Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLie point symmetries and ODEs passing the Painleve test
93
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painleve property are explored for ODEs of order $n =2, dots ,5$. Among the 6 ODEs identifying the Painleve transcendents only $P_{III}$, $P_V$ and $P_{VI}$ have nontrivial symmetry algebras and that only for very special values of the parameters. In those cases the transcendents can be expressed in terms of simpler functions, i.e. elementary functions, solutions of linear equations, elliptic functions or Painleve transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painleve test only very partial classifications have been published. We consider many examples that exist in the literature and show how their symmetry groups help to identify those that may define genuinely new transcendents.
rate research
Read More
This short survey presents the essential features of what is called Painleve analysis, i.e. the set of methods based on the singularities of differential equations in order to perform their explicit integration. Full details can be found in textit{The Painleve handbook} or in various lecture notes posted on arXiv.
A theoretical foundation for a generalization of the elliptic difference Painleve equation to higher dimensions is provided in the framework of birational Weyl group action on the space of point configurations in general position in a projective space. By introducing an elliptic parametrization of point configurations, a realization of the Weyl group is proposed as a group of Cremona transformations containing elliptic functions in the coefficients. For this elliptic Cremona system, a theory of $tau$-functions is developed to translate it into a system of bilinear equations of Hirota-Miwa type for the $tau$-functions on the lattice.
We consider a class of generalized Kuznetsov--Zabolotskaya--Khokhlov (gKZK) equations and determine its equivalence group, which is then used to give a complete symmetry classification of this class. The infinite-dimensional symmetry is used to reduce such equations to (1+1)-dimensional PDEs. Special attention is paid to group-theoretical properties of a class of generalized dispersionless KP (gdKP) or Zabolotskaya--Khokhlov equations as a subclass of gKZK equations. The conditions are determined under which a gdKP equation is invariant under a Lie algebra containing the Virasoro algebra as a subalgebra. This occurs if and only if this equation is completely integrable. A similar connection is shown to hold for generalized KP equations.
We compare the results of our two papers with the results of the paper Aratyn H., Gomes J.F., Zimerman A.H., Higher order Painleve equations and their symmetries via reductions of a class of integrable models, J. Phys. A: Math. Theor., V. 44} (2011), Art. No. 235202.
We consider systems of ordinary differential equations (ODEs) of the form ${cal B}{mathbf K}=0$, where $cal B$ is a Hamiltonian operator of a completely integrable partial differential equation (PDE) hierarchy, and ${mathbf K}=(K,L)^T$. Such systems, whilst of quite low order and linear in the components of $mathbf K$, may represent higher-order nonlinear systems if we make a choice of $mathbf K$ in terms of the coefficient functions of $cal B$. Indeed, our original motivation for the study of such systems was their appearance in the study of Painleve hierarchies, where the question of the reduction of order is of great importance. However, here we do not consider such particular cases; instead we study such systems for arbitrary $mathbf K$, where they may represent both integrable and nonintegrable systems of ordinary differential equations. We consider the application of the Prelle-Singer (PS) method --- a method used to find first integrals --- to such systems in order to reduce their order. We consider the cases of coupled second order ODEs and coupled third order ODEs, as well as the special case of a scalar third order ODE; for the case of coupled third order ODEs, the development of the PS method presented here is new. We apply the PS method to examples of such systems, based on dispersive water wave, Ito and Korteweg-de Vries Hamiltonian structures, and show that first integrals can be obtained. It is important to remember that the equations in question may represent sequences of systems of increasing order. We thus see that the PS method is a further technique which we expect to be useful in our future work.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
