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Register a new userScaling, self-similar solutions and shock waves for V-shaped field potentials
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We investigate a (1+1)-dimensional nonlinear field theoretic model with the field potential $V(phi)| = |phi|.$ It can be obtained as the universal small amplitude limit in a class of models with potentials which are symmetrically V-shaped at their minima, or as a continuum limit of certain mechanical system with infinite number of degrees of freedom. The model has an interesting scaling symmetry of the on shell type. We find self-similar as well as shock wave solutions of the field equation in that model.
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In this lecture we outline the main results of our investigations of certain field-theoretic systems which have V-shaped field potential. After presenting physical examples of such systems, we show that in static problems the exact ground state value of the field is achieved on a finite distance - there are no exponential tails. This applies in particular to soliton-like object called the topological compacton. Next, we discuss scaling invariance which appears when the fields are restricted to small amplitude perturbations of the ground state. Evolution of such perturbations is governed by nonlinear equation with a non-smooth term which can not be linearized even in the limit of very small amplitudes. Finally, we briefly describe self-similar and shock wave solutions of that equation.
This is a brief review of the Schrodingers factorization method and its relations to supersymmetric quantum mechanics and its nonlinear (parastatistical, etc) modifications, self-similar infinite soliton potentials, quantum algebras, coherent states, Ising chains, discretized random matrices and 2D lattice Coulomb gases.
Self-similarity in the network traffic has been studied from several aspects: both at the user side and at the network side there are many sources of the long range dependence. Recently some dynamical origins are also identified: the TCP adaptive congestion avoidance algorithm itself can produce chaotic and long range dependent throughput behavior, if the loss rate is very high. In this paper we show that there is a close connection between the static and dynamic origins of self-similarity: parallel TCPs can generate the self-similarity themselves, they can introduce heavily fluctuations into the background traffic and produce high effective loss rate causing a long range dependent TCP flow, however, the dropped packet ratio is low.
We investigate strongly correlated non-Abelian plasmas out of equilibrium. Based on numerical simulations, we establish a self-similar scaling property for the time evolution of spatial Wilson loops that characterizes a universal state of matter far from equilibrium. Most remarkably, it exhibits a generalized area law which holds for sufficiently large ratio of spatial area and fractional power of time. Performing calculations also for the perturbative regime at higher momenta, we are able to characterize the full nonthermal scaling properties of SU(2) and SU(3) symmetric plasmas from short to large distance scales in terms of two independent universal exponents and associated scaling functions.
We consider the quantum mechanical problem of the motion of a spinless charged relativistic particle with mass$M$, described by the Klein-Fock-Gordon equation with equal scalar $S(vec{r})$ and vector $V(vec{r})$ Coulomb plus ring-shaped potentials. It is shown that the system under consideration has both a discrete at $left|Eright|<Mc^{2} $ and a continuous at $left|Eright|>Mc^{2} $ energy spectra. We find the analytical expressions for the corresponding complete wave functions. A dynamical symmetry group $SU(1,1)$ for the radial wave equation of motion is constructed. The algebra of generators of this group makes it possible to find energy spectra in a purely algebraic way. It is also shown that relativistic expressions for wave functions, energy spectra and group generators in the limit $cto infty $ go over into the corresponding expressions for the nonrelativistic problem.
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