48,567 research outputs found
Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation
We consider the symmetry properties of an integro-differential
multidimensional Gross-Pitaevskii equation with a nonlocal nonlinear (cubic)
term in the context of symmetry analysis using the formalism of semiclassical
asymptotics. This yields a semiclassically reduced nonlocal Gross-Pitaevskii
equation, which can be treated as a nearly linear equation, to determine the
principal term of the semiclassical asymptotic solution. Our main result is an
approach which allows one to construct a class of symmetry operators for the
reduced Gross-Pitaevskii equation. These symmetry operators are determined by
linear relations including intertwining operators and additional algebraic
conditions. The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii
equation. The symmetry operators are found explicitly, and the corresponding
families of exact solutions are obtained
Exponentially accurate solution tracking for nonlinear ODEs, the higher order Stokes phenomenon and double transseries resummation
We demonstrate the conjunction of new exponential-asymptotic effects in the context of a second order nonlinear ordinary differential equation with a small parameter. First, we show how to use a hyperasymptotic, beyond-all-orders approach to seed a numerical solver of a nonlinear ordinary differential equation with sufficiently accurate initial data so as to track a specific solution in the presence of an attractor. Second, we demonstrate the necessary role of a higher order Stokes phenomenon in analytically tracking the transition between asymptotic behaviours in a heteroclinic solution. Third, we carry out a double resummation involving both subdominant and sub-subdominant transseries to achieve the two-dimensional (in terms of the arbitrary constants) uniform approximation that allows the exploration of the behaviour of a two parameter set of solutions across wide regions of the independent variable. This is the first time all three effects have been studied jointly in the context of an asymptotic treatment of a nonlinear ordinary differential equation with a parameter. This paper provides an exponential asymptotic algorithm for attacking such problems when they occur. The availability of explicit results would depend on the individual equation under study
An application of the Maslov complex germ method to the 1D nonlocal Fisher-KPP equation
A semiclassical approximation approach based on the Maslov complex germ
method is considered in detail for the 1D nonlocal
Fisher-Kolmogorov-Petrovskii-Piskunov equation under the supposition of weak
diffusion. In terms of the semiclassical formalism developed, the original
nonlinear equation is reduced to an associated linear partial differential
equation and some algebraic equations for the coefficients of the linear
equation with a given accuracy of the asymptotic parameter. The solutions of
the nonlinear equation are constructed from the solutions of both the linear
equation and the algebraic equations. The solutions of the linear problem are
found with the use of symmetry operators. A countable family of the leading
terms of the semiclassical asymptotics is constructed in explicit form.
The semiclassical asymptotics are valid by construction in a finite time
interval. We construct asymptotics which are different from the semiclassical
ones and can describe evolution of the solutions of the
Fisher-Kolmogorov-Petrovskii-Piskunov equation at large times. In the example
considered, an initial unimodal distribution becomes multimodal, which can be
treated as an example of a space structure.Comment: 28 pages, version accepted for publication in Int. J. Geom. Methods
Mod. Phy
Global well-posedness of Kirchhoff systems
The aim of this paper is to establish the global well-posedness for
Kirchhoff systems. The new approach to the construction of solutions is based
on the asymptotic integrations for strictly hyperbolic systems with
time-dependent coefficients. These integrations play an important role to
setting the subsequent fixed point argument. The existence of solutions for
less regular data is discussed, and several examples and applications are
presented.Comment: 24 page
Continuous Symmetries of Difference Equations
Lie group theory was originally created more than 100 years ago as a tool for
solving ordinary and partial differential equations. In this article we review
the results of a much more recent program: the use of Lie groups to study
difference equations. We show that the mismatch between continuous symmetries
and discrete equations can be resolved in at least two manners. One is to use
generalized symmetries acting on solutions of difference equations, but leaving
the lattice invariant. The other is to restrict to point symmetries, but to
allow them to also transform the lattice.Comment: Review articl
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