Related papers: The mKdV equation on a finite interval
The initial boundary value problem on a half-line for the KdV equation with the boundary conditions $u|_{x=0}=a\leq0$, $u_{xx}|_{x=0}=3a^2$ is integrated by means of the inverse scattering method. In order to find the time evolution of the…
We study the existence of solutions of mixed Riemann-Hilbert or Cherepanov boundary value problem with simply connected fibers on the unit disk ${\Delta}$. Let $L$ be a closed arc on $\partial{\Delta}$ with the end points $\omega_{-1},…
We consider well-posedness of the boundary value problem in presence of an inclusion with complex conductivity $k$. We first consider the transmission problem in $\mathbb{R}^d$ and characterize solvability of the problem in terms of the…
This paper considers to the problems of diffraction of electromagnetic waves on a half-plane, which has a finite inclusion in the form of a Lipschitz curve. The diffraction problem formulated as boundary value problem for Helmholtz…
In 1974 P. Lax introduced an algebro-analytic mechanism similar to the Lax L-A pair. Using it we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded…
We consider an initial mixed-boundary value problem for anisotropic fractional type degenerate parabolic equations posed in bounded domains. Namely, we consider that the boundary of the domain splits into two parts. In one of them, we…
The unified transform method is used to analyze the initial-boundary value problem for the coupled derivative nonlinear Schr\"odinger(CDNLS) equations on the half-line. In this paper, we assume that the solution $u(x,t)$ and $v(x,t)$ of…
An initial-value problem for an ordinary differential equation of the first order, is considered. It is supposed that the right-hand side of the equation is a continuous function defined on a set consisting of an open set and a part of its…
It is proved the existence of multivalent solutions for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The…
The solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann-Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are…
Parameter-elliptic boundary-value problems are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the Hilbert Sobolev scale. The latter are the H\"ormander…
A new matrix modified Korteweg-de Vries (mmKdV) equation with a $p\times q$ complex-valued potential matrix function is first studied via Riemann-Hilbert approach, which can be reduced to the well-known coupled modified Korteweg-de Vries…
We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a…
We present a Riemann-Hilbert problem formalism for the initial-boundary value problem for the three-wave equation: \[p_{ij,t}-\frac{b_i-b_j}{a_i-a_j}p_{ij,x}+\sum_k(\frac{b_k-b_j}{a_k-a_j}-\frac{b_i-b_k}{a_i-a_k})p_{ik}p_{kj}=0,\quad…
For $\nu,\nu_i,\mu_j\in(0,1)$, we analyze the semilinear integro-differential equation on the one-dimensional domain $\Omega=(a,b)$ in the unknown $u=u(x,t)$ \[…
We derive special representation for Weyl functions for finite and semi-infinite Jacobi matrices with bounded entries based on a relationship between spectral problem for Jacobi matrices and initial-boundary value problem for auxiliary…
The inverse spectral transform for the Zakharov-Shabat equation on the semi-line is reconsidered as a Hilbert problem. The boundary data induce an essential singularity at large k to one of the basic solutions. Then solving the inverse…
This article explores fundamental properties of convex interval-valued functions defined on Riemannian manifolds. The study employs generalized Hukuhara directional differentiability to derive KKT-type optimality conditions for an…
In the paper boundary-value problem for a multidimensional system of partial differential equations with fractional derivatives in Riemann-Liouville sense with constant coefficients is studied in a rectangular domain. The existence and…
The Riemann-Hilbert boundary value problem is studied for a class of planar complex vector fields $L$ in a simply connected open set $\Om\subset\R^2$. The first integrals of $L$ are used to reduce the problem into a collection of classical…