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Register a new userOrbital Stability of Localized Structures via Backlund Transformations
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The Backlund Transform, first developed in the context of differential geometry, has been classically used to obtain multi-soliton states in completely integrable infinite dimensional dynamical systems. It has recently been used to study the stability of these special solutions. We offer here a dynamical perspective on the Backlund Transform, prove an abstract orbital stability theorem, and demonstrate its utility by applying it to the sine-Gordon equation and the Toda lattice.
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The geometry of an admissible Backlund transformation for an exterior differential system is described by an admissible Cartan connection for a geometric structure on a tower with infinite--dimensional skeleton which is the universal prolongation of a $|1|$--graded semi-simple Lie algebra.
We construct Backlund transformations (BTs) for the Kirchhoff top by taking advantage of the common algebraic Poisson structure between this system and the $sl(2)$ trigonometric Gaudin model. Our BTs are integrable maps providing an exact time-discretization of the system, inasmuch as they preserve both its Poisson structure and its invariants. Moreover, in some special cases we are able to show that these maps can be explicitly integrated in terms of the initial conditions and of the iteration time $n$. Encouraged by these partial results we make the conjecture that the maps are interpolated by a specific one-parameter family of hamiltonian flows, and present the corresponding solution. We enclose a few pictures where the orbits of the continuous and of the discrete flow are depicted.
In this work we give a mechanical (Hamiltonian) interpretation of the so called spectrality property introduced by Sklyanin and Kuznetsov in the context of Backlund transformations (BTs) for finite dimensional integrable systems. The property turns out to be deeply connected with the Hamilton-Jacobi separation of variables and can lead to the explicit integration of the underlying model through the expression of the BTs. Once such construction is given, it is shown, in a simple example, that it is possible to interpret the Baxter Q operator defining the quantum BTs us the Greens function, or propagator, of the time dependent Schrodinger equation for the interpolating Hamiltonian.
In this paper, we study the existence, stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise. We first prove the existence and uniqueness of tempered random attractors for the stochastic equations and characterize the structures of the attractors by random complete solutions. We then examine the existence and stability of random complete quasi-solutions and establish the relations of these solutions and the structures of tempered attractors. When the stochastic equations are incorporated with periodic forcing, we obtain the existence and stability of random periodic solutions. For the stochastic Chafee-Infante equation, we further establish the multiplicity and stochastic bifurcation of complete and periodic solutions.
Normal form stability estimates are a basic tool of Celestial Mechanics for characterizing the long-term stability of the orbits of natural and artificial bodies. Using high-order normal form constructions, we provide three different estimates for the orbital stability of point-mass satellites orbiting around the Earth. i) We demonstrate the long term stability of the semimajor axis within the framework of the $J_2$ problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining $H_{J_2}$ . ii) We demonstrate the stability of the eccentricity and inclination in a secular Hamiltonian model including lunisolar perturbations (the geolunisolar Hamiltonian $H_{gls}$), after a suitable reduction of the Hamiltonian to the Laplace plane. iii) We numerically examine the convexity and steepness properties of the integrable part of the secular Hamiltonian in both the $H_{J_2}$ and $H_{gls}$ models, which reflect necessary conditions for the holding of Nekhoroshevs theorem on the exponential stability of the orbits. We find that the $H_{J_2}$ model is non-convex, but satisfies a three-jet condition, while the $H_{gls}$ model restores quasi-convexity by adding lunisolar terms in the Hamiltonians integrable part.
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