Conditional symmetries and applications to exact solutions of reaction-diffusion systems

Speaker: Ian Marquette (La Trobe University)

Time: 2026-01-09 16:00-18:00
Location: 1010, SIMIS
Zoom Meeting ID: 826 8629 3394 Passcode: SIMIS

摘要：

Conditional symmetries and applications to exact solutions of reaction-diffusion systems

Symmetries can take different forms depending on the context and allow to obtain different types of insight on a given system. In classical and quantum mechanics symmetries have different nature such as dynamical, hidden or accidental symmetry. They can relate to a well-known underlying Lie algebra or one of its generalizations. Here we will first discuss the concept of Lie symmetry from a mathematical point of view for nonlinear differential equations and in particular for systems of partial differential equations. They can allow to reduce the number of variables and obtain exact solutions with some Ansatz which is constructed. The solutions can be expressed in terms of functions satisfying other well known PDEs, ODEs or even be explicit and expressed in terms of special functions. This also allow via their Lie group to get a new solution from known ones. They have been widely used since the 70ties in different contexts and in various equations of mathematical physics. They allowed to obtain exact solutions for KdV, nonlinear Schrodinger equation and plethora of other ones. Those solutions would be otherwise difficult to reveal.  
The talk will be devoted to the case of conditional and generalized Lie symmetries and their formulation in context of reaction-diffusion system arising in population dynamics and Lotka-Volterra systems. Conditional and generalised Lie symmetries are not associated with Lie algebras but nevertheless allow to obtain exact solution that would not be obtainable via the Lie symmetries. Those are much harder to construct and classify. Unlike the case of Lie symmetries for which systems of determining equations are linear differential equations, here it leads to systems of nonlinear equations. However, such system can still be completely solved in many cases. I will explain a general algorithm for finding Q-conditional symmetries of nonlinear evolution systems.
I will present recent results on a system of two cubic reaction-diffusion equations. Depending on values of coefficients, all possible Lie and Q-conditional (nonclassical) symmetries are identified. A wide range of new exact solutions is constructed, including those expressible in terms of a Lambert function and not obtainable by Lie symmetries. An example of a new real-world application of the system is discussed consisting of two independent gene frequencies arising in population dynamics. I present some results on conditional symmetries for a system of three component Lotka-Volterra systems and discussion its applications.