A PLETHORA OF MULTI-PULSED SOLUTIONS FOR A BOUSSINESQ SYSTEM MIN CHEN Department f Mathematics, Penn State University University Park, PA68, USA. Abstract. This paper studies traveling-wave slutins f the regularized Bussinesq system which mdel waves in a hrizntal water channel traveling in bth directins. An exact slutin with a single trugh is fund analytically. Interesting new multi-pulsed traveling-wave slutins which cnsist f an arbitrary number f trughs are fund numerically. The bifurcatin diagrams fr N-trugh slutins with respect t phase speed are presented.. Regularized Bussinesq System In this reprt, we study the regularized Bussinesq system () η t + u x +(ηu) x 6 η xxt =, u t + η x + uu x 6 u xxt =, which describes apprximately the tw-dimensinal prpagatin f surface waves in a unifrm hrizntal channel f length L filled with an irrtatinal, incmpressible, invisid fluid which in its undisturbed state has depth h. The nn-dimensinal variables η(x, t)andu(x, t) represent, respectively, the deviatin f the water surface frm its undisturbed psitin and the hrizntal velcity at water level 3 h. Amng a class f frmally equivalent mdels, system () is particular interesting because that the dispersin relatin is stabilizing fr all wave numbers and it is reasnably straightfrward t implement the initial-bundaryvalue prblems that arise in labratry experiments. Mrever, the regularized Bussinesq system is much simpler than the full Euler equatins and can be used n a wider class f prblems than a mdel equatin with the assumptin f waves traveling in nly ne directin. Since we are interested in mdeling wave mtin in a wave-tank experiment which is initiated by wave makers at bth ends f the channel, the initial- and bundary-cnditins are () η(,t)=h (t), u(,t)=v (t), η(x, ) = f (x), η(l, t) =h (t), u(l, t) =v (t), u(x, ) = f (x). The cnsistency requirements are the bvius nes dictated by cntinuity cnsideratins, namely (3) h () = f (), h () = f (L), v () = f (), v () = f (L). Key wrds and phrases. Bussinesq system fr water waves, traveling waves, multi-pulsed slutins, hmclinic rbits.
MIN CHEN It is shwn in [3] that this initial- and bundary-value prblem ()-()-(3) is well-psed. Therem. Let f = (f,f ) C (,L), h = (h,h ), v = (v,v ) C (,T) fr sme T, L >. Suppse f, h and v t satisfy the cmpatibility cnditins (3). Define f =max{ f C(,L), f C(,L) }, h = max{ h C(,T ), h C(,T ) }, and v = max{ v C(,T ), v C(,T ) }. Then there is a T = T (T,L, f, h, v ) T and an unique slutin pair (η, u) in C (,T ; C (,L)) that satisfies (). T ur knwledge, this is the nly existence and uniqueness result available fr this physically relevant initial- and nn-hmgeneus Dirichlet-bundary-value prblem fr a system describing tw-way prpagatin f water waves. We nte that ne f the difficulties with the KdV-equatin psed n a bunded dmain is that it requires three bundary cnditins, which is physically unfavrable.. A Nn-trivial Exact Slutin Letting ξ = x kt where k is the speed f the traveling-wave slutin, ne can write the slutin in the frm η(x, t) = η(ξ) andu(x, t) = u(ξ). Suppsing the slutin decays at large distance frm its crests r trughs, it is natural t impse the bundary cnditins (4) (η (n) (ξ),u (n) (ξ)) asξ ±, fr at least n =,,. In the mving frame, the functins η and u satisfy the rdinary differential equatins 6 kη + u kη + uη =, () 6 ku ku + η + u =, where the derivatives are with respect t ξ. It is clear that a hmclinic slutin abut the rigin f (), will lead t a traveling-wave slutin f (). Therefre, the prblem f finding traveling-wave slutins becmes that f finding hmclinic rbits f (). It is easy t check that system () can be written as a single equatin (6) u + k uu u + 6 k (u ) + 8 k u3 4 k u +36u 36 k u =. Instead f slving u(ξ) frm (6) directly, which is very difficult if it is nt impssible, the technique used by Kichennassamy and Olver in [7] fr a single th-rder equatin is adpted here. In the present case, the rdinary differential equatin has cefficients depending n k which are part f the unknwns. Assuming that u(ξ) can be recnstructed as the slutin f a simple first-rder rdinary differential equatin (7) w(u) =(u ), ne finds that fr u (8) (u ) = w, u = w, u = ww + 4 w w, where the primes n w indicate derivatives with respect t u. Fr a slutin u f the frm (9) u(ξ) =A sech (λξ), λ >, the crrespnding functin w(u) must be a cubic plynmial: w(u) =4λ (u A u3 ). Substituting w(u) int (8) and using the resulting relatinships in (6), ne btains a degreethree hmgeneus plynmial in u which has t be zer. In rder fr u t be a nntrivial slutin, all the cefficients have t be zer which yields an algebraic system f equatins
MULTI-PULSED SOLUTIONS 3 η(ξ) 8 (b) u(ξ) 7 6 4 3 3 4 Figure. An exact slutin f regularized Bussinesq system (k =.) n k, λ and A. Substituting the slutin A =,λ = 9 and k = 4 f the algebraic system int (9) and () ne finds a pair f exact traveling-wave slutins u(x, t) =± ( 3 sech (x + x ) t), () η(x, t) = ( ( 3 sech (x + x ) ( 3 4 t) 3sech 4 (x + x )) t). The right-mving exact slutin is pltted in Figure and it is clear that η has a trugh with depth 3.7. 3. Existence f slitary-wave slutins fr k> Since the full Euler equatins have slitary-wave slutins [, ] and the system () is its apprximatins, and as slitary waves play a central rle in evlutin f certain types f disturbances, it is imprtant t shw that the system () has slitary-wave slutins. The fllwing therem is prved in []. Therem. The regularized Bussinesq system pssesses a hmclinic rbit abut the rigin with any phase speed k> and it has the additinal prperties: (a) (u(ξ),η(ξ)) is an even slutin, namely (u(ξ),η(ξ)) = (u( ξ),η( ξ)), (u (ξ),η (ξ)) = (u ( ξ),η ( ξ)) fr any ξ IR, (b) (u(ξ),η(ξ)) is psitive and mntnically decreasing fr ξ>, (c) k < u(ξ) L = u() < (k ); (k ) < η(ξ) L = η() <k, k (d) u() = (k ) + O((k ) ); η() = (k ) + O((k ) ) as k, (e) u() = η()(k + k η()), +η() (f) (k k )u(ξ) <η(ξ) < (k + k )u(ξ). It is wrth t nte that in cntrast t the KdV-type equatins where a traveling-wave slutin satisfying (4) is unique fr a fixed k, such slutins are nt unique fr the regularized Bussinesq system. In particular, an analytical slutin () and a slitary-wave slutin exist fr phase speed k =.. Fr phase speed k >., the nn-uniqueness is clearly seen in ur numerical results presented in next sectin. 4. Even multi-pulsed slutins T btain a general picture f the multi-pulsed slutins f (), it is helpful t study whle slutin branches btained by cntinuatin frm a given apprximate slutin. The
4 MIN CHEN 3 4 7 6 H nrm f (u, η) 8 6 4 4 3 L nrm f (u, η) 4 3 Slitary wave 4 3 4 6 7 8 9 3 4 6 7 8 9 3 3 max(η) min(η) 3 4 6 7 8 9 3 3 4 6 7 8 9 Figure. Bifurcatin diagrams. slutin branches can be calculated numerically by using a tlbx HmCnt in AUTO [6, 4] and the precise quantitative prperties f these branches can be analyzed. In Figure, the bifurcatin diagrams f multi-pulsed slutins with t trughs (slid lines) and the slitary-wave slutins (dashed line) are presented. The H -nrm and the L -nrm f the slutin (u, η), the height f the crests max η(ξ) and the depth f the trughs min η(ξ) with respect t k are pltted. The numbers marked in Figure indicate the number f trughs. Frm Figure, ne can see that the nly branch which cntinues thrugh the entire range (, + ) is the slitary-wave branch. Other branches pssess a turning pint at a pint k c > and the branches cease t exist belw that. There is an ne-parameter family f slitary waves with wave height ranging frm t + which cnfirms the result in Therem. Nte that the multi-pulsed slutins exist when the phase speed and amplitude are large, and the depth f the trughs is at least. measured frm the still water level. Our numerical results shw that the slutins with ne trugh exist fr k.47 and the waves with tw trughs exist fr k 3.46. A slutin frm the upper branch has larger H -nrm, and als has larger L -nrm and larger wave height but smaller magnitude f velcity u, than the crrespnding slutin frm the lwer branch with the same k. The velcity u is always psitive. At any given k, if there is a slutin with N trughs, then there is a slutin with N trughs where N is any psitive integer smaller than N. In Figure 3, five three-trugh slutins alng the branch are presented. The lcatins f these slutins in Figure can be determined by the values f the phase speed k.
MULTI-PULSED SOLUTIONS η u 6 4 4 6 ξ 6 4 4 6 ξ Figure 3. Slutins alng the three-trugh slutin branch in Figure, is near the turning pint with k =4.3, + and x are frm the upper branch with k =6andk = 8, and are frm the lwer branch with k =6andk =8. Since the KdV-type equatins d nt pssess multi-pulsed slutins, ur results shw that there is a significant difference between the ne-way mdel equatins and the tw-way mdel systems, althugh they are frmally apprximatins f the same rder t the Euler equatin (cf. [3]). Hwever, figure reveals that the magnitude f all multi-trugh slutins is quite large, s the difference happens in regimes where the slutin is n lnger small. Thus, ur results d nt call int questin the validity f ne r the ther f KdV-type equatins and Bussinesq systems, but des emphasize that when these mdels are used t simulate waves in a practical situatin, they cannt bth be accurate fr large amplitude waves. Indeed, this interesting anmaly can be viewed as anther cautinary lessn; with the cnclusin that it is best t use these mdels well within the range they were derived t describe. It is als interesting t nte that even thugh the regularized Bussinesq system has the same frmal accuracy as the uni-directinal wave equatins such as the KdV-equatin and the regularized lng wave equatin, the slutin set f traveling waves f this bi-directinal wave system is much mre cmplex. References [] C. J. Amick and K. Kirchgassner, A thery f slitary water-waves in the presence f surface tensin, Arch. Ratinal Mech. Anal., (989), pp. 49. [] T. B. Benjamin, J. L. Bna, and D. K. Bse, Slitary-wave slutins f nnlinear prblems, Phil. Trans. Ryal Sc. Lndn Ser. A, 33 (99), pp. 9 44. [3] J. Bna and M. Chen, A Bussinesq system fr tw-way prpagatin f nnlinear dispersive waves, Physica D., 6 (998), pp. 9 4. [4] A. R. Champneys, Y. A. Kuznetsve, and B. Sandstede, A numerical tlbx fr hmclinic bifurcatin analysis, Internat. J. Bifur. Chas Appl. Sci. Engrg., 6 (996), pp. 867 887. [] M. Chen, Slitary-wave and multi-pulsed traveling-wave slutins f bussinesq systems, (submitted). [6] E. J. Dedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsv, B. Sandstede, and X. J. Wang, Cntinuatin and bifurcatin sftware fr rdinary differential equatins, FTP frm pub/dedel/aut, (997). [7] S. Kichenassamy and P. J. Olver, Existence and nnexistence f slitary wave slutins t higherrder mdel evlutin equatins, SIAM J. Math. Anal, 3 (99), pp. 4 66. E-mail address: chen m@math.psu.edu