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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
HomCont
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