Weyl functions and the boundary value problem for a matrix nonlinear Schr"odinger equation on a semi-strip

Rectangular matrix solutions of the defocusing nonlinear Schr"odinger equation (dNLS) are considered on a semi-strip. Evolution of the corresponding Weyl function is described in terms of the initial-boundary conditions. Then initial condition is recovered from the boundary conditions. Thus, solutions of dNLS are recovered from the boundary conditions.

💡 Research Summary

The paper studies the matrix defocusing nonlinear Schrödinger equation (dNLS)
(v_t = i(v_{xx} - 2 v v^{} v)) with an (m_1 \times m_2) matrix‑valued potential (v(x,t)) on the semi‑infinite strip (D = {(x,t): 0 \le x < \infty,; 0 \le t < a}). The authors use the Lax pair formulation, where the auxiliary linear system (y_x = G y), (y_t = F y) with
(G = i(z j + j V)), (F = -\frac{i}{2}\bigl(z^2 j + z j V - i V_x - j V^2\bigr)),
(j = \operatorname{diag}(I_{m_1}, -I_{m_2})), (V = \begin{pmatrix}0 & v \ v^{} & 0\end{pmatrix}),
satisfies the zero‑curvature condition (G_t - F_x +