3,512 research outputs found
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
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
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
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
Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations
A complete group classification of a class of variable coefficient
(1+1)-dimensional telegraph equations , is
given, by using a compatibility method and additional equivalence
transformations. A number of new interesting nonlinear invariant models which
have non-trivial invariance algebras are obtained. Furthermore, the possible
additional equivalence transformations between equations from the class under
consideration are investigated. Exact solutions of special forms of these
equations are also constructed via classical Lie method and generalized
conditional transformations. Local conservation laws with characteristics of
order 0 of the class under consideration are classified with respect to the
group of equivalence transformations.Comment: 23 page
Group classification of the Sachs equations for a radiating axisymmetric, non-rotating, vacuum space-time
We carry out a Lie group analysis of the Sachs equations for a time-dependent
axisymmetric non-rotating space-time in which the Ricci tensor vanishes. These
equations, which are the first two members of the set of Newman-Penrose
equations, define the characteristic initial-value problem for the space-time.
We find a particular form for the initial data such that these equations admit
a Lie symmetry, and so defines a geometrically special class of such
spacetimes. These should additionally be of particular physical interest
because of this special geometric feature.Comment: 18 Pages. Submitted to Classical and Quantum Gravit
Perturbative Symmetry Approach
Perturbative Symmetry Approach is formulated in symbolic representation.
Easily verifiable integrability conditions of a given equation are constructed
in the frame of the approach. Generalisation for the case of non-local and
non-evolution equations is disscused. Application of the theory to the
Benjamin-Ono and Camassa-Holm type equations is considered.Comment: 16 page
Nonlocal aspects of -symmetries and ODEs reduction
A reduction method of ODEs not possessing Lie point symmetries makes use of
the so called -symmetries (C. Muriel and J. L. Romero, \emph{IMA J.
Appl. Math.} \textbf{66}, 111-125, 2001). The notion of covering for an ODE
is used here to recover -symmetries of as
nonlocal symmetries. In this framework, by embedding into a
suitable system determined by the function ,
any -symmetry of can be recovered by a local symmetry of
. As a consequence, the reduction method of Muriel and
Romero follows from the standard method of reduction by differential invariants
applied to .Comment: 13 page
Laplace Invariants for General Hyperbolic Systems
We consider the generalization of Laplace invariants to linear differential
systems of arbitrary rank and dimension. We discuss completeness of certain
subsets of invariants
On Darboux Integrable Semi-Discrete Chains
Differential-difference equation
with unknown
depending on continuous and discrete variables and is studied.
We call an equation of such kind Darboux integrable, if there exist two
functions (called integrals) and of a finite number of dynamical
variables such that and , where is the operator of total
differentiation with respect to , and is the shift operator:
. It is proved that the integrals can be brought to some
canonical form. A method of construction of an explicit formula for general
solution to Darboux integrable chains is discussed and for a class of chains
such solutions are found.Comment: 19 page
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