3,886 research outputs found
Lattice Dynamics in the Half-Space, II. Energy Transport Equation
We consider the lattice dynamics in the half-space. The initial data are
random according to a probability measure which enforces slow spatial variation
on the linear scale . We establish two time regimes. For
times of order , , locally the measure
converges to a Gaussian measure which is time stationary with a covariance
inherited from the initial measure (non-Gaussian, in general). For times of
order , this covariance changes in time and is governed by a
semiclassical transport equation.Comment: 35 page
Classification of conservation laws of compressible isentropic fluid flow in n>1 spatial dimensions
For the Euler equations governing compressible isentropic fluid flow with a
barotropic equation of state (where pressure is a function only of the
density), local conservation laws in spatial dimensions are fully
classified in two primary cases of physical and analytical interest: (1)
kinematic conserved densities that depend only on the fluid density and
velocity, in addition to the time and space coordinates; (2) vorticity
conserved densities that have an essential dependence on the curl of the fluid
velocity. A main result of the classification in the kinematic case is that the
only equation of state found to be distinguished by admitting extra
-dimensional conserved integrals, apart from mass, momentum, energy, angular
momentum and Galilean momentum (which are admitted for all equations of state),
is the well-known polytropic equation of state with dimension-dependent
exponent . In the vorticity case, no distinguished equations of
state are found to arise, and here the main result of the classification is
that, in all even dimensions , a generalized version of Kelvin's
two-dimensional circulation theorem is obtained for a general equation of
state.Comment: 24 pages; published version with misprints correcte
Ordinary differential equations which linearize on differentiation
In this short note we discuss ordinary differential equations which linearize
upon one (or more) differentiations. Although the subject is fairly elementary,
equations of this type arise naturally in the context of integrable systems.Comment: 9 page
Nonlinear self-adjointness and conservation laws
The general concept of nonlinear self-adjointness of differential equations
is introduced. It includes the linear self-adjointness as a particular case.
Moreover, it embraces the strict self-adjointness and quasi self-adjointness
introduced earlier by the author. It is shown that the equations possessing the
nonlinear self-adjointness can be written equivalently in a strictly
self-adjoint form by using appropriate multipliers. All linear equations
possess the property of nonlinear self-adjointness, and hence can be rewritten
in a nonlinear strictly self-adjoint. For example, the heat equation becomes strictly self-adjoint after multiplying by
Conservation laws associated with symmetries can be constructed for all
differential equations and systems having the property of nonlinear
self-adjointness
A tree of linearisable second-order evolution equations by generalised hodograph transformations
We present a list of (1+1)-dimensional second-order evolution equations all
connected via a proposed generalised hodograph transformation, resulting in a
tree of equations transformable to the linear second-order autonomous evolution
equation. The list includes autonomous and nonautonomous equations.Comment: arXiv version is already officia
On the hierarchy of partially invariant submodels of differential equations
It is noticed, that partially invariant solution (PIS) of differential
equations in many cases can be represented as an invariant reduction of some
PIS of the higher rank. This introduce a hierarchic structure in the set of all
PISs of a given system of differential equations. By using this structure one
can significantly decrease an amount of calculations required in enumeration of
all PISs for a given system of partially differential equations. An equivalence
of the two-step and the direct ways of construction of PISs is proved. In this
framework the complete classification of regular partially invariant solutions
of ideal MHD equations is given
A Necessary Condition for existence of Lie Symmetries in Quasihomogeneous Systems of Ordinary Differential Equations
Lie symmetries for ordinary differential equations are studied. In systems of
ordinary differential equations, there do not always exist non-trivial Lie
symmetries around equilibrium points. We present a necessary condition for
existence of Lie symmetries analytic in the neighbourhood of an equilibrium
point. In addition, this result can be applied to a necessary condition for
existence of a Lie symmetry in quasihomogeneous systems of ordinary
differential equations. With the help of our main theorem, it is proved that
several systems do not possess any analytic Lie symmetries.Comment: 15 pages, no figures, AMSLaTe
The model equation of soliton theory
We consider an hierarchy of integrable 1+2-dimensional equations related to
Lie algebra of the vector fields on the line. The solutions in quadratures are
constructed depending on arbitrary functions of one argument. The most
interesting result is the simple equation for the generating function of the
hierarchy which defines the dynamics for the negative times and also has
applications to the second order spectral problems. A rather general theory of
integrable 1+1-dimensional equations can be developed by study of polynomial
solutions of this equation under condition of regularity of the corresponding
potentials.Comment: 17
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