27 research outputs found
On the paper "Symmetry analysis of wave equation on sphere" by H. Azad and M. T. Mustafa
Using the scalar curvature of the product manifold S^{2}X R and the complete
group classification of nonlinear Poisson equation on (pseudo) Riemannian
manifolds, we extend the previous results on symmetry analysis of homogeneous
wave equation obtained by H. Azad and M. T. Mustafa [H. Azad and M. T. Mustafa,
Symmetry analysis of wave equation on sphere, J. Math. Anal. Appl., 333 (2007)
1180--1888] to nonlinear Klein-Gordon equations on the two-dimensional sphere.Comment: Version accepted in J. Math. Anal. App
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
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
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
On the classification of conditionally integrable evolution systems in (1+1) dimensions
We generalize earlier results of Fokas and Liu and find all locally analytic
(1+1)-dimensional evolution equations of order that admit an -shock type
solution with .
To this end we develop a refinement of the technique from our earlier work
(A. Sergyeyev, J. Phys. A: Math. Gen, 35 (2002), 7653--7660), where we
completely characterized all (1+1)-dimensional evolution systems
\bi{u}_t=\bi{F}(x,t,\bi{u},\p\bi{u}/\p x,...,\p^n\bi{u}/\p x^n) that are
conditionally invariant under a given generalized (Lie--B\"acklund) vector
field \bi{Q}(x,t,\bi{u},\p\bi{u}/\p x,...,\p^k\bi{u}/\p x^k)\p/\p\bi{u} under
the assumption that the system of ODEs \bi{Q}=0 is totally nondegenerate.
Every such conditionally invariant evolution system admits a reduction to a
system of ODEs in , thus being a nonlinear counterpart to quasi-exactly
solvable models in quantum mechanics.
Keywords: Exact solutions, nonlinear evolution equations, conditional
integrability, generalized symmetries, reduction, generalized conditional
symmetries
MSC 2000: 35A30, 35G25, 81U15, 35N10, 37K35, 58J70, 58J72, 34A34Comment: 8 pages, LaTeX 2e, now uses hyperre
Generalized conditional symmetries of evolution equations
We analyze the relationship of generalized conditional symmetries of
evolution equations to the formal compatibility and passivity of systems of
differential equations as well as to systems of vector fields in involution.
Earlier results on the connection between generalized conditional invariance
and generalized reduction of evolution equations are revisited. This leads to a
no-go theorem on determining equations for operators of generalized conditional
symmetry. It is also shown that up to certain equivalences there exists a
one-to-one correspondence between generalized conditional symmetries of an
evolution equation and parametric families of its solutions.Comment: 23 pages, extended versio
Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach
Five types of blow-up patterns that can occur for the 4th-order semilinear
parabolic equation of reaction-diffusion type
u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1,
\quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For
the semilinear heat equation , various blow-up patterns
were under scrutiny since 1980s, while the case of higher-order diffusion was
studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure
Ordinary Differential Operators Possessing Invariant Subspaces of the Power Type
Ordinary dierential operators possessing invariant subspaces spanned by the functions x , i = 0; 1; : : : ; n 1, are considered. A complete description of operators of the submaximal order n 2 is obtained. The dimension C 2n 1 of the linear space of such operators is conjectured to be the upper bound for the operators possessing arbitrary n-dimensional invariant subspace. A general representation for translation-invariant operators is found