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A new class of inverse problems is considered. In the context of classical theory, inverse problems are concerned with finding a model that has the observed measurements. It is well known that such problems usually are ill-posed. At the same time, it is often the case when there is some a priori information about the system. This naturally leads to the following inverse optimal problem : find data F ˆ of a model which is the nearest to a priori given data F 0 and sufficient to ensure the model has the observed measurements S. In this note, an approach to a complete solution to such a problem is developed. Within the framework of this approach, we consider a model problem of recovering the potential field V ˆ from the m observed eigenvalues of the Schrödinger operator, provided that such potential field is at the minimum distance from a priori given potential V a. In the main result, we establish a new type of relationship between the linear spectral problems and systems of nonlinear differential equations which enables us to find a solution to the inverse optimal spectral problem and obtain novel results on the existence of solutions to nonlinear problems as well. • A new class of inverse optimal spectral problem. • The relationship between the spectral problems and the nonlinear problem. • Existence result for the system of NLS equations. [ABSTRACT FROM AUTHOR]

We study the solvability of the Zakharov equation Δ 2 u + (κ − ω 2) Δ u − κ div e − | ∇ u | 2 ∇ u = 0 in a bounded domain under homogeneous Dirichlet or Navier boundary conditions. This problem is a consequence of the system of equations derived by Zakharov to model the Langmuir collapse in plasma physics. Assumptions for the existence and nonexistence of a ground state solution as well as the multiplicity of solutions are discussed. Moreover, we consider formal approximations of the Zakharov equation obtained by the Taylor expansion of the exponential term. We illustrate that the existence and nonexistence results are substantially different from the corresponding results for the original problem. [ABSTRACT FROM AUTHOR]