Pattern formation with a conservation law

Shock Wave Formation Process for a Multidimensional Scalar Conservation Law

We construct a global smooth approximate solution to a multidimensional scalar conservation law describing the shock wave formation process for initial data with small variation. In order to solve the problem, we modify the method of characteristics by introducing “new characteristics”, nonintersecting curves along which the (approximate) solution to the problem under study is constant. The pro...

Spatial Pattern Formation in Asynchronous Cellular Automata with Mass Conservation

The pattern formation of asynchronous cellular automata with mass conservation is explored. A cellular automaton rule causing a specified kind of pattern was manually designed, and the mechanism of pattern formation was found similar to real chemical process. In addition, a methodology to automatically search for pattern-forming cellular automata is proposed, and its performance is validated. T...

A new universal law for the Liesegang pattern formation.

Classical regularities describing the Liesegang phenomenon have been observed and extensively studied in laboratory experiments for a long time. These have been verified in the last two decades, both theoretically and using simulations. However, they are only applicable if the observed system is driven by reaction and diffusion. We suggest here a new universal law, which is also valid in the ca...

Formation of Large Structures in the Acceleration Universe with a Hybrid Expansion Law

In the current paper, we have studied the effect of dark energy on for-mation where dark energy exists in the background. For this purpose, we used both WMAP9 and Planck data to study how the radius changes with redshift in these mod-els. We used different data sets to fix the cosmological parameters to obtain a solution for a spherical region under collapse. The mechanism of structure formation f...

The Conservation Law