinds of solion soluions under he inhoogeneous boundary condiion li jj! ¼ B, where B is nonzero consan. One is coninuous and piecewise sooh W/M -shape-peaks soliary soluion and he oher one-single-peak solion. The wo new kinds of peaked solions canno be equivalen o he regular peakon: ce j cj, where c is a consan. There is no sooh solion found for he new Eq. (). We will ake soe graphs o show how hese hree peaks solions and one-single-peak solions look like.. Hailonian srucure and inegrabiliy Eq. () can be cas in he following Hailonian srucure: ¼ ¼ J dh þ d ¼ K dh þ d ; ðþ where J ¼ oo o; ð3þ K ¼ o 3 o; o ¼ o o ; Z H þ ¼ X d; Z H þ ¼ X 4 þ þ Þd; X ¼ð ; þ T Þ or X ¼ ð ; þþ is he doain of ha needs o be periodic wih T or o approach he sae consan as goes o, and H þ, H þ are wo Hailonian funcions. Boh operaor K and operaor J are Hailonian, and furherore our Eq. () is bi-hailonian (see Reark ). Reark. Apparenly, he operaor K ¼ o 3 o is Hailonian (see [], chaper 7) because of consan coefficiens and skew-syeric propery. Fro Ref. [], we also know ha he operaor J is Hailonian if and only if PrV Jh ða J Þ¼, where A J ¼ Z ðh ^ JhÞd is he associaed bi-vecor R of J, and h is a basic uni-vecor corresponding o. Le P ¼ o h, hen P ¼ h, Jh ¼ ðpþ, A ¼ h ^ðpþ d, and PrV Jh ða J Þ¼ Z Z h ^ Jh ^ðpþ h ^ Jh ^ P d ¼ h ^ðpþ ^ðpþ þ h ^ð P þ P Þ^P d ¼ Z h ^ð P þ h Þ^P d ¼ : So, J is Hailonian. In a siilar way, we can prove ha K þ J is also Hailonian. Therefore, K and J for a Hailonian pair. In order o show he inegrabiliy of his equaion, le us consider he following specral proble w ¼ k! w w w k Uð; kþ ; ð5þ w w where k is a specral paraeer, is a scalar poenial funcion periodic or approaching he sae consan a boh infiniies, and w ¼ðw ; w Þ T is he specral funcion corresponding o he specral paraeer k. Then, we have ð4þ Krk ¼ k Jrk; where rk ¼ k ðw þ w Þ. ð6þ

4 Reark. Eq. (6) plays a very iporan role in he discussions of he periodic soluions of he new wave equaion (), which we will deal wih in a subsequen paper [6]. Acually, on he basis of hose wo operaors, following our earlier ehod [,4] we are able o generae a new inegrable hierarchy. A direc calculaion leads o he following saeen. Eq. () has he following La pair: w w ¼ Uð; kþ ; ð7þ w w w w ¼ V ð; kþ ; ð8þ w w Z. Qiao, L. Liu / Chaos, Solions and Fracals 4 (9) where Uð; kþ ¼ k! k ; V ð; kþ ¼ k k k þ ð 4 A: k þ ðþþ 3 k 4 w In fac, one can use aheaical sofware Maple o check ha he copaibiliy condiion w U V þ½u; V Š¼ ¼ w, naely w generaes equaion (). So he wave equaion () is accordingly copleely inegrable by he Inverse Scaering Transforaion []. 3. W/M-shape-peaks solions and new one-single-peak solions 3.. Traveling wave seing Le ð; Þ ¼p ffiffiffiffiffiffiffi, hen Eq. () becoes vð;þ o vð; Þ o vð; Þ ¼ o 3 ð3=þ o vð; Þ o vð; Þ: 3 o ð9þ Le us consider he raveling wave soluions of he Eq. (9) hrough a generic seing vð; Þ ¼UðnÞ, where n ¼ c, and c is he wave speed. Subsiuing i ino Eq. (9) yields he following ODE: U nnn U n ¼ cu 3= U n : ðþ Apparenly U ¼ consan is a soluion, which is no ineresing for us. Le us find non-rivial soluions. Taking indefinie inegral wice on boh sides of he ODE (), we obain c p ffiffiffiffi þ U nn U þ C ¼ ; U ðþ p 4c ffiffiffiffi U þ C U U þ U n þ C ¼ ; ðþ where C and C are wo consans o be deerined. To have soliary raveling wave soluions, we se U ¼ V and ipose he boundary condiion li V ¼ A; n! A > ; ð3þ which iplies! as approaches (see paper [5,7] for ore deails). Subsiuing he boundary condiion (3) A ino he ODEs () and () generaes he following wo consans C ¼ A c A ; ð4þ C ¼ A4 ca: ð5þ

5 59 Z. Qiao, L. Liu / Chaos, Solions and Fracals 4 (9) So he ODE () becoes U ¼ U þ ðc A3 Þ A 3.. W/M-shape-peaks solions p U 8c ffiffiffiffi U þ A 4 þ 4cA: ð6þ Seing U ¼ V and aking inegral on boh sides of he ODE (6), we arrive a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lnða þ V þ ða þ V Þ þ 4c pffiffiffiffiffi A 3 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi B A A þ c lnþ ln A3 þ c þ A V þ ða 3 þ cþðav þ A V þ A 3 þ 4cÞ C AðV AÞ A ¼ jnj: pffiffiffi A In general, we can no ge an eplici for of V. Bu, if pffiffiffiffiffiffiffi 3 ¼, naely, c ¼ 3, hen we have A 3 þc 4A pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lnða þ V þ V þ AV A Að A þ V þ V þ AV A Þ Þ ln ln ¼ jnj; V A which iplies qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ A 3 þ X þ 3X 3ð3 þ X þ 3X ÞðX Þ ; 4X X ¼ e jnjþln ; n ¼ þ 3 4 A3 : Since ¼, we denoe B ¼, hen! B as n!, herefore we obain he following eplici soluion of Eq. (): V A ð; Þ ¼ B þ p ffiffi sinh j s 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j p ; 3 cosh s þ s ¼ þ 3 ð7þ 4B 3 ln ; whose 3D and D graphs are ploed in Fig. for B ¼. This soluion is of W-shape-peaks solion [5,6] and has hree peaks, and is profile looks like a W ype wave. So, we called i W-shape-peaks solion. Three peaks occur a ¼ 3 4, ¼ 3 4 ln, ¼ 3 4 þ ln, for each ie. See graph for ore deails. We can also se ¼ and ake negaive B as is infiniies lii. The graph, corresponding o B ¼ and he V soluion for (7), is a M-shape-peaks solion soluion of Eq. (), see Fig.. a b Fig.. (a) 3D graph of he eplici soluion ð; Þ defined by (7) when B ¼, wave speed c ¼ 3=4, and inervals of,, : , 6 6, 6 6. (b) D graph of he eplici soluion ð; Þ defined by (7) a ¼. This is a W-shape-peaks solion soluion.

6 Z. Qiao, L. Liu / Chaos, Solions and Fracals 4 (9) a b Fig.. (a) 3D graph of soluion (7) when wave speed c ¼ 3=4, and inervals of,, : , 6 6, 6 6. (b) D graph of soluion (7) a ¼. This is a M-shape-peaks solion soluion One-single-peak solions We already know ha Eq. () has hree peaks (eiher W-shape-peaks or M-shape-peaks) solion soluions. Le us consider he soluion ð; Þ, defined by (7), wihou he absolue value of þ 3. So, we creae 4B 3 Mð; Þ ¼B pffiffi þ 3 X X þ A; 3X þ X þ 3 ð8þ X ¼ e þ 3 4B 3 ; B > : Noe no absolue value in X s epression. A direc verificaion reveals ha Mð; Þ sill saisfies Eq. (). We view soluion (8) as a funcion of n ¼ þ 3. Then apparenly, MðnÞ has he following properies: 4B 3 MðÞ ¼ pffiffi pffiffiffi B; M 6 ðþþ ¼ 8 B; M 6 ð Þ ¼ 8 B: So, we found a coninuous and piecewise-sooh (bu no sooh) solion soluion for our new Eq. (). See he graphs of Mð; Þ in Fig. 3. Regarding negaive B <, le us ake B ¼ as a represenaive. In his case, we have MðÞ ¼, M p ðþþ ¼ ffiffi 6, 8 M p ð Þ ¼ ffiffi 6 which iply ha MðnÞ is an ani-peaked coninuous and piecewise-sooh solion. See Fig. 4 for ore 8 deails. a b Fig. 3. (a) 3D graph of he eplici soluion Mð; Þ defined by (8) when B ¼, wave speed c ¼ 3=4, and inervals of,, M: , 6 6, 6 M 6. (b) D graph of he eplici soluion Mð; Þ defined by (8) a ¼. This is a single peak solion soluion.

7 59 Z. Qiao, L. Liu / Chaos, Solions and Fracals 4 (9) a b Fig. 4. 3D and D graphs of a coninuous and piecewise-sooh solion soluion for Eq. () wih negaive apliude B ¼. This is a single peak solion soluion. 4. Conclusions and open probles In he paper, we presen a new inegrable equaion (). Through he regular raveling wave seing for our Eq. (),we develop wo new ypes of solion soluions: one is W-shape-peaks / M-shape-peaks solion (hree peaks, coninuous and piecewise sooh, bu no sooh, see Figs. and ), and he oher is one-single-peak solion soluion (also coninuous and piecewise sooh, bu no sooh, see Figs. 3 and 4). Those soluions are apparenly differen fro regular peakons. No sooh solions are found for our equaion, bu our equaions are copleely inegrable. Naely, in his paper we provide an inegrable syse wih no sooh solions. We ry o consruc he ineracion (boh collision and chase) of wo single peaked solions, wo W-shape-peaks solions (WW), wo M-shape-peaks solions (MM), WM, MW, and one single peaked and he oher M/W solions. Bu, ha is a really hard procedure because of he following wo ajor reasons:. So far we have an effecive nuerical schee o solve our PDE (). The soluions are no sooh, e.g. our solions (8) and (9) have hree peaks and one peak, respecively. We ried using he superposiion of wo single solions (9) wih sae A and differen wave speed c as an iniial condiion of he PDE (). However, he usual Finie Difference schees could neiher capure he collision nor he chase. The auhors have he ipression ha in order o capure he wave ineracions nuerically soe special echniques need o be developed for his specific Eq. (). This is he ask of our fuure sudy.. We do no have a heoreical ansaz o deal wih he peaked -solion or N-solion soluions of our equaion like he CH equaion wih P N j¼ p jðþe j qjðþj, alhough we are seeking for. We ried eending our peaked solion soluions (8) and (9) o he for of P N j¼ p jðþðe j qjðþj Þ for he purpose of uli-solion soluions. However, ha is no he case for our equaion. There is no sooh solion for our equaion hough i is copleely inegrable. This causes difficuly o discuss uli-solions. Finding wha ansaz is appropriae for our equaion will be a crucial work for discussing ineracion of wo peaked solions. Furherore, we sugges a ore general parial differenial equaion: ¼ k k ð9þ wih a consan k R. When k ¼ ; =; ;, he equaion is inegrable. For k ¼, ha is rivial case; for k ¼, linear case; for k ¼ =, Harry-Dy ype case; for k ¼, we already discussed in his paper. Any oher inegrable cases? We will sudy in he near fuure. The ODE (6) has a physical eaning and can be p cas ino he Newon equaion U ¼ SðUÞ SðA Þ of a paricle wih a new poenial SðUÞ ¼U þ ðc A3 Þ U 8c ffiffiffiffi U A, or can be convered o V ¼ T ðv Þ TðAÞ wih U ¼ V, T ðv Þ¼ V c þ AðA3 þ4cþ. In he paper, we successfully solved his new Newon syse wih new one-single-peak solions 4 V 4V and M/W-shape-peaks solions. The new Newon syse igh have poenial applicaions in he sudy of engineering of loop solions on a vore filaen wih aial flow [6,7].

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