Solitary wave Solutions of a fifth order model equation for steady capillary-gravity waves over a bump with the Bond number near 1/3

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1 Solitary wave Solutions of a fifth order odel euation for steady capillary-gravity waves over a bup with the Bond nuber near /3 hung-hsien Tsai Departent of Inforation Technology Overseas hinese University, iao Guang Rd., Taichung 47, Taiwan, R.O.. ctsai@ocu.edu.tw Shy-Jen Guo Departent of International Trade National Taichung Institute of Technology ch86997@yahoo.co Abstract: The objective of this paper is to study the existence of solitary wave solutions of a fifth order odel euation for steady capillary-gravity waves over a bup with the Bond nuber near /3. We proved that there exist solitary wave solutions of the odel euation for a sufficiently sooth copact bup. Key-Words: Steady capillary-gravity wave, Solitary wave solution, opact bup. Introduction Progressive capillary-gravity waves on an irrotional incopressible inviscid fluid of constant density with surface tension in a two-diensional channel of finite depth have been studied since nineteen century. Assue that a coordinate syste oving with the wave at a speed is chosen so that in reference to it the wave otion is steady. Let H be the depth of water, g the acceleration of gravity, T the coefficient of surface tension, and ρ the constant density of the fluid. Then there are two nondiensional nubers which are iportant and defined as F = c ( gh), the Froude nuber, andτ = T ( ρgh ), the Bond nuber. When F is not close to, the linear theory of water waves is applicable. But when F approaches to, the solutions of linearized euations of water waves will grow to infinity (Peters and Stoker []). Therefore for F close to nonlinear effect ust be taken into account and thus F = is a critical value. The first study of a solitary wave on water with surface tension is due to Korteweg and DeVries [] after who the K-dV euation with surface tension effect is naed. A stationary K-dV euation with Bond nuber not near 3 can also be forally derived by different approaches. However, ifτ is close to, the foral derivation of the stationary K-dV euation fails. Thusτ = 3is also a critical value. It becoes apparent that the probles for F near and forτ near 3depend on each other and are difficult because they are not only strongly nonlinear, but also very delicate. Since the full nonlinear euations for the water waves are too coplicated to study, it is of interest to study odel ISSN: ISBN:

2 euations. In Hunter and Vanden-Broeck s work [8], a fifth order ordinary differential euation considered as a perturbed stationary K-dV euation was obtained with the assuption that F = + Fє, τ = 3+ τє and є is a sall positive paraeter. By integrating the fifth order ordinary differential euation once and set the con-stant of integration to be zero, then the odel euation becoes 3 Fη η + τη xx ηxxxx = () 45 Euation () has been studied extensively by any authors [-7, 9] and several types of solutions have been found, such as periodic solutions [, 5, 6, 7], solitary wave solutions [-7], generalized solitary wave solutions (solitary waves with osciallatory tails at infinity) in the paraeter regionτ < and F > [,9], etc. Proble Forulation We add a bup y = b() x at the botto of the two-diensional ideal fluid flow and then derive a forced odel euation 3 Fη η + τη xx ηxxxx = b () 45 We follow Zufiria [] to construct a Hailtonian associated to (). Whenb =, we rewrite () as 35 ηxxxx 45τη xx 9Fη+ η =. (3) We ultiply ηx to (3) and integrate the resulting euation, then euation (3) has first integral as H = 45Fη + ηxx ηxxxηx + τη x η, (4) H is a constant. Introducing the change of variables then (4) becoes and we have = η, p = ηxxx 45τη x, = ηxx, p = ηx, H(,,p,p) =45F pp τp, (5) dz = J zh( z) = Az+ g( z) f( z, μ), (6) dx μ = ( τ, F ) R, and 4 z = R, J =, (7) p p 45τ A =,g() z =.(8) 35 9F Therefore (5) is a two degree of freedo Hailtonian with two paraetersτ and F. We also found that euation (6) is reversible and has two 4 euilibriu points, z = (,,,) and z = ( F,,,), i.e. there is a unitary ap R on R 3 4 R, R I, = I, with f ( Rz, μ) = Rf ( z, μ) for all z and μ, R =, and f( z, μ) = f( z, μ) =, for all μ. Because different paraeters ( τ, F) in (5) give rise to different eigenvalues λ for the linearized ISSN: ISBN:

3 syste of (6) at the origin, we divide the paraeter plane( τ, F ) into following nine cases ase ( τ =,F = ): λ=,,,. ase ( τ R,F > ): λ= ± r, ± wi; r, w>. ase ( τ <,F = ): λ=,, ± wi; w>. ase 3 ( τ <,F <, (45 τ ) + 36F > ): ase 4 ase 5 ase 6 ase 7 λ =± wi, ± wi; w > w >. ( τ <,F <, (45 τ ) + 36F = ): λ =± wi, ± wi; w > ( τ R,F <, (45 τ ) + 36F < ): λ = ± a± bi; a, b> ( τ >,F <, (45 τ ) + 36F = ): λ = ± r, ± r; r > ( τ >,F <, (45 τ ) + 36F > ): λ = ± r, ± r ; r > r > ase 8 ( τ>,f = ): λ=,, ± r; r >. We found solitary wave solutions in ase 5, 6, and 7. 3 Proble Solution In this section, we shall construct solitary wave solutions of the odel euation (). In what follows except specified otherwise, the bup b(x) is sufficiently sooth and has a copact support on the inteval [x, x ] with b(x ) = b(x ) =. We rewrite () as follows, 3 ηxxxx 45τη xx 9Fη = 45( b ( x)) + η ) f, (9) The existence of solitary wave solutions are proved by contraction apping theore in ase 5,6, and 7. We consider the proble 35 ηxxxx 45τη xx 9 Fη = b % ( x) η, < x< () η( ) = η( ) = () b % ( x) = 45 b( x) is even. Since we are interested in even solutions, we shall only consider x [, ) hereafter. ase 7: τ >, F = ρτ, ρ ( 45 8,) First, we change () and () to an integral euation by constructing the Green s function Gxs (, ) of ηxxxx 45τη xx 9Fη =, < x < () η () = η () = η( ) = η ( ) =. (3) x xxx x and obtain Gxs = e + e r x s r (, ) ( x+ s ) r( r r ) + e + e r ( r r ) r x s r ( x+ s ) r = (45 τ + (45 τ ) + 36 F ), (4) r = (45 τ (45 τ ) + 36 F ), r > r >.(5) Hence () and (3) is euivalent to 35 η() x = Gxs (,)(() b % s η ()) s ds= I() x+ I() x (6) I ( x) = G( x, s) b % ( s) ds, 35 I( x) = G( x, s)( η ( s)) ds (7) We denote by H the Banach space of even functions f (, ) with the nor k ( k ) f = sup f ( x % ) (8) H d f ( k) ( ) ( x ) sup( e dx % k f ( x ) ) d% =, x< and d % is a constant to be specified later. Lea If gs () H and Y( x) = G( x, s) g( s) ds, then Y( x) H+4 and Y H 4 7 g( x), + H 7 = ( r d % ) ISSN: ISBN:

4 Hereafter we shall use as a generic positive constant, which is independent of Y and g. Let { η η } S7 = H H M, 4, (9) M is positive and will be specified later. We also define an operator 35 Q7 ( η)( x) = G( x, s) ( ( s) η ( s)) ds. b % () We want to show that the operator Q 7 aps S7 into S 7 and it is a contraction. Then, () subject to () has a solitary wave solution. Lea Assueb% H, 4 < d% < r with M and b% satisfying, H 4 ± 4MM M, + M < M < M, M ± = M = 35( ), M = b% b%. H 4 35( ) 7 H 4 then the oper- atorq7 aps Sinto 7 S. 7 Furtherore, we have Lea 3 By following the sae assuptions in Lea, if (9) and () hold, then Q 7 : S 7 S7 is a contraction. M < in{ M +, }. (9) 35( ) ax{, r } < d% < r. () 8( 4 MM ) For ase 5 and ase 6, the proofs of the existence of a solitary wave solution of euation () are siilar to ase 7. 4 onclusion We proved that there exist solitary wave solutions of the odel euation () for a sufficiently sooth copact bup by contraction apping theore in ase 5,6, and 7. Reference: [] Aick,.J. & Mcleod, J.B., A singular perturbation proble in waterwaves, Stability Appl. Anal. ontinuous Media, 7-48 (99) [] Aick,.J. & Toland, J.F., Solitary waves with surface tension I: Trajectories hooclinic to periodic orbits in four diensions, Arch. Rat. Mech. Anal. 8, (99) [3] Aick,.J. & Toland, J.F., Hooclinic orbits in the dynaic phasespace analogy of an elastic strut, Euro. Jnl of Applied Matheatics 3, 97-4 (99) [4] B. Buffoni, A.R. hapneys, A.R., Toland, J.F., Bifurcation and colescence of a plethora of hooclinic orbits for a Hailtonian syste, J.Dyn. Differential Euations 8, -8 (996) [5] hapneys, A.R. Hooclinic orbits in reversible systes and their applications in echanics, fluids and optics, Physica D, (998) [6] oppel, W.A. Stability and Asyptotic Behavior of Differential euations, D.. HEATH AND OMPANY, (965) [7] oppola, V.T. & Rand, R.H. oputer Algebra Ipleention of Lie Transfors for Hailtonian Systes: Application to the Nonlinear Stability of L4, ZAMM.Z.angew. Math. Mech. 69, (989) [8] Hunter J.K., and Vanden-Broeck, J.M., Solitary and periodic gravity-capillary waves of finite aplitude, J. Fluid Mech. 34, 5-9 (983) [9] Iooss, G. & Kirchgässner, K. Water waves for sall surface tension: an approach via noral for, Proceedings of the Royal of Edinburgh A, (99) ISSN: ISBN:

5 [] Korteweg, P.J. and de Vries, G. On the change of the for of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. 39, (895) [] Peters, A.D.; Stoker, J.J. Solitary waves in liuids having non-constant density., o. Pure Appl. Math. 3, 5-64 (96) [] Zufiria, J.A. Weakly nonlinear non-syetric gravity waves on water of finite depth, J. fluid Mech. 8, (987) ISSN: ISBN: