Generalization of the system of chemotaxis and Van der Waals equations to the case of a three-dimensional vector field

Abstract

The system of chemotaxis equations for the two-dimensional vector field $\tilde{U}\in\mathbb{R}^2$ is generalized to the case of a three-dimensional vector field $\tilde{U}\in\mathbb{R}^3$ preserving invariance with respect to the generalized Galilean algebra and the quadratic nonlinearities of the reaction and convection terms. By using the Cole – Hopf transformation, we reduce the obtained system to a system analogous to the Van der Waals system of equations in the case $U\in\mathbb{R}^3$. Three types of nonlocal equivalence transformations are obtained for an arbitrary system of convection–diffusion equations with $U\in\mathbb{R}^3$. These transformations are used for constructing nonlocal ansatzs, performing reductions, and finding exact solutions of the Van der Waals and chemotaxis systems of equations.