Cnoidal wave solutions to Boussinesq systems

Transcription

1 IOP PUBLISHING Noninearity 0 007) NONLINEARITY doi:0.088/ /0/6/007 Cnoida wave soutions to Boussinesq systems Hongqiu Chen, Min Chen and Nghiem V Nguyen Department of Mathematica Sciences, University of Memphis, Memphis, TN 385, USA Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA E-mai: hchen@memphis.edu, chen@math.purdue.edu and nvnguyen@math.purdue.edu Received 7 August 006, in fina form Apri 007 Pubished 3 May 007 Onine at stacks.iop.org/non/0/443 Recommended by S Nonnenmacher Abstract In this paper, two different techniques wi be empoyed to study the cnoida wave soutions of the Boussinesq systems. First, the existence of periodic traveing-wave soutions for a arge famiy of systems is estabished by using a topoogica method. Athough this resut guarantees the existence of cnoida wave soutions in a parameter region in the period and phase speed pane, it does not provide the uniqueness nor the non-existence of such soutions in other parameter regions. The expicit soutions are then found by using the Jacobi eiptic function series. Some of these expicit soutions fa in the parameter region where the cnoida wave soutions are proved to exist, and others do not; so the method with Jacobi eiptic functions provides additiona cnoida wave soutions. In addition, the expicit soutions can be used in many ways, such as in testing numerica code and in testing the stabiity of these waves. Mathematics Subject Cassification: 34L30, 35Q5, 35Q53, 35S5,4A6, 46N0, 65L0, 65T40, 65T50, 76B03, 76B5, 76B5 Some figures in this artice are in coour ony in the eectronic version). Introduction In this paper, the existence of periodic traveing-wave soutions to a restricted four-parameter famiy of Boussinesq systems η t + u x + ηu) x + au xxx bη xxt = 0,.) u t + η x + uu x + cη xxx du xxt = 0, that was put forward by Bona, Chen and Saut see [4,5]) to approximate the motion of smaampitude ong waves on the surface of an idea fuid under the force of gravity and in situations where the motion is sensiby two-dimensiona, wi be discussed. The independent variabe x is /07/ $ IOP Pubishing Ltd and London Mathematica Society Printed in the UK 443

2 444 H Chen et a proportiona to distance in the direction of propagation whie t is proportiona to eapsed time, with time scae h 0 /g, where g is the gravitationa force and h 0 scaed to ) the undisturbed water depth. The dependent variabes η and u have the foowing physica interpretation. The quantity ηx, t) is deviation reative to the undisturbed surface, so ηx, t) + h 0 corresponds to the tota depth of the iquid at x, t) whie ux, t) is the horizonta veocity fied at the height θh 0, where 0 θ. From the derivation of.), the parameters a, b, c, d are not independenty specified but must obey the consistency conditions a + b = θ ) and c + d = 3 θ ) 0..) If a 0 connotes a typica wave ampitude and µ a typica waveength, the condition of sma ampitude and ong waveength just mentioned amounts to α = a 0, β = h 0 h 0 µ, α β = a 0µ h 3. 0 As with one-way modes, there are potentiay many different but formay equivaent Boussinesq systems. The pethora of possibiities is owed in the main to the choice of the dependent variabe u at different water depths and to the fact that the ower-order reations can be used systematicay to ater the higher-order terms without disturbing the forma eve of approximation. Systems in.) are first order approximations in α and β to Euer s equation, justified rigorousy by Bona, Coin and Lannes cf [6]). We refer the reader to the papers [4, 5, ] for a further discussion about the derivation and we-posedness of these systems. In this paper, we extend the resuts obtained in [7] see aso [,]) for a singe equation to systems of equations which are suitabe for more genera physica situations namey, the waves are no onger assumed to be uni-directiona). It is noted ater in remark 3. that, even though we wi be abe to transform the system of ordinary differentia equations in the traveing frame to a singe equation, the technique used in [7] sti does not appy. We aso empoy a competey different approach, namey the Jacobi eiptic function series, to find expicit soutions. These two approaches are then compared at the end and they do compement each other. In this paper, attention wi be speciay given to the couped Benjamin, Bona and Mahony BBM)-system: η t + u x + ηu) x 6 η xxt = 0, u t + η x + uu x 6 u xxt = 0,.3) which is when a = c = 0 and b = d = /6. This system is we-posed and with nice properties, such as the presence of the operator 6 x, the existence of Hamitonian and we-deveoped numerica schemes see [3 5]). The paper is organized as foows. Section recas definitions that wi be used and gives a brief review of the topoogica degree theory for positive operators. In section 3, the theory is first appied to the couped BBM-system to show the existence of periodic traveing-wave soutions ηx, t), ux, t)) of the form ηx, t) = ηx ωt) = η n e inπ/)x ωt), ux, t) = ux ωt) = u n e inπ/)x ωt),.4)

3 Cnoida wave soutions to Boussinesq systems 445 where and ω connote the haf-period and the phase speed, respectivey. It is proved that for any ω > and for any arge enough, there exists an infinitey smooth non-trivia soution in the form of.4). The section ends with genera resuts on systems.) and.), where b, d > 0 and a,c 0. It is proved that for any ω > max{, ac bd } or ω < min{, a+c b+d, ac }, and for bd any arge enough, there exists an infinitey smooth non-trivia soution in the form of.4). In section 4, attention wi be directed to expicit periodic traveing-wave soutions of the couped BBM-system. The expicit series soutions in terms of Jacobi eiptic functions are found for ω, ω)), where ω is in 0, 0.39) or in 5, ). The soutions in this form aso exist for other systems in.) and detais wi appear esewhere.. Preiminaries and notation In this section, we reca definitions that wi be used and give a brief review of the topoogica degree theory for positive operators. For p<+ and an open set in R, et L p ) be the usua Banach space of rea or compex-vaued, Lebesgue measurabe functions defined on with the norm f p L p ) = f p dx and L ) be the space of measurabe, essentiay bounded functions with the norm f L = ess sup fx). x When it introduces no confusion, L p ) is simpy written as L p. Simiary, et C denote the compex fied and p be the usua Banach space { } p u ={u n } : u n C, u n p < with the norm u p p = u n p, whereas is defined as { u ={u n } : u n C, sup <n< } u n < with its usua norm u = sup <n< u n. The foowing eementary facts from anaysis are recaed. Any f ={f n } defines a periodic function f of period, where fx)= f n e i nπx..) Vice versa, if f L,), then f can be expanded amost everywhere as a series in the form.), with f n = fx)e inπx/) dx. In this sense, one can identify f L,) with the sequence of its Fourier coefficients f ={f n }. Moreover, f L = ) f. For any u and v in, the convoution u v is defined as u v ={u v) n }, where u v) n = u n kv k. Since u v u v, it foows that u v.

4 446 H Chen et a For the convenience of the reader, a brief review of the topoogica degree theory for positive operators on Banach spaces is given here and we refer the reader to the works of Krasnose skii [0, ], Granas [9] and Benjamin et a [] for detais. Let X be a Banach space equipped with the norm X. We define a cosed subset K X as a cone, if the foowing conditions are satisfied: i) λk {λf : f K} K for a λ 0, ii) K + K {f + g : f, g K} K, iii) K { K} K { f : f K} ={0}. For any 0 <r<r<, denote B r ={f X : f X <r}, K r = K B r, K r = K B r B r ={f X : f X = r}, and Kr R ={f K : r< f X <R}. An operator A defined on K is said to be positive if AK K. A positive operator A is compact if AK r ) has a compact cosure. Note that the operator A is not necessariy inear. In fact, for the remaining of our paper A wi be noninear. A tripe K, A,U)is caed admissibe if i) K is a convex subset of X, ii) U K is open in the reative topoogy on K, iii) A : K K is continuous and AU) is a subset of a compact set in K and iv) A has no fixed point on U, the boundary of the open set U in the reative topoogy on K. Denote the set of a admissibe tripes by T. Let K, A,U) T and A be a constant mapping on K, namey there is a point a K such that Au = a for every u K. The fixed point index of the positive operator A on U is defined as { ifa U, ik, A, U) = 0 ifa/ U. We mention here, among the many properties of ik, A,U), the three that wi be of use in our current probem. a) Homotopy invariance) If two tripes K, A,U)and K, B,U) T and A is homotopic to B on U, then ik, A,U)= ik, B,U). b) Fixed point property) If K, A,U) T and ik, A,U) 0, then A has at east one fixed point in U. c) Additivity) If K, A,U) T and U,U,...,U n is a coection of mutuay disjoint open subsets of U such that Au u for a u U \ n j= U j, then ik, A,U) = n j= ik, A,U j ). The foowing three emmas are taken directy from [] in which K is a cone, the operator A is positive, continuous and compact on K. Lemma.. Suppose that 0 <ρ< and that either a) Ax x/ K for a x K ρ or b) tax x for a x K ρ and a t [0, ]. Then ik, A,K ρ ) =.

5 Cnoida wave soutions to Boussinesq systems 447 Lemma.. Suppose that 0 <ρ< and that either ã) x Ax / K for a x K ρ or b) there exists a non-zero x K such that x Ax λ x for a x K ρ and a λ 0. Then ik, A,K ρ ) = 0. Lemma.3. Let K, A, U) be admissibe. If there exists a non-zero x K such that x Ax λ x for a x U and a λ 0, then ik,a,u)= 0. The foowing theorem is an immediate consequence of the first two emmas. Theorem.4. Suppose that either a) or b) hods for an r satisfying 0 <r< and that either ã) or b) hods for an R satisfying r<r<. Then A has at east one fixed point in Kr R {f K,r < f X <R}. Moreover, ik, A,Kr R) =. The theory described above wi be utiized in section 3 to estabish the existence of cnoida wave soutions for.) as foows. By substituting.4) into systems.) and equating the Fourier coefficients, one obtains an infinite system which can be posed as a fixed point probem on a certain cone. Using the theory above, the index of the operator associated with this fixed point probem is shown to be non-zero hence, there must exist at east one soution in the cone). The anaysis is compicated a bit by the fact that the trivia constant) soution ies in the cone. By choosing the haf-period arge enough, however, one can then excude this trivia soution. 3. Existence theorem 3.. The couped BBM-system Let ηx, t) = ηx ωt), ux, t) = ux ωt) be the traveing-wave soution. Substituting it into.3) yieds ωη + u + ηu) + 6 ωη = 0, ωu + η + u ) + 6 ωu = 0, 3.) where the primes denote the derivatives with respect to the moving frame ξ = x ωt. By integrating once and etting the constants of integration be C and C, the system becomes ωη + u + ηu + 6 ωη = C, ωu + η + u + 6 ωu = C. Assuming f, g) is a constant soution, we see that ωf + g + fg C = 0, ωg + f + g C = 0. Soving f from the second equation and substituting it into the first, one sees that g satisfies a cubic equation which is for sure to have a rea root. Hence there is at east one constant soution.

6 448 H Chen et a By introducing the new dependent variabes η = η f and ū = u g, the system in η and ū then reads as ω η + ū + ηū) + 6 ω η + f ū + g η = 0, ωū + η + ū + 6 ωū + gū = 0. 3.) For simpicity in notation, we wi study the case with f = g = 0, namey the case where C = C = 0. The system then becomes ωη + u + ηu) + 6 ωη = 0, ωu + η + u + 6 ωu = ) It wi become cear ater that the method used in 3.3) can be extended to some other cases with non-zero C and C. Remark 3.. Notice that the resut obtained here wi ony be an existence theory due to severa factors, such as the method of index theory and the integrating constants being taken to be zero. It is easy to see that there are other soutions. For exampe, η = c, u = c with c and c being any constants are soutions to 3.), but 3.3) ony admits three constant soutions, namey 0, 0), ω 4 ± ω ω +8)/4,3ω ω +8)/), for any rea ω. The study of the existence of periodic soutions to 3.3) is carried out as foows. Substituting.4) into 3.3) and equating the Fourier coefficients yied the foowing infinite system: ωη n + u n ω ) nπ η n = η u) n, ωu n + η n ω ) nπ u n = 3.4) u u) n, where η ={η n }, u ={u n } and <n<. The system 3.4) can be put into a more convenient matrix form [ ] ηn η u) n T n = u n u u), 3.5) n where ω + ) nπ ) T n = ω + ) nπ ). 3.6) For the phase speed ω >, T n is invertibe for a n with T ω + ) nπ ) n = ω + ) nπ ) ω + ) nπ ). 3.7)

7 Cnoida wave soutions to Boussinesq systems 449 The norm of T n is defined as T n = max j i= T n i, j) = ω + nπ ) ), where Tn i, j) is the i, j)th entry of the matrix Tn. To set up the probem as a fixed point probem, a set K = X is defined by K ={η, u) ={η n,u n )} X : η n,u n ) = η n,u n ), η 0 η 0, u 0 u 0}. One can easiy verify that K is indeed a cone in X equipped with the norm η, u) X = η n + u n ). An operator A on K is now defined as foows: for any w η, u) ={η n,u n )} K, Aw ={Aw) n }, where η u) n Aw) n = T n u u). 3.8) n Thus 3.5) can be written in the form w = Aw and the fixed points of operator A in the cone K are soutions of 3.5). Remark 3.. Since 3.3) can be transformed into a singe equation, by soving for η from the second equation and substituting it into the first equation, it is natura to wonder if the existing theorems for the singe equation such as the ones in [7] can be appied directy. Unfortunatey, this attempt does not succeed since the noninear terms in the resuting equation are not of one sign. ω u 4) + u ω)ωu +8u 36wu) +6ωu ) 36u = 0 3.9) Remark 3.3. The invertibiity of T n hinges on the condition that ω >. Thus the existence resut we are aiming to estabish is ony for the case when ω >. Notice that if ηx ωt), ux ωt)) is a soution, ηx + ωt), ux + ωt)) is aso a soution; thus the existence of a cnoida wave soution for ω> wi impy the existence of a cnoida wave soution for ω<. For the rest of the section 3., ω> wi be assumed. Lemma 3.4. For ω>, A is a continuous, positive and compact operator on the cone K. Proof. The proof foows the same ines as in [7] and is reproduced here for the reader s convenience. a) A is a positive operator on K; i.e. A maps K into itsef. For any w = η, u) K, et τ n η u) n = η n ku k. It is easy to verify that for a n 0, τ n = η n k u k = η n+k u k = η n k) u k = η n m u m = τ n m=

8 450 H Chen et a and τ n τ n+ = η n k u k k=0 η n+ k u k = η n k u k + η n+k+ u k+ k=0 k=0 η n++k u k η n k u k+ = η n k η n++k )u k u k+ ) 0. Therefore, τ n is a decreasing function of n and 0 τ n τ 0 = η u) 0 η u w X. 3.0) Since each entry Tn i, j) of Tn is positive and even in n, decreasing with respect to n, the sequence { Tn } is square-summabe, i.e. Tn < ; it foows immediatey that AK K. b) A is continuous. Let w = η, u) and w = η, ū) be two arbitrary eements in K. For a n, the difference Aw) n A w) n can be bounded componentwise, namey, η u) n η ū) n u ū η + ū η η, u u) n ū ū) n u ū ū + u ). Hence, it foows that Aw A w X γ [ u ū η + ū + u ) + ū η η ] where [ γ = k=0 k=0 γ η + u + η + ū ) η η + u ū ) 4γ η + u + η + ū ) w w X γ D w w X, ] Tn = ω + 6 ) nπ ) 3.) and D = η + u + η + ū ). The operator A is now readiy seen to be continuous from K into itsef. c) A is compact. Consider a bounded set M in X, say M {w = η, u) X : w X B}. For each N,a cut-off operator A N is defined as foows: { Aw)n, for N n N, A N w) n = 3.) 0, otherwise. Then A N is a compact operator having a rank of N +) as A is continuous. Now, for w M, Aw) n Tn η u + u η u + u. 3.3) Thus, A N w Aw X 4B4 γ N 4B4 γ,

9 Cnoida wave soutions to Boussinesq systems 45 [ where γ N = n N ] Tn. Consequenty, sup A N w Aw X 0asN. w M Thus A is compact as it is the uniform imit of compact operators on bounded sets. Attention is now turned to the fixed points of A. A fixed point p, q) ={p n,q n )} is said to be trivia if p n = q n = 0 for a n 0. It is fairy straightforward to check that for a trivia fixed point of 3.5), p 0 = 0 if and ony if q 0 = 0. A trivia soution with p 0 = q 0 = 0 corresponds to the origin whie p 0,q 0 0 corresponds to a non-zero constant soution. Remark 3.5. As mentioned in remark 3. the operator A has three trivia fixed points, but ony two of them are in K. They are the origin and the constant traveing-wave soution p, q ), where p =, 0,p 0, 0, ) and q =, 0,q 0, 0, ) with p 0 = 4 ω 4+ω ω +8) and q 0 = 3ω ω +8. The other does not beong to K as 4 ω 4 ω ω +8)<0. Proposition 3.6. Let γ be as defined in 3.). Then for any r satisfying { } 0 <r<r 0 min γ, p, q ) X, 3.4) w taw for a w K r and for a t [0, ]. Proof. Suppose there exist a w K r and a t [0, ] such that w = taw. Then using 3.3) on a Aw) n yieds w X = r = t Aw) n ) 4r 4 γ, 3.5) which impies that r γ, a contradiction. Proposition 3.7. For any { ω ) R>R 0 max γ ω ) + 3), p, q ) X }, 3.6) ω there exists a non-zero w K such that w Aw λ w, for a w K R and a λ 0. [ ] ηn Proof. Foowing the same idea as in [7], et w ={ η, ũ) n } be given by = [ ]. u n +n Ceary w 0 and w K. Again, suppose to the contrary that there exist a w = η, u) K R and a λ 0 such that for a n, [ ] ηn ω + ) nπ ) η u) n + u u) n = nπ + λ w. u n ω + 6 ) ) ω + 6 nπ ) ) u u) n + η u) n 3.7)

10 45 H Chen et a In particuar, η 0 = ω η ω k u k + u u 0 = η ω k u k + ω u η n T n η k u k + u ) + λ, ) + λ. 3.8) Since w = 0 / K R, and that η 0 = 0 if and ony if u 0 = 0, η 0 0 and u 0 0. Consequenty, we obtain from 3.8) the bounds 0 <η 0 ω, 0 <u 0 ω ω,0 λ ω and ω η k u k + u ω ). 3.9) ω One can see from 3.7) and from the fact that τ n is decreasing in n that ) + Therefore, R = ηn + u n u n T n 4 η k u k + u { T n T n λ +n, ) λ + +n. η k u k + u η k u k + u ) +4 ) } λ + +n λ +n ). Hence, from 3.), 3.9) and using the fact that +n ) 3, the concusion R ω ) γ ω ) + 3) ω 3.0) is drawn, which contradicts the assumption on R. Theorem 3.8. Let r and R be as above. Then the fixed point index of A on K R r ={w K : r< w X <R} is ik, A,K R r ) =. Proof. This foows immediatey from theorem.4 and propositions 3.6 and 3.7. An immediate consequence of theorem 3.8 is that there must be at east one fixed point of A in Kr R. However, the anaysis is not yet compete since the constant periodic soution p, q ) Kr R coud be the ony fixed point in KR r. This case is excuded through the foowing. Let ɛ = ɛ) > 0 be an arbitrariy fixed, sufficienty sma number whose vaue wi be determined ater. Let K ɛ p, q ) ={w = η, u) K : η, u) p, q ) X <ɛ} and K ɛ p, q ) ={w = η, u) K : η, u) p, q ) X = ɛ}.

11 Cnoida wave soutions to Boussinesq systems 453 Lemma 3.9. If p, q ) is the ony fixed point of A in Kr R, then when the haf-period >0is chosen arge enough, ik, A,K ɛ p, q )) = 0. Proof. The ɛ wi be chosen sma enough so K ɛ is in Kr R. Therefore, the emma is proved, owing to emma.3, if one can show that I A) K ɛ p, q ) omits the ray {λ w : λ 0}, where w K is defined as in proposition 3.7. Now suppose that there are a w = η, u) K ɛ p, q ) and a λ 0 such that w Aw = λ w. Then for a n Z, [ ] ηn ω + ) nπ ) η u) n + u u) n = u n ω + ) nπ ) ω + ) nπ ) + λ w. u u) n + η u) n 3.) In particuar, for n =, η Aη 0 u + η u 0 ) + Bu 0 u + λ/, 3.) u Au 0 u + Bη 0 u + η u 0 ) + λ/, where ω + ) π ) A = ω + 6 π ) ) and B = ω + 6 π ) ). Thus η + u u 0 A + B)η + u ) + A + B)η 0 u + λ. 3.3) Since w K ɛ p, q ), it can be written as w = η, u) = p, q ) + ɛ η, ũ), where η, ũ) X =. Note that for n { { ηn = η n /ɛ 0, ηn η n+, and ũ n = u n /ɛ 0, ũ n ũ n+. In terms of the new variabes η, ũ),3.3) can be written as ɛ ɛ η + ũ ) ω + ω ) π q 0 + ɛũ 0 ) η + ũ ) + ɛũ p 0 + ɛ η 0 ) ω + ω ) π + λ. 3.4) The haf-period can be chosen arge enough so that q 0 = 3ω ω +8 >ω+ ω ) π. 3.5) The expicit condition for is >L 0 π ωω ++ ω +8). 3.6) ω ) Notice that p 0 q 0,so p 0 >ω+ ω ) π. 3.7)

12 454 H Chen et a The number ɛ can then be chosen sma enough to satisfy q 0 + ɛũ 0 >ω+ ω ) π and p 0 + ɛ η 0 >ω+ ω ) π. It foows immediatey from 3.4) that λ = 0 and η =ũ = 0, which impy that η n =ũ n = 0 for a n 0. Using 3.) for n = 0 yieds that η 0 =ũ 0 = 0. This contradicts the assumption that w = η, u) K ɛ p, q). Theorem 3.0. For w>, if the haf-period is chosen arge enough as in 3.6), then the operator A has a non-trivia fixed point w = η, ū) in the cone segment Kr R. Moreover, i) η n, ū n > 0 for every n Z and ii) for any σ 0, the sequences { n σ η n } and { n σ ū n } are in. Therefore, the non-trivia fixed point soution is infinitey smooth. Proof. The existence of a non-trivia fixed point is the consequence of theorem 3.8 and emma 3.9. It is eft to estabish i) and ii). Let n = N be the smaest non-negative integer such that either η N or ū N is zero. Notice that N>since the soution is non-trivia and η = 0 if and ony if u = 0, which wi ead to the trivia soution. Reca that as a fixed point of A, w = η, ū) = X is given by [ ] ηn ω + ) nπ ) η ū) n + ū ū) n = ū n ω + ) nπ ) ω + ) nπ ). 3.8) ū ū) n + η ū) n If η N = 0, it foows from 3.8) and from both η k and ū k > 0 for every k [ N +,N ] that 0 = ω + ) Nπ ) η ū) N + ū ū) N ω + ) Nπ ) η N ū + ūn ū > 0 which is certainy not true. The case ū N = 0 can be handed simiary. Thus η n, ū n > 0 for every n Z and i) is proved. Since η ū) n η ū < and ū ū) n ū <, it foows that for n, there exists a constant C 0 independent of n satisfying η n C Tn C 0 n and ū n C Tn C 0 n. 3.9) Therefore, { η n } and {ū n } are in. The continuous bootstrapping argument can now be utiized to show that for any σ 0, the sequences { n σ η n } and { n σ ū n } are in. The arguments start with noting from 3.9) that for n, and n η n C 0 n, n ū n C 0 n + n ) η n C, + n ) ū n C. 3.30)

13 Cnoida wave soutions to Boussinesq systems 455 Therefore η ū) n = Simiary, η n k ū k = + n ū ū) n = + n k ) η n k + k )ū k + n k )+ k ) + n k ) η n k + k )ū k C + n. ū n k ū k C + n. It now foows that for n, η n Tn η ū) n + ) ū ū) n C n 3, ū n Tn η ū) n + ) ū ū) n C n ) Therefore the sequences { n η n } and { n ū n } are in. Simiary, one can show that for any σ and for n, n σ η n C n, n σ ū n C n, +n ) σ η n C, +n ) σ ū n C and hence η n C σ and ū n σ + n C σ n σ + 3.3) which eads to the concusion that the sequences { n σ η n } and { n σ ū n } are in. The resuts are now summarized in the foowing theorem. Theorem 3.. For any phase speed ω > and for any >L 0 where L 0 is defined in 3.6), there exists an infinitey smooth non-trivia cnoida wave soution with period in the form of ηx, t) = ηx ωt) = ux, t) = ux ωt) = Moreover, the norm of the soution satisfies ) r0 < η, u) L L <) R0, η n e inπ/)x ωt), u n e inπ/)x ωt). where r 0 and R 0 are functions of ω and defined in 3.4) and 3.6).

14 456 H Chen et a 3.. The genera case Attention is now turned to the genera four-parameter famiy of Boussinesq systems.) and.). Upon substituting the form of the soution.4) into.) and carrying out simiar cacuations, the foowing infinite system is obtained: ] ] D n [ ηn u n [ η u)n = u u) n, 3.33) where ) nπ ) ) nπ ω +b a ) D n = ) nπ ) nπ ) c ). ω +d To use the same argument as for the couped BBM-system, the matrix ) nπ ) ) nπ ) Dn ω +d a = detd n ) ) nπ ) ) nπ ), c ω +b where ) nπ ) nπ 4 detd n ) = ω + ω b + d) + a + c)) + ω bd ac) is required to have positive and even entries for a n and each entry is to be squaresummabe and decreasing with respect to n. Therefore ω has two admissibe regions, namey ω > max{, ac bd } and ω < min{, a+c b+d, ac }. When a or c is zero, which incudes the bd case of the couped BBM-system, the two regions degenerate to one which is ω >. An operator B is now defined as foows: for any w ={η, u) n } X, Bw ={Bw) n }, where η u) n Bw) n = Dn u u). n System 3.33) can be written in the form w = Bw and the fixed points of operator B are soutions of 3.33). Hence, one arrives at the foowing concusions. Theorem 3.. For a Boussinesq system with b, d > 0, { a,c 0 which satisfy the consistency condition.) and for any phase speed ω > max, ac } and for > L 0, where L 0 bd depends on ω and the dispersive constants a, b, c, d, there exists an infinitey smooth nontrivia cnoida wave soution with period in the form of ηx, t) = ηx ωt) = η n e inπ/)x ωt), ux, t) = ux ωt) = u n e inπ/)x ωt). By introducing the new variabes η, ū) = η, u) and appying the same arguments for sma phase veocity, namey for ω < min{, a + c)/b + d),ac/bd)}, the foowing existence theorem is a straightforward consequence.

15 Cnoida wave soutions to Boussinesq systems 457 Theorem 3.3. For a Boussinesq system with b, d > { 0, a,c 0 which satisfy the consistency condition.) and for any phase speed ω < min, a + c b + d, ac } and for >L 0, where bd L 0 depends on ω and the dispersive constants a, b, c, d, there exists an infinitey smooth non-trivia cnoida wave soution with period in the form of ηx, t) = ηx ωt) = η n e inπ/)x ωt), ux, t) = ux ωt) = 4. Expicit cnoida wave soutions u n e inπ/)x ωt). In this section, the expicit cnoida wave soutions of the couped BBM-system are found by using the Jacobi eiptic function series on 3.9). The soutions in this form aso exist for other systems in.) and detais wi appear esewhere. For competeness, the definition of the Jacobi eiptic functions is recaed here. Let φ v = dt for 0 m, 0 m sin t which is denoted as v = Fφ,m). Then φ = F v, m). The two basic Jacobi eiptic functions cnv, m) and snv, m) are defined as snv, m) = sinφ) = sinf v, m)) and cnv, m) = cosφ) = cosf v, m)), where m is known as the eiptic moduus. It is easy to see that cnv, m) and snv, m) are periodic functions in v with period π/ 4 dt. 0 m sin t Moreover, these functions are generaizations of the trigonometric and hyperboic functions which satisfy snv, 0) = sinv), cnv, 0) = cosv), cnv, ) = sechv), snv, ) = tanhv). To search for expicit soutions of 3.3), a function u in the form of the Jacobi eiptic function series satisfying 3.9) is sought. The deviation η can then be evauated from the second equation in 3.3). Let M uξ) = a j cn j λξ, m) j=0 and substitute it into 3.9). Using the basic facts, such as cn v, m) = snv, m) m + m cn v, m) and 4.) cn v, m) +snv, m) =, an equation in terms of cn is obtained. By baancing the terms in the highest order, it yieds M =. Assuming, for simpicity, that u is an even soution, it then takes the form uξ) = a 0 + a cn λξ, m). 4.)

16 458 H Chen et a The corresponding haf-period of u can therefore be computed using an eiptic integra = λ π 0 dt, 4.3) m sin t by noting that in 4.) there are no odd power terms of cn, so the fundamenta period is haf of that for cnλξ, m). Substituting 4.) into 3.9) and aso using cn ) v, m) = m +4m )cn v, m) 6m cn 4 v, m), one obtains an equation in the form of c 0 + c cnλξ, m) + c 4 cnλξ, m) 4 + c 6 cnλξ, m) 6 = 0, where c i are functions of a 0,a,ω,λand m. By requiring a the coefficients c i = 0, i = 0,, 4 and 6, one arrives at a noninear agebraic system in a 0,a,ω,λand m. With the hep of the software Mape, a set of non-trivia soutions is found where with λ being a root of a 0 = 9 ω9 0m λ +0λ ), a = 0 3 ωλ m, Fm,λ ) 60m )m )m +)λ m 4 m +)λ 4 79 = 0 4.4) and ω satisfying where 35m )m )m +)ω = G m,λ ), G m,λ ) = 70m )m )m +) + 367λ m 4 m +) +440λ 4 m )m )m +)m 4 m +). 4.5) Therefore, for a fixed m [0, ] and m, the cnoida wave soutions exist if there are rea positive soutions λ to Fm,λ ) = 0 which satisfy ω m,λ G m,λ ) ) = ) 35m )m )m +) It is worth noting that without oss of generaity, ony the positive soutions of λ and ω need to be considered because λ and λ offer the same soution and ω and ω offer a pair of soutions, one is right propagating and the other is eft propagating. In figure, the curves where Fm,λ ) = 0 are potted and the regions are separated by the signs of ω m,λ ) which is cacuated from 4.6). It is shown that for each fixed m >, there are two positive λ satisfying Fm,λ ) = 0. By denoting the two soutions of F = 0as λ m ) and λ m ) with λ λ, it is cear that λ is in the ω -positive region. This is aso the case for λ with 0.50 m. Figure b) zooms in a region near the curve λ m ) to show this fact. For m in 0.5, 0.50), the oss of significant digits occurs in the computation. Specificay, when m = 0.50, λ.4 03 and ω The situation is more severe as m + with the resuting λ and ω very cose to 0. The sign of ω, which dictates the existence of the second branch of cnoida wave soutions, is hard to identify. It is worth noting that we are not particuary interested in this region since as λ and

17 Cnoida wave soutions to Boussinesq systems a) F=0 λ m ) w <0 w = b) w < λ w <0 w >0 F=0 λ m ) λ w 3.8 =0 0 w < m m w >0 F=0 λ m ) w =0 Figure. a) The ocation Fm,λ ) = 0 and the region where ω m,λ ) 0 are indicated; b) same as a) with a zoom in the boxed region. 7 6 m m a) cnoida wave exists 7 6 b) 5 5 -haf period 4 3 branch λ -haf period 4 3 branch λ m / m / ω phase veocity ω phase veocity Figure. a) Location where the cnoida wave soutions exist: at any point above the soid ine existence theory) and at any point on dashed ines expicit formua); b) same as a) with a zoom near the origin. m +, the corresponding haf-period, according to 4.3), is approaching zero. Therefore, the wave ength is sma which is outside the vaidity region of the Boussinesq systems. The corresponding haf-period and the phase veocity ω for the soutions corresponding to branches λ and λ are potted in figure a). The branch corresponding to λ has phase veocity from 5 to, asm varies from to, with the corresponding haf-period from + to around.95. As m + which corresponds to the right endpoint, λ 9 and ω. When m = which corresponds to the eft endpoint, one recovers the known soitary wave soution uξ) =± 5 ) 3 sech ξ, 0 ηξ) = 5 ) )) 4.7) 3 3 sech ξ 3sech 4 ξ, 4 0 0

18 460 H Chen et a 3 ω=0.3 and haf period=.5 0 ω=7. and haf period=.0 η soid ine, u dashed ine 0 η soid ine, u dashed ine ξ=x ωt ξ=x ωt Figure 3. Exampes of soutions. with ω = 5, found in [3, 8]. This cacuation provides an expicit and concrete proof that the soitary wave 4.7) is a imit of cnoida wave soutions as approaches infinity, a fact which may not be easy to prove otherwise. It is expected that this is aso the case for other branches of cnoida wave soutions. The branch corresponding to λ are cnoida wave soutions with the phase veocity between 0 and c 0 = ) It is worth noting that these soutions are not in the parameter range where the index theory in section 3 is appicabe. As m +, the phase veocity and haf-period appear to approach zero and the detai around that point is shown in figure b). At m =, this is again a soitary wave soution found in [8]. Figure a) aso shows the region, which is above the soid curve, where for any point in the region, namey for a specified haf-fundamenta-period and a phase speed ω, there is a cnoida wave soution corresponding to that point the resut of section 3). The dashed ines indicate the parameters where the expicit soutions exist. The soutions corresponding to the part of branch λ ocated beow the soid curve and the ones corresponding to branch λ are not covered by theorem 3. in section 3 which is non-constructive). Therefore it demonstrates that the condition 3.6) for existence of the cnoida wave soution is sufficient, but not necessary. In figure 3, two pairs of expicit soutions, one with ω = 0.3 and another with ω = 7., are potted. The soid ines are for ηx, t) = ηξ) and the dashed ines are for ux, t) = uξ). It is noted that these soutions are not in the regime covered by theorem 3. according to figure a). The above resuts are summarized in the foowing theorem. Theorem 4.. For any fixed m with <m, there are two positive roots, denoted by λ and λ with λ λ,offm,λ ) = 0, where F is defined in 4.4). Denoting ξ = x ωt, the corresponding cnoida wave soutions in the form of ux, t) = uξ) = a 0 + a cn λξ, m) 4.8) exist for λ = λ m) with <m and for λ = λ m) with 0.50 m. The phase speed ω is defined by 4.5) and 4.6) and a 0 = 9 ω9+0m λ +0λ ), a = 0 3 ωλ m. Moreover, the soution ηx, t) has the form ηx, t) = ωu u ω 6 λ u ξξ = b 0 + b cn λξ, m) + b 4 cn 4 λξ, m),

19 Cnoida wave soutions to Boussinesq systems 46 where b 0 = ω 0 8 ω λ 4 m 4 m +5), b = 40 7 ω λ 4 m m ), b 4 = 0 9 ω λ 4 m 4. In summary, for the couped BBM-system, we proved the existence of cnoida waves in a D region on the ω pane, where is the haf-period and ω the phase speed, by a topoogica method and find expicit cnoida wave soutions for = ω) which is determined by 4.3) 4.6). The region and the function = ω) are shown in figure a). For Boussinesq systems with b, d > 0 and a,c 0 which satisfy.), we proved, using a topoogica method, the existence of cnoida waves in two regions, which degenerate to one when a or c is 0, on the ω pane. Acknowedgments N V Nguyen gratefuy acknowedges support by the US Nationa Science Foundation under grant DMS References [] Benjamin T B, Bona J L and Bose D K 990 Soitary-wave soutions of noninear probems Phi. Trans. R. Soc. Lond. Ser. A [] Bona J and Chen H 00 Soitary waves in noninear dispersive systems Discrete Contin. Dyn. Syst. Ser. B [3] Bona J L and Chen M 998 A Boussinesq system for two-way propagation of noninear dispersive waves Physica D [4] Bona J L, Chen M and Saut J-C 00 Boussinesq equations and other systems for sma-ampitude ong waves in noninear dispersive media: I. Derivation and the inear theory J. Nonin. Sci [5] Bona J L, Chen M and Saut J-C 004 Boussinesq equations and other systems for sma-ampitude ong waves in noninear dispersive media: II. Noninear theory Noninearity [6] Bona J L, Coin T and Lannes D 005 Long wave approximations for water waves Arch. Ration. Mech. Ana [7] Chen H 004 Existence of periodic traveing-wave soutions of noninear, dispersive wave equations Noninearity [8] Chen M 998 Exact soutions of various Boussinesq systems App. Math. Lett [9] Granas A 97 The Leray-Schauder index and fixed point theory for arbitrary ANRs Bu. Soc. Math. Fr [0] Krasnose skiĭ M A 964 Positive Soutions of Operator Equations ed L F Boron Groningen: P. Noordhoff Ltd) Transated from the Russian by R E Faherty) [] Krasnose skiĭ M A 964 Topoogica Methods in the Theory of Noninear Integra Equations ed J Burak NY, USA: Pergamon Press) Transated by A H Armstrong) [] Peregrine D H 967 Long waves on a beach J. Fuid Mech