Initial-Boundary Value Problem for the Camassa�CHolm Equation with Linearizable Boundary Condition

We present a Riemann?CHilbert problem formalism for the initial boundary value problem for the Camassa?CHolm equation on the half-line x > 0 with homogeneous Dirichlet boundary condition at x = 0. We show that, similarly to the problem on the whole line, the solution of this problem can be obtained in parametric form via the solution of a Riemann?CHilbert problem determined by the initial data via associated spectral functions. This allows us to apply the non-linear steepest descent method and to describe the large-time asymptotics of the solution.     

Analysis of the Global Relation for the Nonlinear Schrödinger Equation on the Half-line

It has been shown recently that the unique, global solution of the Dirichlet problem of the nonlinear Schrödinger equation on the half-line can be expressed through the solution of a 2×2 matrix Riemann–Hilbert problem. This problem is specified by the spectral functions {a(k),b(k)} which are defined in terms of the initial condition q(x,0)=q ₀(x), and by the spectral functions {A(k),B(k)} which are defined in terms of the specified boundary condition q(0,t)=g ₀(t) and the unknown boundary value q _x (0,t)=g ₁(t). Furthermore, it has been shown that given q ₀ and g ₀, the function g ₁ can be characterized through the solution of a certain 'global relation' coupling q ₀, g ₀, g ₁, and (t,k), where satisfies the t-part ofthe associated Lax pair evaluated at x=0. We show here that, by using a Gelfand–Levitan–Marchenko triangular representation of , the global relation can be explicitly solved for g ₁.     

The Ostrovsky–Vakhnenko equation: A Riemann–Hilbert approach

We present an inverse scattering transform approach for the (differentiated) Ostrovsky–Vakhnenko equation:

u_txx−3u_x+3u_xu_xx+uu_xxx=0.$u_{txx}-3u_x+3u_x{u}_{xx}+u{u}_{xxx}=0.$

This equation can also be viewed as the short-wave model for the Degasperis–Procesi equation. The approach is based on an associated Riemann–Hilbert problem, which allows us to give a representation for the classical (smooth) solution of the Cauchy problem, to get the principal term of its long-time asymptotics, and also to find, in a natural way, loop soliton solutions.     

Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation

We consider the initial boundary value (IBV) problem for the focusing nonlinear Schrödinger equation in the quarter plane x>0, t >0 in the case of periodic initial data, u(x,0) = α exp(?2iβx) (or asymptotically periodic, u(x, 0) =α exp(?2iβx)→0 as x→∞), and a Robin boundary condition at x = 0: u_x(0, t)+qu(0, t) = 0, q ≠ 0. Our approach is based on the unified transform (the Fokas method) combined with symmetry considerations for the corresponding Riemann-Hilbert (RH) problems. We present the representation of the solution of the IBV problem in terms of the solution of an associated RH problem. This representation also allows us to determine an initial value (IV) problem, of a shock type, a solution of which being restricted to the half-line x > 0 is the solution of the original IBV problem. In the case β < 0, the large-time asymptotics of the solution of the IBV problem is presented in the “rarefaction” sector, demonstrating, in particular, an oscillatory behavior of the boundary values in the case q > 0, contrary to the decay to 0 in the case q < 0.     

The short pulse equation by a Riemann–Hilbert approach

We develop a Riemann–Hilbert approach to the inverse scattering transform method for the short pulse (SP) equation

$$\begin{aligned} u_{xt}=u+\tfrac{1}{6}(u^3)_{xx} \end{aligned}$$

with zero boundary conditions (as \(|x|\rightarrow \infty \)). This approach is directly applied to a Lax pair for the SP equation. It allows us to give a parametric representation of the solution to the Cauchy problem. This representation is then used for studying the longtime behavior of the solution as well as for retrieving the soliton solutions. Finally, the analysis of the longtime behavior allows us to formulate, in spectral terms, a sufficient condition for the wave breaking.

     

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

We present a Riemann–Hilbert problem formalism for the initial value problem for the Camassa–Holm equation $u_t?u_{txx}+2\omega u_x+3u{u}_x=2u_x{u}_{xx}+u{u}_{xxx}$ 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 ω, this soliton train is close to a train of peakons, which are piecewise smooth solutions of the CH equation for $\omega =0$. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Decaying Long-Time Asymptotics for the Focusing NLS Equation with Periodic Boundary Condition

We consider the initial boundary value problem for the focusingnonlinear Schrödinger equation in the quarter plane , inthe case of decaying initial data (for , as )and Dirichlet boundary data (for ) approaching a periodic (single-frequency) background as .We first provide admissibility conditions for the normal derivativeof the solution on the boundary, under the assumption that itbehaves asymptotically in a similar (single-frequency) manner.We then show that for the range , the long-time asymptotics of the solution inside the quarterplane exhibits decaying oscillations of Zakharov–Manakovtype.     

The Camassa–Holm equation on the half-line

We study the initial-boundary-value problem for the Camassa–Holm equation on the half-line by associating to it a matrix Riemann–Hilbert problem in the complex k-plane; the jump matrix is determined in terms of the spectral functions corresponding to the initial and boundary values. We prove that if the boundary values $u(0,t)$ are ?0 for all t then the corresponding initial-boundary-value problem has a unique solution, which can be expressed in terms of the solution of the associated RH problem. In the case $u(0,t)<0$, the compatibility of the initial and boundary data is explicitly expressed in terms of an algebraic relation to be satisfied by the spectral functions. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

A Riemann–Hilbert Problem for Propagation of Electromagnetic Waves in an Inhomogeneous,Dispersive Ω Waveguide

We consider the inverse scattering problem for a model of electromagnetic wave propagation in a rectangular waveguide filled with dispersive material. The waveguide is inhomogeneous in the longitudinal direction but homogeneous in the transverse directions. Dispersive properties of the material are described by a single-resonance Lorentz model. By reformulating the scattering problem in the frequency domain as a Riemann–Hilbert problem, we prove that the constitutive parameters of the inhomogeneous waveguide are reconstructed uniquely from the scattering data.     

Uniqueness in a frequency-domain inverse problem of a stratified uniaxial bianisotropic medium

An inverse problem for a stratified uniaxial bianisotropic slab is considered in the frequency domain. The problem is treated as an analytic factorization problem in the frequency complex plane. Uniqueness in the parameter reconstruction is discussed and illustrated under a particular choice of a priori information.