Complicated travelling wave solutions of a modelling system describing media with memory and spatial nonlocality
This paper deals with a system of balance equation \[ \begin{aligned} \rho\frac{du}{dt}+ \frac{\partial p}{\partial x} &= \rho g,\\ \frac{\partial \rho}{\partial t}+ \rho \frac{\partial u}{\partial x}&= 0, \tag{1}\\ \tau \biggl( \frac{dp}{dt}- \chi \frac{d\rho}{\partial t}\biggr) &= \kappa\rho- p+ \sigma \Biggl\{ \frac{\partial^2 p}{\partial x^2}+ \frac{2\eta}{\rho} \biggl( \frac{\partial p}{\partial x}\biggr)^2+ \frac{\varepsilon}{\rho} \biggl( \frac{\partial\rho} {\partial x}\biggr) \biggl( \frac{\partial p}{\partial x}- \eta \frac{\partial \rho}{\partial x}\biggr)- \eta \frac{\partial^2\rho} {\partial x^2} \Biggr\} \end{aligned} \] for mass and momentum closed by the dynamic equation of state. Here \(u\) is velocity, \(p\) denotes pressure, \(\rho\) is the density; \(g\) is acceleration by an external force; \[ \frac{d}{dt}= \frac{\partial}{\partial t}+u \frac{\partial}{\partial x}; \] \(\tau\) is the relaxation time; \(x\) denotes voluminal viscosity, \(\sqrt{\kappa}\) is the equilibrium sound velocity; \(\sigma\) defines the effective length of the nonlocal effects; \(\varepsilon= \pm 1\), \(\eta\) is a constant. In this paper the authors investigate a set of travelling wave solutions of (1), and show the existence of solutions of the corresponding ODE within the class of periodic, quasi-periodic and stochastic solutions.
scientific article; zbMATH DE number 1486482