Q-conditional Symmetries Research Articles

Symmetries and Exact Solutions of the Diffusive Holling–Tanner Prey-Predator Model

We consider the classical Holling–Tanner model extended on 1D space by introducing the diffusion term. Making a reasonable simplification, the diffusive Holling–Tanner system is studied by means of symmetry based methods. Lie and Q-conditional (nonclassical) symmetries are identified. The symmetries obtained are applied for finding a wide range of exact solutions, their properties are studied and a possible biological interpretation is proposed. 3D plots of the most interesting solutions are drown as well.

The Shigesada–Kawasaki–Teramoto model: Conditional symmetries, exact solutions and their properties

We study a simplification of the well-known Shigesada–Kawasaki–Teramoto model, which consists of two nonlinear reaction–diffusion equations with cross-diffusion. A complete set of Q-conditional (nonclassical) symmetries is derived using an algorithm adopted for the construction of conditional symmetries. The symmetries obtained are applied for finding a wide range of exact solutions, possible biological interpretation of some of which being presented. Moreover, an alternative application of the simplified model related to the polymerization process is suggested and exact solutions are found in this case as well.

A Hunter-Gatherer–Farmer Population Model: New Conditional Symmetries and Exact Solutions with Biological Interpretation

New Q-conditional (nonclassical) symmetries and exact solutions of the hunter-gatherer-farmer population model proposed by Aoki, Shida and Shigesada (Theor. Popul. Biol. 1996;50:1-17) are constructed. The main method used for the aforementioned purposes is an extension of the nonclassical method for system of partial differential equations. An analysis of properties of the exact solutions obtained and their biological interpretation are carried out. New results are compared with those derived in recent studies devoted to the same model.

New Conditional Symmetries and Exact Solutions of the Diffusive Two-Component Lotka–Volterra System

The diffusive Lotka–Volterra system arising in an enormous number of mathematical models in biology, physics, ecology, chemistry and society is under study. New Q-conditional (nonclassical) symmetries are derived and applied to search for exact solutions in an explicit form. A family of exact solutions is examined in detail in order to provide an application for describing the competition of two species in population dynamics. The results obtained are compared with those published earlier as well.

Consistent approximate Q-conditional symmetries of PDEs: application to a hyperbolic reaction-diffusion-convection equation

Within the theoretical framework of a recently introduced approach to approximate Lie symmetries of differential equations containing small terms, which is consistent with the principles of perturbative analysis, we define accordingly approximate Q-conditional symmetries of partial differential equations. The approach is illustrated by considering the hyperbolic version of a reaction-diffusion-convection equation. By looking for its first order approximate Q-conditional symmetries, we are able to explicitly determine a large set of non-trivial approximate solutions.

Conditional symmetries and exact solutions of a nonlinear three-component reaction-diffusion model

AbstractQ-conditional (non-classical) symmetries of the known three-component reaction-diffusion (RD) system [K. Aoki et al. Theor. Popul. Biol. 50, 1–17 (1996)] modelling interaction between farmers and hunter-gatherers are constructed for the first time. A wide variety of Q-conditional symmetries are found, and it is shown that these symmetries are not equivalent to the Lie symmetries. Some operators of Q-conditional (non-classical) symmetry are applied for finding exact solutions of the RD system in question. Properties of the exact solutions (in particular, their asymptotic behaviour) are identified and possible biological interpretation is discussed.

Lie and Q-Conditional Symmetries of Reaction-Diffusion-Convection Equations with Exponential Nonlinearities and Their Application for Finding Exact Solutions

This review is devoted to search for Lie and Q-conditional (nonclassical) symmetries and exact solutions of a class of reaction-diffusion-convection equations with exponential nonlinearities. A complete Lie symmetry classification of the class is derived via two different algorithms in order to show that the result depends essentially on the type of equivalence transformations used for the classification. Moreover, a complete description of Q-conditional symmetries for PDEs from the class in question is also presented. It is shown that all the well-known results for reaction-diffusion equations with exponential nonlinearities follow as particular cases from the results derived for this class of reaction-diffusion-convection equations. The symmetries obtained for constructing exact solutions of the relevant equations are successfully applied. The exact solutions are compared with those found by means of different techniques. Finally, an application of the exact solutions for solving boundary-value problems arising in population dynamics is presented.

Q-Conditional Symmetries and Exact Solutions of Nonlinear Reaction–Diffusion Systems

A wide range of reaction–diffusion systems with constant diffusivities that are invariant under Q-conditional operators is found. Using the symmetries obtained, the reductions of the corresponding systems to the systems of ODEs are conducted in order to find exact solutions. In particular, the solutions of some reaction–diffusion systems of the Lotka–Volterra type in an explicit form and satisfying Dirichlet boundary conditions are obtained. An biological interpretation is presented in order to show that two different types of interaction between biological species can be described.

Nonlinear reaction–diffusion systems with a non-constant diffusivity: Conditional symmetries in no-go case

Q-conditional symmetries (nonclassical symmetries) for a general class of two-component reaction–diffusion systems with non-constant diffusivities are studied. The work is a natural continuation of our paper “Conditional symmetries and exact solutions of nonlinear reaction–diffusion systems with non-constant diffusivities” (Cherniha and Davydovych, 2012) [1] in order to extend the results on so-called no-go case. Using the notion of Q-conditional symmetries of the first type, an exhaustive list of reaction–diffusion systems admitting such symmetry is derived. The results obtained are compared with those derived earlier. The symmetries for reducing reaction–diffusion systems to two-dimensional dynamical systems (ODE systems) and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for nonlinear reaction–diffusion systems with an arbitrary power-law diffusivity are constructed and their properties for possible applicability are established.

Lie and conditional symmetries of the three-component diffusive Lotka–Volterra system

Lie and Q-conditional symmetries of the classical three-component diffusive Lotka–Volterra system in the case of one space variable are studied. The group-classification problems for finding Lie symmetries and Q-conditional symmetries of the first type are completely solved. Notably, non-Lie symmetries (Q-conditional symmetry operators) for a multi-component nonlinear reaction–diffusion system are constructed for the first time. The results are compared with those derived for the two-component diffusive Lotka–Volterra system. The conditional symmetry obtained for the non-Lie reduction of the three-component system used for modeling competition between three species in population dynamics is applied and the relevant exact solutions are found. Particularly, the exact solution describing different scenarios of competition between three species is constructed.

New conditional symmetries and exact solutions of reaction–diffusion–convection equations with exponential nonlinearities

A complete description of Q-conditional symmetries for a class of reaction–diffusion–convection equations with exponential diffusivities is derived. It is shown that all the known results for reaction–diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa. The symmetries obtained for constructing exact solutions of the relevant equations are successfully applied. The exact solutions are compared with those found earlier by means of different approaches. An application of the solutions for solving a boundary-value problem arising in population dynamics is presented.

Conditional symmetries and exact solutions of nonlinear reaction–diffusion systems with non-constant diffusivities

Q-conditional symmetries (nonclassical symmetries) for the general class of two-component reaction–diffusion systems with non-constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type, an exhausted list of reaction–diffusion systems admitting such symmetry is derived. The results obtained for the reaction–diffusion systems are compared with those for the scalar reaction–diffusion equations. The symmetries found for reducing reaction–diffusion systems to two-dimensional dynamical systems, i.e., ODE systems, and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for a nonlinear reaction–diffusion system with an arbitrary diffusivity are constructed. Finally, the application of the exact solutions for solving a biologically and physically motivated system is presented.

New conditional symmetries and exact solutions of reaction–diffusion systems with power diffusivities

A wide range of new Q-conditional symmetries for reaction–diffusion systems with power diffusivities are constructed. The relevant non-Lie ansätze to reduce the reaction–diffusion systems to ODE systems and examples of exact solutions are obtained. The relation of the solutions obtained with the development of spatially inhomogeneous structures is discussed.

New conditional symmetries and exact solutions of nonlinear reaction–diffusion–convection equations

A complete description of Q-conditional symmetries for two classes of reaction–diffusion–convection equations with power diffusivities is derived. It is shown that all the known results for reaction–diffusion equations with power diffusivities follow as particular cases from those obtained here but not vice versa. The symmetries obtained for constructing exact solutions of the relevant equations are successfully applied. In the particular case, new exact solutions of nonlinear reaction–diffusion–convection equations arising in application and their natural generalizations are found.

Conditional symmetries and exact solutions of one reaction-diffusion-convection equation

We give a complete description of Q-conditional symmetries for a nonlinear reaction-diffusion-convection equation. The obtained Q-conditional operators are used for the construction of exact solutions of some equations encountered in applied problems.

New Q-conditional symmetries and exact solutions of some reaction–diffusion–convection equations arising in mathematical biology

A theorem giving a complete description of Q-conditional symmetries of a class of nonlinear reaction–diffusion–convection equations is proved. Furthermore the Q-conditional symmetries obtained and the method of additional generating conditions are applied for finding exact solutions of the generalized Fisher, Fitzhugh–Nagumo and Kolmogorov–Petrovskii–Piskunov equations. The symmetries and solutions constructed are compared with those obtained by other authors. In particular, it was established that the known travelling wave solutions of these equations are particular cases of more general (non-Lie) solutions. The relation between Q-conditional symmetries and generalized conditional symmetries is also shown.

Lie Symmetries, Q-Conditional Symmetries, and Exact Solutions of Nonlinear Systems of Diffusion-Convection Equations

A complete description of Lie symmetries is obtained for a class of nonlinear diffusion-convection systems containing two Burgers-type equations with two arbitrary functions. A nonlinear diffusion-convection system with unique symmetry properties that is simultaneously invariant with respect to the generalized Galilei algebra and the operators of Q-conditional symmetries with cubic nonlinearities relative to dependent variables is found. For systems of evolution equations, operators of this sort are found for the first time. For the nonlinear system obtained, a system of Lie and non-Lie ansatze is constructed. Exact solutions, which can be used in solving relevant boundary-value problems, are also found.

Nonlinear systems of the Burgers-type equations: Lie and Q-conditional symmetries, Ansätze and solutions

A class of nonlinear diffusion–convection systems containing two Burgers-type equations is considered. A complete description of Lie symmetries is obtained for these systems. New results of finding Q-conditional symmetries are also presented. Moreover, a variety of Lie and non-Lie Ansätze and exact solutions of a diffusion–convection system invariant under the generalized Galilei algebra are constructed.