34 research outputs found
Classical and nonclassical symmetries of a generalized Boussinesq equation
We apply the Lie-group formalism and the nonclassical method due to Bluman
and Cole to deduce symmetries of the generalized Boussinesq equation, which has
the classical Boussinesq equation as an special case. We study the class of
functions for which this equation admit either the classical or the
nonclassical method. The reductions obtained are derived. Some new exact
solutions can be derived
Nonlinear Dirac and diffusion equations in 1 + 1 dimensions from stochastic considerations
We generalize the method of obtaining the fundamental linear partial
differential equations such as the diffusion and Schrodinger equation, Dirac
and telegrapher's equation from a simple stochastic consideration to arrive at
certain nonlinear form of these equations. The group classification through one
parameter group of transformation for two of these equations is also carried
out.Comment: 18 pages, Latex file, some equations corrected and group analysis in
one more case adde
Strichartz estimates for the water-wave problem with surface tension
Strichartz-type estimates for one-dimensional surface water-waves under
surface tension are studied, based on the formulation of the problem as a
nonlinear dispersive equation. We establish a family of dispersion estimates on
time scales depending on the size of the frequencies. We infer that a solution
of the dispersive equation we introduce satisfies local-in-time Strichartz
estimates with loss in derivative:
where depends on and on the norms of the
initial data in . The proof uses the frequency analysis
and semiclassical Strichartz estimates for the linealized water-wave operator.Comment: Fixed typos and mistakes. Merged with arXiv:0809.451
Study of the risk-adjusted pricing methodology model with methods of Geometrical Analysis
Families of exact solutions are found to a nonlinear modification of the
Black-Scholes equation. This risk-adjusted pricing methodology model (RAPM)
incorporates both transaction costs and the risk from a volatile portfolio.
Using the Lie group analysis we obtain the Lie algebra admitted by the RAPM
equation. It gives us the possibility to describe an optimal system of
subalgebras and correspondingly the set of invariant solutions to the model. In
this way we can describe the complete set of possible reductions of the
nonlinear RAPM model. Reductions are given in the form of different second
order ordinary differential equations. In all cases we provide solutions to
these equations in an exact or parametric form. We discuss the properties of
these reductions and the corresponding invariant solutions.Comment: larger version with exact solutions, corrected typos, 13 pages,
Symposium on Optimal Stopping in Abo/Turku 200
Group classification of variable coefficient KdV-like equations
The exhaustive group classification of the class of KdV-like equations with
time-dependent coefficients is carried out using
equivalence based approach. A simple way for the construction of exact
solutions of KdV-like equations using equivalence transformations is described.Comment: 8 pages; minor misprints are corrected. arXiv admin note: substantial
text overlap with arXiv:1104.198
On asymptotic nonlocal symmetry of nonlinear Schr\"odinger equations
A concept of asymptotic symmetry is introduced which is based on a definition
of symmetry as a reducibility property relative to a corresponding invariant
ansatz. It is shown that the nonlocal Lorentz invariance of the free-particle
Schr\"odinger equation, discovered by Fushchych and Segeda in 1977, can be
extended to Galilei-invariant equations for free particles with arbitrary spin
and, with our definition of asymptotic symmetry, to many nonlinear
Schr\"odinger equations. An important class of solutions of the free
Schr\"odinger equation with improved smoothing properties is obtained
Symmetry justification of Lorenz' maximum simplification
In 1960 Edward Lorenz (1917-2008) published a pioneering work on the `maximum
simplification' of the barotropic vorticity equation. He derived a coupled
three-mode system and interpreted it as the minimum core of large-scale fluid
mechanics on a `finite but unbounded' domain. The model was obtained in a
heuristic way, without giving a rigorous justification for the chosen selection
of modes. In this paper, it is shown that one can legitimate Lorenz' choice by
using symmetry transformations of the spectral form of the vorticity equation.
The Lorenz three-mode model arises as the final step in a hierarchy of models
constructed via the component reduction by means of symmetries. In this sense,
the Lorenz model is indeed the `maximum simplification' of the vorticity
equation.Comment: 8 pages, minor correction
Equivalence of conservation laws and equivalence of potential systems
We study conservation laws and potential symmetries of (systems of)
differential equations applying equivalence relations generated by point
transformations between the equations. A Fokker-Planck equation and the Burgers
equation are considered as examples. Using reducibility of them to the
one-dimensional linear heat equation, we construct complete hierarchies of
local and potential conservation laws for them and describe, in some sense, all
their potential symmetries. Known results on the subject are interpreted in the
proposed framework. This paper is an extended comment on the paper of J.-q. Mei
and H.-q. Zhang [Internat. J. Theoret. Phys., 2006, in press].Comment: 10 page
Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source
A new approach to group classification problems and more general
investigations on transformational properties of classes of differential
equations is proposed. It is based on mappings between classes of differential
equations, generated by families of point transformations. A class of variable
coefficient (1+1)-dimensional semilinear reaction-diffusion equations of the
general form () is studied from the
symmetry point of view in the framework of the approach proposed. The singular
subclass of the equations with is singled out. The group classifications
of the entire class, the singular subclass and their images are performed with
respect to both the corresponding (generalized extended) equivalence groups and
all point transformations. The set of admissible transformations of the imaged
class is exhaustively described in the general case . The procedure of
classification of nonclassical symmetries, which involves mappings between
classes of differential equations, is discussed. Wide families of new exact
solutions are also constructed for equations from the classes under
consideration by the classical method of Lie reductions and by generation of
new solutions from known ones for other equations with point transformations of
different kinds (such as additional equivalence transformations and mappings
between classes of equations).Comment: 40 pages, this is version published in Acta Applicanda Mathematica
Dynamics of the Universe with global rotation
We analyze dynamics of the FRW models with global rotation in terms of
dynamical system methods. We reduce dynamics of these models to the FRW models
with some fictitious fluid which scales like radiation matter. This fluid
mimics dynamically effects of global rotation. The significance of the global
rotation of the Universe for the resolution of the acceleration and horizon
problems in cosmology is investigated. It is found that dynamics of the
Universe can be reduced to the two-dimensional Hamiltonian dynamical system.
Then the construction of the Hamiltonian allows for full classification of
evolution paths. On the phase portraits we find the domains of cosmic
acceleration for the globally rotating universe as well as the trajectories for
which the horizon problem is solved. We show that the FRW models with global
rotation are structurally stable. This proves that the universe acceleration is
due to the global rotation. It is also shown how global rotation gives a
natural explanation of the empirical relation between angular momentum for
clusters and superclusters of galaxies. The relation is obtained
as a consequence of self similarity invariance of the dynamics of the FRW model
with global rotation. In derivation of this relation we use the Lie group of
symmetry analysis of differential equation.Comment: Revtex4, 22 pages, 5 figure