25 research outputs found
Numerical study of a multiscale expansion of the Korteweg de Vries equation and Painlev\'e-II equation
The Cauchy problem for the Korteweg de Vries (KdV) equation with small
dispersion of order \e^2, \e\ll 1, is characterized by the appearance of a
zone of rapid modulated oscillations. These oscillations are approximately
described by the elliptic solution of KdV where the amplitude, wave-number and
frequency are not constant but evolve according to the Whitham equations.
Whereas the difference between the KdV and the asymptotic solution decreases as
in the interior of the Whitham oscillatory zone, it is known to be
only of order near the leading edge of this zone. To obtain a
more accurate description near the leading edge of the oscillatory zone we
present a multiscale expansion of the solution of KdV in terms of the
Hastings-McLeod solution of the Painlev\'e-II equation. We show numerically
that the resulting multiscale solution approximates the KdV solution, in the
small dispersion limit, to the order .Comment: 20 pages, 14 figure
Multiple Factorizations of Bivariate Linear Partial Differential Operators
We study the case when a bivariate Linear Partial Differential Operator
(LPDO) of orders three or four has several different factorizations.
We prove that a third-order bivariate LPDO has a first-order left and right
factors such that their symbols are co-prime if and only if the operator has a
factorization into three factors, the left one of which is exactly the initial
left factor and the right one is exactly the initial right factor. We show that
the condition that the symbols of the initial left and right factors are
co-prime is essential, and that the analogous statement "as it is" is not true
for LPDOs of order four.
Then we consider completely reducible LPDOs, which are defined as an
intersection of principal ideals. Such operators may also be required to have
several different factorizations. Considering all possible cases, we ruled out
some of them from the consideration due to the first result of the paper. The
explicit formulae for the sufficient conditions for the complete reducibility
of an LPDO were found also
Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit
We study the small dispersion limit for the Korteweg-de Vries (KdV) equation
in a critical scaling regime where
approaches the trailing edge of the region where the KdV solution shows
oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an
asymptotic expansion for the KdV solution in a double scaling limit, which
shows that the oscillations degenerate to sharp pulses near the trailing edge.
Locally those pulses resemble soliton solutions of the KdV equation.Comment: 25 pages, 4 figure
On a class of three-dimensional integrable Lagrangians
We characterize non-degenerate Lagrangians of the form
Z f(ux, uy, ut) dx dy dt
such that the corresponding Euler-Lagrange equations (fux )x+(fuy )y+(fut )t = 0 are integrable by
the method of hydrodynamic reductions. The integrability conditions constitute an over-determined
system of fourth order PDEs for the Lagrangian density f, which is in involution. The moduli
space of integrable Lagrangians, factorized by the action of a natural equivalence group, is threedimensional.
Familiar examples include the dispersionless Kadomtsev-Petviashvili (dKP) and the
Boyer-Finley Lagrangians, f = u3
x/3 + u2
y − uxut and f = u2
x + u2
y − 2eut , respectively. A complete
description of integrable cubic and quartic Lagrangians is obtained. Up to the equivalence
transformations, the list of integrable cubic Lagrangians reduces to three examples,
f = uxuyut, f = u2
xuy + uyut and f = u3
x/3 + u2
y − uxut (dKP).
There exists a unique integrable quartic Lagrangian,
f = u4
x + 2u2
xut − uxuy − u2t
.
We conjecture that these examples exhaust the list of integrable polynomial Lagrangians which are
essentially three-dimensional (it was verified that there exist no polynomial integrable Lagrangians
of degree five).
We prove that the Euler-Lagrange equations are integrable by the method of hydrodynamic
reductions if and only if they possess a scalar pseudopotential playing the role of a dispersionless
‘Lax pair’
Intertwining Laplace Transformations of Linear Partial Differential Equations
We propose a generalization of Laplace transformations to the case of linear
partial differential operators (LPDOs) of arbitrary order in R^n. Practically
all previously proposed differential transformations of LPDOs are particular
cases of this transformation (intertwining Laplace transformation, ILT). We
give a complete algorithm of construction of ILT and describe the classes of
operators in R^n suitable for this transformation.
Keywords: Integration of linear partial differential equations, Laplace
transformation, differential transformationComment: LaTeX, 25 pages v2: minor misprints correcte
Integrable Systems and Metrics of Constant Curvature
In this article we present a Lagrangian representation for evolutionary
systems with a Hamiltonian structure determined by a differential-geometric
Poisson bracket of the first order associated with metrics of constant
curvature. Kaup-Boussinesq system has three local Hamiltonian structures and
one nonlocal Hamiltonian structure associated with metric of constant
curvature. Darboux theorem (reducing Hamiltonian structures to canonical form
''d/dx'' by differential substitutions and reciprocal transformations) for
these Hamiltonian structures is proved
On the local systems Hamiltonian in the weakly nonlocal Poisson brackets
We study in this work the important class of nonlocal Poisson Brackets (PB)
which we call weakly nonlocal. They appeared recently in some investigations in
the Soliton Theory. However there was no theory of such brackets except very
special first order case. Even in this case the theory was not developed
enough. In particular, we introduce the Physical forms and find Casimirs,
Momentum and Canonical forms for the most important Hydrodynamic type PB of
that kind and their dependence on the boundary conditions.Comment: 45 pages, late
Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions
We investigate multi-dimensional Hamiltonian systems associated with constant
Poisson brackets of hydrodynamic type. A complete list of two- and
three-component integrable Hamiltonians is obtained. All our examples possess
dispersionless Lax pairs and an infinity of hydrodynamic reductions.Comment: 34 page
Non-homogeneous systems of hydrodynamic type possessing Lax representations
We consider 1+1 - dimensional non-homogeneous systems of hydrodynamic type
that possess Lax representations with movable singularities. We present a
construction, which provides a wide class of examples of such systems with
arbitrary number of components. In the two-component case a classification is
given.Comment: 22 pages, latex, minor change