1,259 research outputs found
On the equivalence of different approaches for generating multisoliton solutions of the KPII equation
The unexpectedly rich structure of the multisoliton solutions of the KPII
equation has been explored by using different approaches, running from dressing
method to twisting transformations and to the tau-function formulation. All
these approaches proved to be useful in order to display different properties
of these solutions and their related Jost solutions. The aim of this paper is
to establish the explicit formulae relating all these approaches. In addition
some hidden invariance properties of these multisoliton solutions are
discussed
Soliton solutions of the Kadomtsev-Petviashvili II equation
We study a general class of line-soliton solutions of the
Kadomtsev-Petviashvili II (KPII) equation by investigating the Wronskian form
of its tau-function. We show that, in addition to previously known line-soliton
solutions, this class also contains a large variety of new multi-soliton
solutions, many of which exhibit nontrivial spatial interaction patterns. We
also show that, in general, such solutions consist of unequal numbers of
incoming and outgoing line solitons. From the asymptotic analysis of the
tau-function, we explicitly characterize the incoming and outgoing
line-solitons of this class of solutions. We illustrate these results by
discussing several examples.Comment: 28 pages, 4 figure
Young diagrams and N-soliton solutions of the KP equation
We consider -soliton solutions of the KP equation,
(-4u_t+u_{xxx}+6uu_x)_x+3u_{yy}=0 . An -soliton solution is a solution
which has the same set of line soliton solutions in both
asymptotics and . The -soliton solutions include
all possible resonant interactions among those line solitons. We then classify
those -soliton solutions by defining a pair of -numbers with , which labels line solitons in the solution. The
classification is related to the Schubert decomposition of the Grassmann
manifolds Gr, where the solution of the KP equation is defined as a
torus orbit. Then the interaction pattern of -soliton solution can be
described by the pair of Young diagrams associated with . We also show that -soliton solutions of the KdV equation obtained by
the constraint cannot have resonant interaction.Comment: 22 pages, 5 figures, some minor corrections and added one section on
the KdV N-soliton solution
The dispersion-managed Ginzburg-Landau equation and its application to femtosecond lasers
The complex Ginzburg-Landau equation has been used extensively to describe
various non-equilibrium phenomena. In the context of lasers, it models the
dynamics of a pulse by averaging over the effects that take place inside the
cavity. Ti:sapphire femtosecond lasers, however, produce pulses that undergo
significant changes in different parts of the cavity during each round-trip.
The dynamics of such pulses is therefore not adequately described by an average
model that does not take such changes into account. The purpose of this work is
severalfold. First we introduce the dispersion-managed Ginzburg-Landau equation
(DMGLE) as an average model that describes the long-term dynamics of systems
characterized by rapid variations of dispersion, nonlinearity and gain in a
general setting, and we study the properties of the equation. We then explain
how in particular the DMGLE arises for Ti:sapphire femtosecond lasers and we
characterize its solutions. In particular, we show that, for moderate values of
the gain/loss parameters, the solutions of the DMGLE are well approximated by
those of the dispersion-managed nonlinear Schrodinger equation (DMNLSE), and
the main effect of gain and loss dynamics is simply to select one among the
one-parameter family of solutions of the DMNLSE.Comment: 22 pages, 4 figures, to appear in Nonlinearit
On a family of solutions of the KP equation which also satisfy the Toda lattice hierarchy
We describe the interaction pattern in the - plane for a family of
soliton solutions of the Kadomtsev-Petviashvili (KP) equation,
. Those solutions also satisfy the
finite Toda lattice hierarchy. We determine completely their asymptotic
patterns for , and we show that all the solutions (except the
one-soliton solution) are of {\it resonant} type, consisting of arbitrary
numbers of line solitons in both aymptotics; that is, arbitrary incoming
solitons for interact to form arbitrary outgoing solitons
for . We also discuss the interaction process of those solitons,
and show that the resonant interaction creates a {\it web-like} structure
having holes.Comment: 18 pages, 16 figures, submitted to JPA; Math. Ge
Noise-induced perturbations of dispersion-managed solitons
We study noise-induced perturbations of dispersion-managed solitons by
developing soliton perturbation theory for the dispersion-managed nonlinear
Schroedinger (DMNLS) equation, which governs the long-term behavior of optical
fiber transmission systems and certain kinds of femtosecond lasers. We show
that the eigenmodes and generalized eigenmodes of the linearized DMNLS equation
around traveling-wave solutions can be generated from the invariances of the
DMNLS equations, we quantify the perturbation-induced parameter changes of the
solution in terms of the eigenmodes and the adjoint eigenmodes, and we obtain
evolution equations for the solution parameters. We then apply these results to
guide importance-sampled Monte-Carlo simulations and reconstruct the
probability density functions of the solution parameters under the effect of
noise.Comment: 12 pages, 6 figure
Deconstruction of the Corso Grosseto viaduct and setup of a testing site for full scale load tests
BRIDGE|50 is a research project recently launched in Italy in the context of the Torino-Ceres construction works jointly with Politecnico di Milano, Politecnico di Torino, public authorities and private companies. The aim of the BRIDGE|50 research project is to investigate the residual structural performance of the Corso Grosseto 50-year-old prestressed concrete bridge through an experimental campaign. The dismantling and demolition procedures of Corso Grosseto viaduct are presented in this paper, including the setup of the field laboratory where several deck beams and pier caps will be tested up to collapse
Building extended resolvent of heat operator via twisting transformations
Twisting transformations for the heat operator are introduced. They are used,
at the same time, to superimpose a` la Darboux N solitons to a generic smooth,
decaying at infinity, potential and to generate the corresponding Jost
solutions. These twisting operators are also used to study the existence of the
related extended resolvent. Existence and uniqueness of the extended resolvent
in the case of solitons with N "ingoing" rays and one "outgoing" ray is
studied in details.Comment: 15 pages, 2 figure
Initial-boundary value problems for discrete evolution equations: discrete linear Schrodinger and integrable discrete nonlinear Schrodinger equations
We present a method to solve initial-boundary value problems for linear and
integrable nonlinear differential-difference evolution equations. The method is
the discrete version of the one developed by A. S. Fokas to solve
initial-boundary value problems for linear and integrable nonlinear partial
differential equations via an extension of the inverse scattering transform.
The method takes advantage of the Lax pair formulation for both linear and
nonlinear equations, and is based on the simultaneous spectral analysis of both
parts of the Lax pair. A key role is also played by the global algebraic
relation that couples all known and unknown boundary values. Even though
additional technical complications arise in discrete problems compared to
continuum ones, we show that a similar approach can also solve initial-boundary
value problems for linear and integrable nonlinear differential-difference
equations. We demonstrate the method by solving initial-boundary value problems
for the discrete analogue of both the linear and the nonlinear Schrodinger
equations, comparing the solution to those of the corresponding continuum
problems. In the linear case we also explicitly discuss Robin-type boundary
conditions not solvable by Fourier series. In the nonlinear case we also
identify the linearizable boundary conditions, we discuss the elimination of
the unknown boundary datum, we obtain explicitly the linear and continuum limit
of the solution, and we write down the soliton solutions.Comment: 41 pages, 3 figures, to appear in Inverse Problem
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