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Explicit solutions of nonlinear evolution equations via nonlocal reductions approach

In the last 30 years a great progress in the investigation of nonlinear partial differential equations of mathematical and theoretical physics was achieved due to application of different approaches mainly based on modern functional and algebraic-geometric methods [1–3]. It gave possibility to study different properties of solutions important for applications in physics, mechanics and other fields of knowledge and to construct exact formulae for solutions of many systems of nonlinear differential equations having numerous applications. In the present communication we consider an approach connected with studying of a system of nonlinear partial differential equations possessing so-called operator Zakharov–Shabat representation which is based on reduction principle. The formula for exact solutions of a system of nonlinear partial differential equations describing resonance interaction of M waves is given. 1. Let Φ be a function of variables x, y, t ∈ R1 satisfying to a system of linear partial differential equations