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Register a new userOn applications of Maupertuis-Jacobi correspondence for Hamiltonians $F(x,|p|)$ in some 2-D stationary semiclassical problems
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We make use of the Maupertuis -- Jacobi correspondence, well known in Classical Mechanics, to simplify 2-D asymptotic formulas based on Maslovs canonical operator, when constructing Lagrangian manifolds invariant with respect to phase flows for Hamiltonians of the form $F(x,|p|)$. As examples we consider Hamiltonians coming from the Schrodinger equation, the 2-D Dirac equation for graphene and linear water wave theory.
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We set up and study a coupled problem on stationary non-isothermal flow of electrorheological fluids. The problem consist in finding functions of velocity, pressure and temperature which satisfy the motion equations, the condition of incompressibility, the equation of the balance of thermal energy and boundary conditions. We introduce the notions of a $P$-generalized solution and generalized solution of the coupled problem. In case of the $P$-generalized solution the dissipation of energy is defined by the regularized velocity field, which leads to a nonlocal model. Under weak conditions, we prove the existence of the $P$ -generalized solution of the coupled problem. The existence of the generalized solution is proved under the conditions on smoothness of the boundary and on smallness of the data of the problem
We present a simple formula for the generating function for the polynomials in the $d$--dimensional semiclassical wave packets. We then use this formula to prove the associated Rodrigues formula.
We are interested in the phenomenon of the essential spectrum instability for a class of unbounded (block) Jacobi matrices. We give a series of sufficient conditions for the matrices from certain classes to have a discrete spectrum on a half-axis of a real line. An extensive list of examples showing the sharpness of obtained results is provided.
In this paper, classical small perturbations against a stationary solution of the nonlinear Schrodinger equation with the general form of nonlinearity are examined. It is shown that in order to obtain correct (in particular, conserved over time) nonzero expressions for the basic integrals of motion of a perturbation even in the quadratic order in the expansion parameter, it is necessary to consider nonlinear equations of motion for the perturbations. It is also shown that, despite the nonlinearity of the perturbations, the additivity property is valid for the integrals of motion of different nonlinear modes forming the perturbation (at least up to the second order in the expansion parameter).
We investigate the exact relation existing between the stability equation for the solutions of a mechanical system and the geodesic deviation equation of the associated geodesic problem in the Jacobi metric constructed via the Maupertuis-Jacobi Principle. We conclude that the dynamical and geometrical approaches to the stability/instability problem are not equivalent.
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