Reduction of partial differential equations to the systems of ordinary differential equations
In our article we develop an approach for constructing particular solutions of differential equations. This approach is based on the use of higher symmetries allowed by partial differential equations and the method of differential constraints proposed by N.N. Yanenko. We restrict ourselves to the study of partial differential equations with two independent variables. Differential constraints and the coefficients of admissible symmetry operators generate ordinary differential equations. The classical Lie theory works well in the case of point and contact transformations. When higher symmetries and higher-order differential constraints are considered then arises the problem of integrating higher-order ordinary differential equations. The solutions of such differential equations are obtained by the inverse scattering problem and finite-zone integration method in the soliton theory. However, this approach has a number of significant difficulties. For example, it is often difficult to sort out real solutions from a set of complex solutions, or solutions are expressed through insufficiently studied functions. Our approach is based on the numerical integration of passive systems. The additional ordinary differential equations are invariant manifolds of evolution equations. This allows us to rewrite an overdetermed system as two systems of ordinary differential equations. Further we sequentially solve these systems by the Runge -Kutta method. We apply this approach to the Korteweg - de Vries equations, Sin-Gordon and Sinh-Gordon equations. The bounded and unrestricted solutions are found and solution images are constructed. This approach can be used for equations with an arbitrary number of independent variables.