10,454 research outputs found
A refined invariant subspace method and applications to evolution equations
The invariant subspace method is refined to present more unity and more
diversity of exact solutions to evolution equations. The key idea is to take
subspaces of solutions to linear ordinary differential equations as invariant
subspaces that evolution equations admit. A two-component nonlinear system of
dissipative equations was analyzed to shed light on the resulting theory, and
two concrete examples are given to find invariant subspaces associated with
2nd-order and 3rd-order linear ordinary differential equations and their
corresponding exact solutions with generalized separated variables.Comment: 16 page
Invariant Modules and the Reduction of Nonlinear Partial Differential Equations to Dynamical Systems
We completely characterize all nonlinear partial differential equations
leaving a given finite-dimensional vector space of analytic functions
invariant. Existence of an invariant subspace leads to a re duction of the
associated dynamical partial differential equations to a system of ordinary
differential equations, and provide a nonlinear counterpart to quasi-exactly
solvable quantum Hamiltonians. These results rely on a useful extension of the
classical Wronskian determinant condition for linear independence of functions.
In addition, new approaches to the characterization o f the annihilating
differential operators for spaces of analytic functions are presented.Comment: 28 pages. To appear in Advances in Mathematic
Conformal flow on and weak field integrability in AdS
We consider the conformally invariant cubic wave equation on the Einstein
cylinder for small rotationally symmetric
initial data. This simple equation captures many key challenges of nonlinear
wave dynamics in confining geometries, while a conformal transformation relates
it to a self-interacting conformally coupled scalar in four-dimensional anti-de
Sitter spacetime (AdS) and connects it to various questions of AdS
stability. We construct an effective infinite-dimensional time-averaged
dynamical system accurately approximating the original equation in the weak
field regime. It turns out that this effective system, which we call the
conformal flow, exhibits some remarkable features, such as low-dimensional
invariant subspaces, a wealth of stationary states (for which energy does not
flow between the modes), as well as solutions with nontrivial exactly periodic
energy flows. Based on these observations and close parallels to the cubic
Szego equation, which was shown by Gerard and Grellier to be Lax-integrable, it
is tempting to conjecture that the conformal flow and the corresponding weak
field dynamics in AdS are integrable as well.Comment: 22 pages, v2: minor revisions, several references added, v3: typos
corrected, v4: typos corrected, one reference added, matches version accepted
by CM
Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces
In this paper we derive structure theorems that characterize the spaces of
linear and non-linear differential operators that preserve finite dimensional
subspaces generated by polynomials in one or several variables. By means of the
useful concept of deficiency, we can write explicit basis for these spaces of
differential operators. In the case of linear operators, these results apply to
the theory of quasi-exact solvability in quantum mechanics, specially in the
multivariate case where the Lie algebraic approach is harder to apply. In the
case of non-linear operators, the structure theorems in this paper can be
applied to the method of finding special solutions of non-linear evolution
equations by nonlinear separation of variables.Comment: 23 pages, typed in AMS-LaTe
Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems
Drawing upon the bursting mechanism in slow-fast systems, we propose
indicators for the prediction of such rare extreme events which do not require
a priori known slow and fast coordinates. The indicators are associated with
functionals defined in terms of Optimally Time Dependent (OTD) modes. One such
functional has the form of the largest eigenvalue of the symmetric part of the
linearized dynamics reduced to these modes. In contrast to other choices of
subspaces, the proposed modes are flow invariant and therefore a projection
onto them is dynamically meaningful. We illustrate the application of these
indicators on three examples: a prototype low-dimensional model, a body forced
turbulent fluid flow, and a unidirectional model of nonlinear water waves. We
use Bayesian statistics to quantify the predictive power of the proposed
indicators
Relative Periodic Solutions of the Complex Ginzburg-Landau Equation
A method of finding relative periodic orbits for differential equations with
continuous symmetries is described and its utility demonstrated by computing
relative periodic solutions for the one-dimensional complex Ginzburg-Landau
equation (CGLE) with periodic boundary conditions. A relative periodic solution
is a solution that is periodic in time, up to a transformation by an element of
the equation's symmetry group. With the method used, relative periodic
solutions are represented by a space-time Fourier series modified to include
the symmetry group element and are sought as solutions to a system of nonlinear
algebraic equations for the Fourier coefficients, group element, and time
period. The 77 relative periodic solutions found for the CGLE exhibit a wide
variety of temporal dynamics, with the sum of their positive Lyapunov exponents
varying from 5.19 to 60.35 and their unstable dimensions from 3 to 8.
Preliminary work indicates that weighted averages over the collection of
relative periodic solutions accurately approximate the value of several
functionals on typical trajectories.Comment: 32 pages, 12 figure
Differential constraints and exact solutions of nonlinear diffusion equations
The differential constraints are applied to obtain explicit solutions of
nonlinear diffusion equations. Certain linear determining equations with
parameters are used to find such differential constraints. They generalize the
determining equations used in the search for classical Lie symmetries
Negative magnetic eddy diffusivities from test-field method and multiscale stability theory
The generation of large-scale magnetic field in the kinematic regime in the
absence of an alpha-effect is investigated by following two different
approaches, namely the test-field method and multiscale stability theory
relying on the homogenisation technique. We show analytically that the former,
applied for the evaluation of magnetic eddy diffusivities, yields results that
fully agree with the latter. Our computations of the magnetic eddy diffusivity
tensor for the specific instances of the parity-invariant flow-IV of G.O.
Roberts and the modified Taylor-Green flow in a suitable range of parameter
values confirm the findings of previous studies, and also explain some of their
apparent contradictions. The two flows have large symmetry groups; this is used
to considerably simplify the eddy diffusivity tensor. Finally, a new analytic
result is presented: upon expressing the eddy diffusivity tensor in terms of
solutions to auxiliary problems for the adjoint operator, we derive relations
between magnetic eddy diffusivity tensors that arise for opposite small-scale
flows v(x) and -v(x).Comment: 29 pp., 19 figures, 42 reference
- …