We show that, under rather general assumptions on the form of the entropy function, the energy balance equation for a system in thermodynamic equilibrium is equivalent to a set of nonlinear equations of hydrodynamic type. This set of equations is integrable via the method of characteristics and it provides the equation of state for the gas. The shock wave catastrophe set identifies the phase tr...

Dynamics of sine-Gordon kinks in the presence of rapidly varying periodic perturbations of different physical origins is described analytically and numerically. The analytical approach is based on asymptotic expansions, and it allows to derive, in a rigorous way, an effective nonlinear equation for the slowly varying field component in any order of the asymptotic procedure as expansions in the ...

We consider dissipative systems on the real axis in situations when the evolution is dominated by a dynamics similar to the one of a linear diffusion equation. It is surprising that such a diffusive behavior occurs in relatively complicated systems. After a discussion of the linear and nonlinear diffusion equation, we give a brief introduction into the methods which are available to describe di...

A survey of results and problems of soliton dynamics in dispersive and hyperbolic nonlinear PDE’s and the related spectral and scattering theory. I focus on the problem of large time behavior of the nonlinear Schrödinger equation, with both solitary and radiative waves appearing in the solution. The equations are nonintegrable in general and in arbitrary dimension. I will formulate the main con...

Abstract{In this paper, we study on behavior control of robot using orbits of nonlinear dynamics. Behavior generation and control using entrainment and synchronization phenomena in nonlinear dynamical systems is discussed. The behavior of the Arnold equation, which is known to show the chaotic behavior of non-compressive perfect uid, is analyzed. Methods to integrate the dynamics into the infor...

We present theoretical and numerical evidence to show that self-defocusing nonlinear optical propagation can be used to compute Euler fluid dynamics and possibly Navier-Stokes fluid dynamics. In particular, the formation of twin vortices and the Kármán vortex street behind an obstacle, two well-known viscous fluid phenomena, is numerically demonstrated using the nonlinear Schrödinger equation. ...

We consider a vector kinetic BGK equation leading to isentropic gas dynamics in the relaxation limit. We prove the existence of a weak solution satisfying a kinetic entropy inequality for any initial data with nite energy. Key-words: isentropic gas dynamics { kinetic BGK model { kinetic entropy Mathematics Subject Classi cation: 35L65, 76N15, 35B35, 82C40 Work partially supported by European TM...