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On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

Tsyfra Ivan

Opuscula Mathematica

2021

Vol. 41, no. 5

685--699

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differen¬tial equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen, then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

Solitary wave solution to the nonlinear evolution equation in cascaded quadratic media beyond the slowly varying envelope approximations

Sarma A K, Biswas A.

Optica Applicata

2014

Vol. 44, nr 2

205--212

We report an exact bright and dark soliton solution to the nonlinear evolution equation derived by MOSES and WISE (Phys. Rev. Lett. 97, 2006, 073903) for cascaded quadratic media beyond the slowly varying envelope approximations. The integrability aspects of the model are addressed. The travelling wave hypothesis as well as the ansatz method are employed to obtain an exact 1-soliton solution. Both bright and dark soliton solutions are obtained. The corresponding constraint conditions are obtained in order for the soliton solutions to exist.

On integrability of quadratic harnesses

Bryc W.

Demonstratio Mathematica

2012

Vol. 45, nr 2

275-282

We investigate integrability properties of processes with linear regressions and quadratic conditional variances. We establish the right order of dependence of which moments are finite on the parameter (…) defined below, raising the question of determining the optimal constant.

On Algebras Associated with Integrable Hamiltonian Systems

Kasperczuk S.

Computational Methods in Science and Technology

2010

Vol. 16, No. 2

161-163

The aim of this paper is to give a general setting, based on quantum deformations, for the explicit construction of certain classes of integrable Hamiltonian systems.