309 research outputs found
On Negative Order KdV Equations
In this paper, based on the regular KdV system, we study negative order KdV
(NKdV) equations about their Hamiltonian structures, Lax pairs, infinitely many
conservation laws, and explicit multi-soliton and multi-kink wave solutions
thorough bilinear B\"{a}cklund transformations. The NKdV equations studied in
our paper are differential and actually derived from the first member in the
negative order KdV hierarchy. The NKdV equations are not only gauge-equivalent
to the Camassa-Holm equation through some hodograph transformations, but also
closely related to the Ermakov-Pinney systems, and the Kupershmidt deformation.
The bi-Hamiltonian structures and a Darboux transformation of the NKdV
equations are constructed with the aid of trace identity and their Lax pairs,
respectively. The single and double kink wave and bell soliton solutions are
given in an explicit formula through the Darboux transformation. The 1-kink
wave solution is expressed in the form of while the 1-bell soliton is in
the form of , and both forms are very standard. The collisions of
2-kink-wave and 2-bell-soliton solutions, are analyzed in details, and this
singular interaction is a big difference from the regular KdV equation.
Multi-dimensional binary Bell polynomials are employed to find bilinear
formulation and B\"{a}cklund transformations, which produce -soliton
solutions. A direct and unifying scheme is proposed for explicitly building up
quasi-periodic wave solutions of the NKdV equations.
Furthermore, the relations between quasi-periodic wave solutions and soliton
solutions are clearly described. Finally, we show the quasi-periodic wave
solution convergent to the soliton solution under some limit conditions.Comment: 61 pages, 4 figure
Binary Bell polynomials approach to the integrability of nonisospectral and variable-coefficient nonlinear equations
Recently, Lembert, Gilson et al proposed a lucid and systematic approach to
obtain bilinear B\"{a}cklund transformations and Lax pairs for
constant-coefficient soliton equations based on the use of binary Bell
polynomials. In this paper, we would like to further develop this method with
new applications. We extend this method to systematically investigate complete
integrability of nonisospectral and variable-coefficient equations. In
addiction, a method is described for deriving infinite conservation laws of
nonlinear evolution equations based on the use of binary Bell polynomials. All
conserved density and flux are given by explicit recursion formulas. By taking
variable-coefficient KdV and KP equations as illustrative examples, their
bilinear formulism, bilinear B\"{a}cklund transformations, Lax pairs, Darboux
covariant Lax pairs and conservation laws are obtained in a quick and natural
manner. In conclusion, though the coefficient functions have influences on a
variable-coefficient nonlinear equation, under certain constrains the equation
turn out to be also completely integrable, which leads us to a canonical
interpretation of their -soliton solutions in theory.Comment: 39 page
Painlev\'e analysis for nonlinear partial differential equations
The Painlev\'e analysis introduced by Weiss, Tabor and Carnevale (WTC) in
1983 for nonlinear partial differential equations (PDE's) is an extension of
the method initiated by Painlev\'e and Gambier at the beginning of this century
for the classification of algebraic nonlinear differential equations (ODE's)
without movable critical points. In these lectures we explain the WTC method in
its invariant version introduced by Conte in 1989 and its application to
solitonic equations in order to find algorithmically their associated
B\"acklund transformation. A lot of remarkable properties are shared by these
so-called ``integrable'' equations but they are generically no more valid for
equations modelising physical phenomema. Belonging to this second class, some
equations called ``partially integrable'' sometimes keep remnants of
integrability. In that case, the singularity analysis may also be useful for
building closed form analytic solutions, which necessarily % Conte agree with
the singularity structure of the equations. We display the privileged role
played by the Riccati equation and systems of Riccati equations which are
linearisable, as well as the importance of the Weierstrass elliptic function,
for building solitary waves or more elaborate solutions.Comment: 61 pages, no figure, standard Latex, to appear in The Painlev\'e
property, one century later, ed. R. Conte, CRM series in mathematical physics
(Springer--Verlag, Berlin, 1998) (Carg\`ese school, 3-22 June 1996
Equivalence transformations in the study of integrability
We discuss how point transformations can be used for the study of
integrability, in particular, for deriving classes of integrable
variable-coefficient differential equations. The procedure of finding the
equivalence groupoid of a class of differential equations is described and then
specified for the case of evolution equations. A class of fifth-order
variable-coefficient KdV-like equations is studied within the framework
suggested.Comment: 14 pages; the version accepted to Physica Script
Open problems for the superKdV equations
After a review of the basic results concerning the supersymmetric
extensions of the Korteweg-de Vries equation, with a pedagogical presentation
of the superspace techniques, we discuss some basic open problems mainly in
relation with the N=2 extensions.Comment: harvmac, 13 p. (b); talk presented at AARMS-CRM Workshop on Backlund
and Darboux Transformations: The geometry of Soliton Theory. June 4-9, 1999
(Halifax, Nova Scotia
Generalized Super Bell Polynomials with Applications to Superymmetric Equations
In this paper, we introduce a class of new generalized super Bell polynomials
on a superspace, explore their properties, and show that they are a natural and
effective tool to systematically investigate integrability of supersymmetric
equations. The connections between the super Bell polynomials and super
bilinear representation, bilinear B\"{a}cklund transformation, Lax pair and
infinite conservation laws of supersymmetric equations are established. We take
supersymmetric KdV equation and supersymmetric sine-Gordon equation to
illustrate this procedure.Comment: 36 page
On a method for constructing the Lax pairs for nonlinear integrable equations
We suggest a direct algorithm for searching the Lax pairs for nonlinear
integrable equations. It is effective for both continuous and discrete models.
The first operator of the Lax pair corresponding to a given nonlinear equation
is found immediately, coinciding with the linearization of the considered
nonlinear equation. The second one is obtained as an invariant manifold to the
linearized equation. A surprisingly simple relation between the second operator
of the Lax pair and the recursion operator is discussed: the recursion operator
can immediately be found from the Lax pair. Examples considered in the article
are convincing evidence that the found Lax pairs differ from the classical
ones. The examples also show that the suggested objects are true Lax pairs
which allow the construction of infinite series of conservation laws and
hierarchies of higher symmetries. In the case of the hyperbolic type partial
differential equation our algorithm is slightly modified; in order to construct
the Lax pairs from the invariant manifolds we use the cutting off conditions
for the corresponding infinite Laplace sequence. The efficiency of the method
is illustrated by application to some equations given in the Svinolupov-Sokolov
classification list for which the Lax pairs and the recursion operators have
not been found earlier
Discrete KP equation with self-consistent sources
We show that the discrete Kadomtsev-Petviashvili (KP) equation with sources
obtained recently by the "source generalization" method can be incorporated
into the squared eigenfunction symmetry extension procedure. Moreover, using
the known correspondence between Darboux-type transformations and additional
independent variables, we demonstrate that the equation with sources can be
derived from Hirota's discrete KP equations but in a space of higher dimension.
In this way we uncover the origin of the source terms as coming from
multidimensional consistency of the Hirota system itself.Comment: 11 pages; one reference added, several typos or grammatical errors
corrected (v2
The method of Poisson pairs in the theory of nonlinear PDEs
The aim of these lectures is to show that the methods of classical
Hamiltonian mechanics can be profitably used to solve certain classes of
nonlinear partial differential equations. The prototype of these equations is
the well-known Korteweg-de Vries (KdV) equation. In these lectures we touch the
following subjects: i) The birth and the role of the method of Poisson pairs
inside the theory of the KdV equation; ii) the theoretical basis of the method
of Poisson pairs; iii) the Gel'fand-Zakharevich theory of integrable systems on
bihamiltonian manifolds; iv) the Hamiltonian interpretation of the Sato picture
of the KdV flows and of its linearization on an infinite-dimensional
Grassmannian manifold. v)the reduction technique and its use to construct
classes of solutions; iv) the role of the technique of separation of variables
in the study of the reduced systems. vii) some relations intertwining the
method of Poisson pairs with the method of Lax pairs.Comment: 55 pages, Latex with amsmath and amssymb. Lectures given by F. Magri
at the 1999 CIME course ``Direct and Inverse Methods in Solving Nonlinear
Evolution Equations '' Cetraro, (Italy) September 199
Deriving conservation laws for ABS lattice equations from Lax pairs
In the paper we derive infinitely many conservation laws for the ABS lattice
equations from their Lax pairs. These conservation laws can algebraically be
expressed by means of some known polynomials. We also show that H1, H2, H3, Q1,
Q2, Q3 and A1 equation in ABS list share a generic discrete Riccati equation.Comment: 16 page
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