On integration of multidimensional version of $n$-wave type equation

Abstract

We represent a version of multidimensional quasilinear partial differential equation (PDE) together with large manifold of particular solutions given in an integral form. The dimensionality of constructed PDE can be arbitrary. We call it the $n$-wave type PDE, although the structure of its nonlinearity differs from that of the classical completely integrable (2+1)-dimensional $n$-wave equation. The richness of solution space to such a PDE is characterized by a set of arbitrary functions of several variables. However, this richness is not enough to provide the complete integrability, which is shown explicitly. We describe a class of multi-solitary wave solutions in details. Among examples of explicit particular solutions, we represent a lump-lattice solution depending on five independent variables. In Appendix, as an important supplemental material, we show that our nonlinear PDE is reducible from the more general multidimensional PDE which can be derived using the dressing method based on the linear integral equation with the kernel of a special type (a modification of the $\bar\partial$-problem). The dressing algorithm gives us a key for construction of higher order PDEs, although they are not discussed in this paper.