A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II

Abstract

A simplified Keller-Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller-Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry classification of the Cauchy problem depending on the initial profile form is presented. The Lie symmetries obtained are used for reduction of the Cauchy problem to that of (1 + 1)-dimensional. Exact solutions of some (1 + 1)-dimensional problems are constructed. In particular, we have proved that the Cauchy problem for the (1 + 1)-dimensional simplified Keller-Segel system can be linearized and solved in an explicit form. Moreover, additional biologically motivated restrictions were established in order to obtain a unique solution. The Lie symmetry classification of the (1 + 2)-dimensional Neumann problem for the simplified Keller-Segel system is derived. Because Lie symmetry of boundary-value problems depends essentially on geometry of the domain, which the problem is formulated for, all realistic (from applicability point of view) domains were examined. Reduction of the the Neumann problem on a strip is derived using the symmetries obtained. As a result, an exact solution of a nonlinear two-dimensional Neumann problem on a finite interval was found.

Symmetries and Conservation Laws for a Class of Fourth-Order Reaction–Diffusion–Advection Equations

“…This paper falls within a set of papers belonging to a wider project in which we take into considerations the symmetry structure of certain classes of reaction-diffusion-advection equations (RDAEs); see, e.g., [16][17][18][19][20], and for classes of the same type where advection is neglected, see [21] and references within. Moreover a significant contribution has been provided by Cherniha and his co-workers in the field of symmetries applied to RDAEs that model several diffusion phenomena [22][23][24][25].…”

Symmetry reductions of a (2 + 1)‐dimensional Keller–Segel model

“…The simplified KS model in (2 + 1) dimensions given by was considered in Cherniha and Didovych 9 and Didovych 10 . Here, t denotes the time variable, x and y are the space variables and the parameters d 1 , χ 0 , α and β are non‐negative constants satisfying χ 0 α ≠ 0, with and , the gradient and the Laplacian in the space variables.…”

“…Moreover, Lie symmetry generators were used for constructing exact solutions of some boundary value problems. In Cherniha and Didovych, 9 the Lie symmetry classification of the (2 + 1)‐dimensional Neumann problem for the simplified KS system () was obtained. Furthermore, the authors derived an exact solution of a nonlinear two‐dimensional Neumann problem on a finite interval.…”

“…was considered in Cherniha and Didovych 9 and Didovych. 10 Here, t denotes the time variable, x and y are the space variables and the parameters d 1 , χ 0 , α and β are non-negative constants satisfying χ 0 α ≠ 0, with r = ∂ x , ∂ y À Á and Δ = ∂ 2…”

New Exact Solutions of the Generalized Benjamin–Bona–Mahony Equation

“…Among them, we can list the traditional methods: the Hirotas bilinear method [10] and the Darboux transformation method [11]. There are also some recent direct and algebraic methods: the variational iteration method [12], the exp-function method [13], various extended tanh-function methods [14] and Lie symmetry analysis [15][16][17].…”