Karpenko I. The modified Camassa–Holm equation with non-vanishing boundary conditions by a Riemann–Hilbert approach

The aim of this thesis is to develop the Inverse Scattering Transform method, in the form of the Riemann-Hilbert (RHP), for the modified Camassa-Holm equation (mCH) m_t +(u^2-u_x^2)m_x = 0, m=u-u_{xx}, with the further aim of studying the properties of solutions to this equation, in particular, the large time asymptotics. A characteristic feature of the Cauchy problems considered for this equation is the non-zero conditions on the behavior of the solution when the spatial variable x tends to infinity. The mCH equation is a modification, with cubic nonlinearity, of the original Camassa-Holm (CH) equation; like the CH equation, it is integrable in the sense that it is a condition for the compatibility of a pair of linear differential equations (Lax pair). Due to the non-zero background, the x-equation of the Lax pair can be considered as a spectral problem with a continuous spectrum. This makes it possible to formulate the inverse scattering problem as a RHP of analytical factorization in the complex plane of the spectral parameter. In Section 2, we consider the Cauchy problem for the mCH equation under the condition that the solution tends to 1 as |x| →∞. For this problem, a formalism of the RHP is developed, which is based on the adaptation of the general idea of using special solutions (Jost solutions) of the Lax pair equations as "blocks" for constructing a matrix RHP to the case of the mCH equation, taking into account the peculiarities of the Lax pair equations associated with this equation. The x-equation of the Lax pair for the mCH equation, known in the literature, as a spectral problem with a spectral parameter λ has two features that significantly affect the analytical properties of the Jost solutions, namely: (a) λ is involved in the coefficient matrix as a product with a "moment" m(x,t), which is an unknown function within the inverse problem; (b) when |x| →∞, m(x,t) tends to a non-zero constant. These features affect the problem of controlling the behavior of the Jost solutions as λ →∞. To solve this problem, (i) by applying calibration transformations, the Lax pair equations are transformed to a form in which the diagonal terms dominate when λ →∞; (ii) a new spatial variable is introduced, which allows us to obtain an explicit description of the behavior of the Jost solutions as λ →∞; (iii) a new spectral parameter μ is introduced, which allows avoiding the irrational dependence of the coefficients in the Lax pair equations on the spectral parameter. Using the developed formalism, a parametric representation of the solution of the Cauchy problem is obtained in terms of the solution of the associated RHP (the data for which are determined by the initial data), analyzing its behavior as λ → 0. The proposed formalism allows us to characterize both regular and irregular one-soliton solutions corresponding to the RHP with trivial jump conditions and appropriately specified residue conditions. In particular, two types of irregular soliton solutions of the mCH equation are characterized: (i) peakon-type solutions, which are functions that are continuous with the first derivative but have unbounded derivatives of orders greater than 2 at the peak; (ii) looped multivalued solutions. In Section 3, using the formalism developed in Section 2, we derive the main terms of the large time asymptotic of the solution of the Cauchy problem for the mCH equation on a constant nonzero background. The study is focused on the soliton-free case. First, the original RHP associated with the mCH equation, which has specific singularities, is reduced to a regular RHP, i.e., one that has only a jump condition and a normalization condition. Further, the solution of the obtained regular RHP is analyzed asymptotically as t →∞ using an appropriate adaptation of the nonlinear steepest descent method. As a result, the main asymptotic terms of the solution u(x,t) of the Cauchy problem are obtained in those sectors of the half-plane (x,t) where the deviation from the background is non-trivial. In Section 4, we consider the Cauchy problem for the mCH equation under the condition that the solution tends to two different constants as x →± ∞. For this problem, we also develop a RHP formalism. For this purpose, we transform the Lax pair equations, which allow us to study in detail the analytical properties of the corresponding Jost solutions and spectral functions, in particular, symmetries and behavior at branch points. Similarly to the case of a constant background, a parametric representation of the Cauchy problem is obtained.