Riemann-Hilbert approach for the Camassa-Holm equation on the line. (English) Zbl 1110.35056
Summary: We present a Riemann-Hilbert problem formalism for the initial value problem for the Camassa-Holm equation
\[
u_{t} - u_{txx}+2\omega u_{x}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}
\]
on the line (CH). We show that: (i) for all \(\omega >0\), the solution of this problem can be obtained in a parametric form via the solution of some associated Riemann-Hilbert problem; (ii) for large time, it develops into a train of smooth solitons; (iii) for small \(\omega\) , this soliton train is close to a train of peakons, which are piecewise smooth solutions of the CH equation for \(\omega =0\).
MSC:
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
| 37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |
| 35B40 | Asymptotic behavior of solutions to PDEs |
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