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Peakon solutions of the shallow water equation. (English) Zbl 0980.35145

This paper deals with the shallow water equation \[ U_t+ 3UU_x= U_{xxt}+ 2U_x U_{xx}- 2KU_x.\tag{1} \] The authors consider singular limits of quasiperiodic solutions when the spectral curve becomes singular and its arithmetic genus drops to zero. The solutions are then expressed in terms of purely exponential \(\tau\)-functions and they describe the finite time interaction of two solitary peakons of (1). To this end a new parametrization of the Jacobi inversion problem is used.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
76B25 Solitary waves for incompressible inviscid fluids
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
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References:

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