On Multidimensional Determinant Differential-Operator Equations

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system’s guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.46698/g9113-3086-1480-k On Multidimensional Determinant Differential-Operator Equations Rakhmelevich, I. V. Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 2. 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. Keywords: determinant differential-operator equation, determinant, linear differential operator, eigenfunction, kernel of an operator, traveling wave type solution. Language: Russian Download the full text For citation: Rakhmelevich, I. V. On Multidimensional Determinant Differential-Operator Equations, Vladikavkaz Math. J., 2020, vol. 22, no. 2, pp.53-69. DOI 10.46698/g9113-3086-1480-k + References 1. Polyanin, A. D. and Zaytsev, V. F. Handbook of Nonlinear Partial Differential Equations, 2nd Ed., Boca Raton-London, Chapman and Hall-CRC Press, 2012. 2. Polyanin, A. D., Zaytsev, V. F. and Zhurov, A. I. Metody resheniya nelineynykh uravneniy matematicheskoy fiziki i mekhaniki [Methods of Solving of Nonlinear Equations of Mathematical Physics and Mechanics], Moscow, Fizmatlit, 2005, 256 p. (in Russian). 3. Zhdanov, R. Z. Separation of Variables in the Nonlinear Wave Equation, Journal of Physics A: Mathematical and General, 1994, vol. 27, no. 9, pp. L291-L297. DOI: 10.1088/0305-4470/27/9/009. 4. Rakhmelevich, I. V. On Two-Dimensional Hyperbolic Equation with Power Non-linearity on the Derivatives, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika [Tomsk State University Journal of Mathematics and Mechanics], 2015, no. 1(33), pp. 12-19 (in Russian). 5. Rakhmelevich, I. V. 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