A rigorous and complete solution of the initial-boundary-value (Goursat) problem for second harmonic generation (and its matrix analog) on the semi-strip is given in terms of the Weyl functions. A wide class of the explicit solutions and their Weyl functions is obtained also.

Abstract In this paper, we study the Cauchy and Goursat problems of spin-$n/2$ zero rest-mass equations on Minkowski spacetime by using conformal geometric method. our strategy, prove wellposedness problem in Einstein's cylinder. Then establish pointwise decays fields energy equalities between null boundaries $\scri^\pm$ hypersurface $\Sigma_0=\left\{ t=0 \right\}$. Finally, partial compactific...

We consider the Goursat problem for linear partial differential equations with constant coefficients in two complex variables. find conditions summable solutions of case when Newton polygon has exactly one side a positive slope.

In present article the self-contained derivation of eigenvalue inverse problem results is given by using a discrete approximation with three-diagonal SturmLiouville operator on a finite interval. It is demonstrated that inverse problem procedure is nothing else than well known Gram-Schmidt orthonormalization in Euclidean space for special vectors numbered by space coordinate index. All the resu...

We show that 1-quasiconformal mappings on Goursat groups are CR or anti-CR mappings. This can reduce the determination of 1-quasiconformal mappings to the determination of CR automorphisms of CR manifolds, which is a fundamental problem in the theory of several complex variables.