ONE SIX-ORDER PARTIAL DIFFERENTIAL EQUATION

One six-order partial differential equation in the presence of the Painleve property is considered in this work. Differential equations are the models of different physical processes such as tasks of nonlinear waves, processes of turbulence, drift waves in plasma, etc. Ablowitz’s hypothesis is widely used that all reductions of completely integrable partial differential equations lead to ordinary differential equations with the Painleve property. The Painleve property is the basis of classification and reduction to the canonical form of nonlinear partial differential equations, just like this property allows one to classify ordinary differential equations. The Painleve property classification of partial differential equations higher than the third order is still far from complete. This is due to the fact that the known methods of research give generally only necessary conditions for existence of the Painleve property. To prove the sufficiency, for example, it is possible to reduce the investigated equation by a suitable replacement to the equation, for which the presence of the Painleve property has already been found. Therefore, of particular interest are the methods allowing one to build the equations with the a priori Painleve property. Introduction contains the definition of the Painleve property for a partial differential equation known in the literature and describes the main method of research — resonance method. In the main part, the resonant structure is investigated and the fulfillment of necessary conditions for the presence of the Painleve property is checked. To achieve this goal, we solved the problems of constructing series representing the solution of the six-order partial differential equations containing six arbitrary functions. The convergence of the obtained series is proved by using majorant series. The terms of lesser weight are found, in the presence of which for the equation a necessary condition for existence of the Painleve property, as well as a suitable substitution reducing the obtained equation to the linear one will be satisfied. Rational solutions are built in terms of negative resonances with respect to the function φ.

1. Weiss J., Tabor M., Carnevale G. The Painleve property for partial differential equations. Journal of Mathematical Physics, 1983, vol. 24, no. 3. pp. 522–526. Doi: 10.1063/1.525721

2. Cosgrove С. Painleve classification of all semilinear partial differential equations of the second order. I. Hyperbolic equations in two independent variables, Studies in Applied Mathematics, 1993, vol. 89, no. 1, pp. 1–61. Doi: 10.1002/sapm19938911

3. Martynov I. P., Berezkina N. S., Pronko V. A. Analytical theory of nonlinear equations and systems. Grodno, Yanka Kupala State University of Grodno, 2009. 395 p. (in Russian).

4. Exton H. On non-linear ordinary differential equations with fixed critical points. Rendiconti di Matematica, 1971, vol. 4, no. 3, pp. 385–628.

5. Sdunek A. G., Martynov I. P., Pronko V. A. On the rational solutions of differential equations. Vesnik Hrodzenskaha Dziarzhaunaha Universiteta Imia Ianki Kupaly. Seryia 2. Matematyka. Fizika. Infarmatyka, Vylichal’naia Tekhnika i Kira-van ne = Vesnik of Yanka Kupala State University of Grodno. Series 2. Mathematics. Physics. Informatics, Сomputer Technology and its Сontrol, 2000, no. 3, pp. 33–39 (in Russian).