R. A. Leo

Numerical analysis of the one-mode solutions in the Fermi-Pasta-Ulam system

The stability of the one-mode nonlinear solutions of the Fermi-Pasta-Ulam beta system is numerically investigated. No external perturbation is considered for the one-mode exact analytical solutions, the only perturbation being that introduced by computational errors in the numerical integration of motion equations. The threshold energy for the excitation of the other normal modes and the dynamics of this excitation are studied as a function of the parameter micro characterizing the nonlinearity, the energy density epsilon and the number N of particles of the system. The results achieved confirm in part previous ones, obtained with a linear analysis of the problem of the stability, and clarify the dynamics by which a one-mode exchanges energy with the other modes with increasing energy density. In a range of energy density near the threshold value and for various values of the number of particles N, the nonlinear one-mode exchanges energy with the other linear modes for a very short time, immediately recovering all its initial energy. This sort of recurrence is very similar to Fermi recurrences, even if in the Fermi recurrences the energy of the initially excited mode changes continuously and only periodically recovers its initial value. A tentative explanation for this intermittent behavior, in terms of Floquets theorem, is proposed. Preliminary results are also presented for the Fermi-Pasta-Ulam alpha system which show that there is a stability threshold, for large N, independent of N.

Vortices and invariant surfaces generated by symmetries for the 3D Navier–Stokes equations

We show that certain infinitesimal operators of the Lie-point symmetries of the incompressible 3D Navier–Stokes equations give rise to vortex solutions with different characteristics. This approach allows an algebraic classification of vortices and throws light on the alignment mechanism between the vorticity ω and the vortex stretching vector Sω, where S is the strain matrix. The symmetry algebra associated with the Navier–Stokes equations turns out to be infinite-dimensional. New vortical structures, generalizing in some cases well-known configurations such as, for example, the Burgers and Lundgren solutions, are obtained and discussed in relation to the value of the dynamic angle φ=arctan|ω→∧Sω→|/ω→·Sω→. A systematic treatment of the boundary conditions invariant under the symmetry group of the equations under study is also performed, and the corresponding invariant surfaces are recognized.

On the Relation between Lie Symmetries and Prolongation Structures of Nonlinear Field Equations : Non-Local Symmetries

An algebraic method is devised to look for non-local symmetries of the pseudopotential type of nonlinear field equations. The method is based on the use of an infinite-dimensional subalgebra of the prolongation algebra L associated with the equations under consideration. Our approach, which is applied by way of example to the Dym and the Korteweg-de Vries equation, allows us to obtain a general formula for the infinitesimal operator of non-local symmetries expressed in terms of elements of L. The method could be exploited to investigate the symmetry properties of other nonlinear field equations possessing nontrivial prolongations.

Group analysis of the three‐wave resonant system in (2+1) dimensions

The three‐wave resonant interaction equations (2D‐3WR) in two spatial and one temporal dimension within a group framework are analyzed. The symmetry algebra of this system, which turns out to be an infinite‐dimensional Lie algebra whose subalgebra is of the Kac–Moody type, is found. The one‐ and two‐dimensional symmetry subalgebras are classified and the corresponding reduction equations are obtained. From these the new invariant and the partially invariant solutions of the original 2D‐3WR equations are obtained.

Thermostatistics in the neighbourhood of the π-mode solution for the Fermi–Pasta–Ulam β system: from weak to strong chaos

We consider a π-mode solution of the Fermi–Pasta–Ulam β system. By perturbing it, we study the system as a function of the energy density from a regime where the solution is stable to a regime where it is unstable, first weakly and then strongly chaotic. We introduce, as an indicator of stochasticity, the ratio ρ (when it is defined) between the second and the first moment of a given probability distribution. We will show numerically that the transition between weak and strong chaos can be interpreted as the symmetry breaking of a set of suitable dynamical variables. Moreover, we show that in the region of weak chaos there is numerical evidence that the thermostatistic is governed by the Tsallis distribution.

A theorem on the existence of symmetries of fractional PDEs

Abstract We propose a theorem that extends the classical Lie approach to the case of fractional partial differential equations (fPDEs) of the Riemann–Liouville type in ( 1 + 1 ) dimensions.

On the isospectral-eigenvalue problem and the recursion operator of the Harry-Dym equation

SummaryThe isospectral-eigenvalue problem for the Herry-Dym equation is derived using a prolongation technique. The eigenvalue equation is then exploited to find a recursion operator.

An analysis of the Transverse Optical Klystron (TOK) and the Free Electron Laser (FEL) through the exact solution of their evolution equation

Abstract We give the exact analytical solution of the evolution equation of an electron beam interacting in a wiggler with an electromagnetic wave. The initial energy distribution is assumed both monochromatic and gaussian. Furthermore, the evolution in a drift space is considered. Some basic characteristics of the FEL and TOK are derived from the mathematical properties of the solution.

A foundational approach to the Lie theory for fractional order partial differential equations

Abstract We provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie symmetries in the case of an arbitrary finite number of independent variables. We also prove the Lie theorem in the case of fractional differential equations, showing that the proper space for the analysis of these symmetries is the infinite dimensional jet space.

Gauge equivalence theory of the noncompact Ishimori model and the Davey-Stewartson equation

The gauge equivalence between a noncompact version of the Ishimori spin model and the Davey–Stewartson equation is established. Explicit relationships connecting the corresponding two sets of fields involved in these systems are obtained via any pair of complex functions satisfying an equation of the Schrodinger type for a free particle. Using these formulas, two examples of classes of nontrivial exact singular solutions to the Davey–Stewartson equation are given. One of them is of the closed stringlike type, while the other is doubly periodic and is expressed in terms of Riemann theta functions. The role played by the symmetry group associated with the gauge equivalent equations under consideration is also clarified.