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Analytical solution of a general boundary value problem for a BKW-equation. (Russian) Zbl 1057.35025

The article is devoted to the study of the one-dimensional stationary linearized BKW-like equation \[ \xi_x\frac{\partial h}{\partial x} = \xi\int_{-1}^1d\mu'\int_0^{\infty} \exp(-\xi^{\prime 2})k(\mu,\xi;\mu',\xi')h(x,\xi',\xi')\,d\xi' - \xi h(x,\mu',\xi'), \] where \(k(\mu,\xi;\mu',\xi') = 1 + (3/2)\mu\xi\mu'\xi' + (1/2)(\xi^{2} - 2) (\xi^{\prime 2} - 2)\) is the kernel of the integral operator. The function \(h\) satisfies the following boundary conditions: \[ \begin{gathered} h(0,\mu,\xi) = h_0(\mu,\xi),\quad 0 < \mu < 1,\\ h(x,\mu,\xi) = h_{as}(x,\mu,\xi) + o(1),\quad x\to\infty,\quad -1 < \mu < 0. \end{gathered} \] Here \(h_{as}\) is a linear combination of discrete solutions to the equation under consideration.
The authors obtain an eigenfunction expansion for a general solution to the above boundary value problem. The solvability conditions make it possible to determine both unknown coefficients of this expansion and free parameters of a solution to the vector Riemann-Hilbert problem.

MSC:

35Q15 Riemann-Hilbert problems in context of PDEs
35P05 General topics in linear spectral theory for PDEs
35C05 Solutions to PDEs in closed form
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