Abstract:
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.
Key words:determinant differential-operator
equation, determinant, linear differential operator, eigenfunction, kernel of an
operator, traveling wave type solution.
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