Riemann–Hilbert approach for the Camassa–Holm equation on the line

Abstract

We present a Riemann–Hilbert problem formalism for the initial value problem for the Camassa–Holm equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mi>ω</mml:mi> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> <mml:mi>u</mml:mi> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> </mml:math> on the line (CH). We show that: (i) for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ω</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:math> , 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 ω , this soliton train is close to a train of peakons, which are piecewise smooth solutions of the CH equation for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ω</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> .