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The geometry of peaked solitons and billiard solutions of a class of integrable PDE’s. (English) Zbl 0808.35124
The following equation governing propagation of small-amplitude waves on a surface of a shallow layer of water is considered: \[ U_ t- U_{xxt}+ kU_ x+ 3UU_ x= 2U_ x U_{xx}+ UU_{xxx}. \tag{1} \] Unlike the famous KdV equation, equation (1) does not assume that the wave length is much greater than the depth of water. The parameter \(k\) is related to a critical wave speed in the shallow water layer, while the variable \(U(x,t)\) represents the \(x\)-component of the velocity. Equation (1) is integrable, possessing a Lax pair and a bi-Hamiltonian structure. Recently, the particular case \(k=0\) was studied in detail. It was shown that in this case equation (1) gives rise to the so-called peakons (peaked solitons). In this work, an objective is to construct classes of exact solutions describing multipeakon states, as well as some other solution, e.g., solitons on a quasiperiodic background. The approach to the problem is based not on the inverse scattering technique, but rather on a technique of complex geometry. The integrable PDE (1) is reduced to integrable finite-dimensional Hamiltonian systems, and then special solutions to the PDE are expressed through solutions to the finite- dimensional systems. This approach allows to obtain, e.g., exact solutions describing a collision of two solitons. For this case, explicit formulas giving phase shifts produced by the collision are obtained (they prove to be essentially different from the well-known analogous formulas for the KdV solitons). Using a special limiting procedure, the so-called billiard solutions to the PDE (1) are obtained, too. These are exact solutions which remain finite everywhere along with the first derivative, but the derivative suffers discontinuities at certain points. It is stated that this is the first example of the billiard solutions which are obtained as weak solutions to an integrable nonlinear PDE.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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