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Solitary-wave and multi-pulsed traveling-wave solutions of Boussinesq systems. (English) Zbl 1034.35108
From the introduction: This paper studies solitary-wave and multi-pulsed solutions of the Boussinesq systems \[ \begin{aligned} & \eta_0+u_x+ (u\eta)_x+ au_{xxx}-b\eta_{xxt}=0,\\ & u_t+\eta_x+uu_x+ c\eta_{xxx}-du_{xxt}=0. \tag{1} \end{aligned} \] The variable \(\eta(x,t)\) is the non-dimensional deviation of water surface from its undisturbed position and \(u(x,t)\) is the non-dimensional horizontal velocity at a height above the bottom of the channel corresponding to \(\theta h_0\), where \(h_0\) is the undisturbed water depth and \(\theta\) lies in \([0,1]\). The real constants \(a,b,c\) and \(d\) satisfy \[ \begin{aligned} a=\tfrac 12\Bigl(\theta^2- \tfrac 13\Bigr)\lambda, \quad & b=\tfrac 12\Bigl(\theta^2-\tfrac 13 \Bigr) (1-\lambda),\\ c=\tfrac 12(1-\theta^2) \mu,\quad & d=\tfrac 12(1-\theta^2) (1-\mu),\tag{2} \end{aligned} \] where \(\lambda\) and \(\mu\) are modeling parameters which can be any real numbers. As explained in J. L. Bona, M. Chen and J.-C. Saut [Boussinesq equations for small-amplitude long wavelength water waves (preprint, 1999)], the systems in (1)–(2) are formally equivalent and have the same formal justification as the classical Boussinesq system for the two-way propagation of small-amplitude long waves. An overall of all the systems in (1)–(2) was presented in (loc. cit.). Previous research was focused on the local and global wellposedness, the regularity of the solutions, and the exact solutions of these systems.
We study the traveling-wave solutions of the systems with general \(a,b,c\) and \(d\). Special attention is given to the regularized Boussinesq system which is the member of (1)–(2) with \(a=c=0\) and \(b=d=\tfrac 16\). This system is particularly interesting because its dispersion relation is stabilizing for all wave numbers and the natural initial-boundary-value problems that arise in laboratory experiments are well-posed J. Bona and M. Chen [Physica D, 116, 191–224 (1998; Zbl 0962.76515)]. Letting \(\xi=x-kt\) where \(k\) is the speed of the traveling-wave solution, one can write the solution in the form \(\eta(x,t)=\eta(\xi)\) and \(u(x,t)=u(\xi)\). Supposing the solution decays at large distance from its crests or troughs, it is natural to impose the boundary conditions \[ \bigl(\eta^{(n)}(\xi), u^{(n)} (\xi)\bigr)\to 0\text{ as }\xi\to\pm\infty,\text{ for at least }n= 0,1,2.\tag{3} \] The functions \(\eta\) and \(u\) satisfy the ordinary differential equations \[ \begin{aligned} & au''+bk\eta'' +u-k\eta+ u\eta=0,\\ & dku''+c\eta''-ku+\eta+ \tfrac 12 u^2=0, \tag{4} \end{aligned} \] where the derivatives are with respect to \(\xi\). It is clear that a homoclinic solution about the origin of (4) will lead to a traveling-wave solution of (1). Therefore, the problem of finding traveling-wave solutions becomes that of finding homoclinic orbits of (4).
The existence of solitary-wave solutions for a number of systems is proved. Interesting new multi-pulsed traveling-wave solutions which consist of an arbitrary number of troughs are found numerically. The bifurcation diagrams of \(N\)-trough solutions with respect to phase speed and system parameters are presented.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B32 Bifurcations in context of PDEs
35Q51 Soliton equations
76B25 Solitary waves for incompressible inviscid fluids
Software:
HomCont
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[1] Amick C.J., J. Differential Equation 54 pp 231– (1984) · Zbl 0557.35074 · doi:10.1016/0022-0396(84)90160-8
[2] Amick C.J., Arch. Rational Mech. Anal 105 pp 1– (1989) · Zbl 0666.76046 · doi:10.1007/BF00251596
[3] Amick C.J., Arch. Rational Mech. Ana 76 pp 9– (1981)
[4] Z. angew Math Phys 43 pp 591– (1992) · Zbl 0766.34027 · doi:10.1007/BF00946252
[5] Benjamin T.B., Phil. Trans. Royal Soc. London 331 pp 195– (1990) · Zbl 0707.35131 · doi:10.1098/rsta.1990.0065
[6] Benjamin T.B., Phil. Trans. Royal Soc. 272 pp 47– (1972) · Zbl 0229.35013 · doi:10.1098/rsta.1972.0032
[7] Bona J.L., Physica D 116 pp 191– (1998) · Zbl 0962.76515 · doi:10.1016/S0167-2789(97)00249-2
[8] Bona, J.L., Chen, M. and Saut, J.C. ”Bousstnesq equations for smali-amplitude long wavelength water waves”.
[9] bona J.L., Math. Proc. Cambridge Phil. Soc 79 pp 167– (1976) · Zbl 0332.76007 · doi:10.1017/S030500410005218X
[10] Boussinesq J.V., Comptes Rendus de l’acadmie de Sciences 72 pp 755– (1871)
[11] Comp. Rend. Hebd. des Scances de L’Acadmie de Sciences 73 pp 256– (1871)
[12] Journal de Mathkmatiques Pures et Appliquées 17 pp 55– (1872)
[13] Buffon B., J. Diff. Eqns 121 pp 109– (1995) · Zbl 0832.34032 · doi:10.1006/jdeq.1995.1123
[14] Champneys A.R., Int. J. Bifurcation and Chaos 4 pp 1447– (1994) · Zbl 0873.34037 · doi:10.1142/S0218127494001143
[15] Champneys A.R., J . Fluid Mech 342 pp 199– (1997) · Zbl 0913.76010 · doi:10.1017/S0022112097005193
[16] Champneys A.R., J. Bifur. Chaos Appl. Sci. Engrg 6 pp 867– (1996) · Zbl 0877.65058 · doi:10.1142/S0218127496000485
[17] Champneys A.R., A shooting technique, Advances in Computational Mathematics 1 pp 81– (1993) · Zbl 0824.65080 · doi:10.1007/BF02070822
[18] Toland J.F., Nonlinearity 6 pp 665– (1993) · Zbl 0789.58035 · doi:10.1088/0951-7715/6/5/002
[19] Chen M., Appl. Math. Lett 11 pp 45– (1998) · Zbl 0942.35140 · doi:10.1016/S0893-9659(98)00078-0
[20] Conte R., system, J. Phys .A 27 pp 2831– (1994) · Zbl 0837.35121 · doi:10.1088/0305-4470/27/8/020
[21] Craig W., Wave Motions 19 pp 367– (1994) · Zbl 0929.76015 · doi:10.1016/0165-2125(94)90003-5
[22] Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B. and Wang, X.J. ”Continuation and bifurcation software for ordinary differential equations”. FTP. doedel/auto
[23] Groves M.D., Nonlinearity 11 pp 341– (1998) · Zbl 0994.34034 · doi:10.1088/0951-7715/11/2/009
[24] Hunt G.W., Proc. Roy. Soc. Lond 425 pp 245– (1989) · Zbl 0697.73043 · doi:10.1098/rspa.1989.0105
[25] Hunt G.W., Proc. R. Sac. Lond 434 pp 485– (1991) · Zbl 0753.73037 · doi:10.1098/rspa.1991.0109
[26] Kaup D.J., Progress of Theoretical Physics 54 pp 396– (1975) · Zbl 1079.37514 · doi:10.1143/PTP.54.396
[27] Krishnan E.V., J. Phys. Soc 51 pp 2391– (1982) · doi:10.1143/JPSJ.51.2391
[28] Pego R., Stud. Appl. Math. 99 (4) pp 311– (1997) · Zbl 0889.35079 · doi:10.1111/1467-9590.00063
[29] Peregrine D.H., proceedings of an advanced seminar conducted by the Mathematics Research Center pp 95– (1972)
[30] Sachs R.L., Physica D 30 pp 1– (1988) · Zbl 0694.35207 · doi:10.1016/0167-2789(88)90095-4
[31] Schonbek M.E., J. Differential Equations 42 pp 325– (1981) · Zbl 0476.35067 · doi:10.1016/0022-0396(81)90108-X
[32] Teng M.H., Methuds Appl. Anal 1 pp 108– (1994)
[33] Toland J.F., Math.Proc. Camb.Phill.Soc 90 pp 343– (1981) · Zbl 0474.76032 · doi:10.1017/S0305004100058801
[34] Toland J.F., Comm. Math. Phys 94 pp 239– (1984) · Zbl 0557.76028 · doi:10.1007/BF01209303
[35] Toland J.F., Proceedings of Symposia in Pure Mathematics 45 pp 447– (1986) · doi:10.1090/pspum/045.2/843631
[36] Ton B.A., Nonlinear Anal 4 pp 15– (1980) · Zbl 0441.76011 · doi:10.1016/0362-546X(80)90032-2
[37] Whitham, G.B. ”Linear and nonlinear waves”. J.Wiley&Sons. · Zbl 0373.76001
[38] Winther R., SIAM J. Numer. Anal 19 pp 561– (1982) · Zbl 0489.76027 · doi:10.1137/0719037
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