Surface tension of contrast structures
The article is devoted to studying the nonlinear evolution of physical fields of various natures where specific configurations called contrast structures appear. A two-dimensional medium is considered where a physical field \(u(x,y,t)\) participates in transfer at a velocity \(\mathbf V\) and in diffusion with a small diffusion coefficients \(\mu\). In addition, the medium is characterized by a growth parameter \(\gamma\) whose dependence on the field strength is nonlinear: \[ \frac{\partial u}{\partial t}+V_x\frac{\partial u}{\partial x}+V_y\frac{\partial u}{\partial y}=\mu\Biggl(\frac{\partial^2 u}{\partial x^2} + \frac{\partial u}{\partial y^2}\Biggr)+\gamma(u)u, \] where \(\gamma > 0\) if \(|u|< U\), \(\gamma = 0\) if \(|u|= U\), and \(\gamma < 0\) if \(|u|> U\). To be specific, it is assumed that \(\gamma(u) = \gamma_0[1-(\frac{u}{U})^2]\). The equation which is considered inside a bounded domain \(D\) with a boundary \(\Sigma\), is augmented by the initial condition \(u|_{t=0} = u_0(x,y)\) and a boundary condition \(u|_{\Sigma} = 0\). The aim of the article is to be specify a factor determining the character of the evolution and lifetime of an unstable nonstationary contrast structure. The authors study the evolution of the initial field specified in the form of one or several circular or elliptical spots that possibly include spots of opposite sign. The authors show that the domain of rapid field variation in a contrast structure has a surface tension which determines the character of evolution in the examples considered, and suggests a formula for evaluating the surface tension. The methods of asymptotic analysis is used; to determine constraints on various parameters, the authors also employ computer experiments by CRAY Y-MP C90 supercomputer.
scientific article; zbMATH DE number 1557595