On multidimensional determinant differential-operator equations

We consider a class of multi-dimensional determinant differential-operator equations, the left side of which represents a determinant with the elements containing a product of linear one-dimensional differential operators of arbitrary order, while the right side of the equation depends on the unknown function and its first derivatives. The homogeneous and inhomogeneous determinant differential-operator equations are investigated separately. Some theorems on decreasing of dimension of equation are proved. The solutions obtained in the form of sum and product of functions in subsets of independent variables, in particular, of functions in one variable. In particular, it is proved that the solution of the equation under considering is the product of eigenfunctions of linear operators contained in the equation. A theorem on interconnection between the solutions of the initial equation and the solutions of some auxiliary linear equation is proved for the homogeneous equation. Also a solution of the homogeneous equation is obtained under the hypotheses that the linear differential operators сontained in the equation have proportional eigenvalues. Traveling wave type solution is obtained, in particular, the solutions of exponential form and also in the form of arbitrary function in linear combination of independent variables. If the linear operators in the equation are homogeneous then the solutions in the form of generalized monomials are also found. Some partial solutions to inhomogeneous equation are obtained provided that the right-hand side contains only either independent variables or power or exponential nonlinearity in unknown function and the powers of its first derivatives.