A Riemann–Hilbert approach to the modified Camassa–Holm equation with nonzero boundary conditions
Anne Boutet de Monvel, Iryna Karpenko, Dmitry Shepelsky
Abstract
This paper aims at developing the Riemann–Hilbert problem approach to the modified Camassa–Holm (mCH) equation in the case when the solution is assumed to approach a non-zero constant at both infinities of the space variable. In this case, the spectral problem for the associated Lax pair equation has a continuous spectrum, which allows formulating the inverse spectral problem as a Riemann–Hilbert factorization problem with jump conditions across the real axis. We obtain a representation for the solution of the Cauchy problem for the mCH equation and also a description of certain soliton-type solutions, both regular and non-regular.
Topics & Concepts
MathematicsCamassa–Holm equationMathematical analysisLax pairVariable (mathematics)Hilbert spaceCauchy problemRiemann hypothesisBoundary value problemRiemann–Hilbert problemConstant (computer programming)Riemann problemInitial value problemPure mathematicsIntegrable systemComputer scienceProgramming languageNonlinear Waves and SolitonsNonlinear Photonic SystemsAlgebraic structures and combinatorial models