38 research outputs found
Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions
We express AKNS hierarchies, admitting reductions to matrix NLS and matrix
mKdV hierarchies, in terms of a bidifferential graded algebra. Application of a
universal result in this framework quickly generates an infinite family of
exact solutions, including e.g. the matrix solitons in the focusing NLS case.
Exploiting a general Miura transformation, we recover the generalized
Heisenberg magnet hierarchy and establish a corresponding solution formula for
it. Simply by exchanging the roles of the two derivations of the bidifferential
graded algebra, we recover "negative flows", leading to an extension of the
respective hierarchy. In this way we also meet a matrix and vector version of
the short pulse equation and also the sine-Gordon equation. For these equations
corresponding solution formulas are also derived. In all these cases the
solutions are parametrized in terms of matrix data that have to satisfy a
certain Sylvester equation
Description of two soliton collision for the quartic gKdV equation
This paper concerns the problem of collision of two solitons for the quartic
generalized Korteweg-de Vries equation. We introduce a new framework to
describe the collision in the special case where one soliton is small with
respect to the other. We prove that the two soliton survive the collision, we
describe the collision phenomenon (computation of the first order of the
resulting shifts on the solitons). Moreover, we prove that in this situation,
there does not exist pure two-soliton solutions
Hamiltonian Structures of the Multi-Boson KP Hierarchies, Abelianization and Lattice Formulation
We present a new form of the multi-boson reduction of KP hierarchy with Lax
operator written in terms of boson fields abelianizing the second Hamiltonian
structure. This extends the classical Miura transformation and the
Kupershmidt-Wilson theorem from the (m)KdV to the KP case. A remarkable
relationship is uncovered between the higher Hamiltonian structures and the
corresponding Miura transformations of KP hierarchy, on one hand, and the
discrete integrable models living on {\em refinements} of the original lattice
connected with the underlying multi-matrix models, on the other hand. For the
second KP Hamiltonian structure, worked out in details, this amounts to finding
a series of representations of the nonlinear \hWinf algebra in terms of
arbitrary finite number of canonical pairs of free fields.Comment: 12 pgs, (changes in abstract, intro and outlook+1 ref added). LaTeX,
BGU-94 / 1 / January- PH, UICHEP-TH/94-
Wave envelopes with second-order spatiotemporal dispersion: II. Modulational instabilities and dark Kerr solitons
A simple scalar model for describing spatiotemporal dispersion of pulses, beyond the classic “slowly-varying envelopes + Galilean boost” approach, is studied. The governing equation has a cubic nonlinearity and we focus here mainly on contexts with normal group-velocity dispersion. A complete analysis of continuous waves is reported, including their dispersion relations and modulational instability characteristics. We also present a detailed derivation of exact analytical dark solitons, obtained by combining direct-integration methods with geometrical transformations. Classic results from conventional pulse theory are recovered as-ymptotically from the spatiotemporal formulation. Numerical simulations test new theoretical predictions for modulational instability, and examine the robustness of spatiotemporal dark solitons against perturbations to their local pulse shape
New Explicit Solutions for Homogeneous Kdv Equations of Third Order by Trigonometric and Hyperbolic Function Methods
In this paper, we study two-component evolutionary systems of the homogeneous KdV equation of the third order types (I) and (II). Trigonometric and hyperbolic function methods such as the sine-cosine method, the rational sine-cosine method, the rational sinh-cosh method, sech-csch method and rational tanh-coth method are used for analytical treatment of these systems. These methods, have the advantage of reducing the nonlinear problem to a system of algebraic equations that can be solved by computerized packages
On Decompositions of the KdV 2-Soliton
The KdV equation is the canonical example of an integrable non-linear partial
differential equation supporting multi-soliton solutions. Seeking to understand
the nature of this interaction, we investigate different ways to write the KdV
2-soliton solution as a sum of two or more functions. The paper reviews
previous work of this nature and introduces new decompositions with unique
features, putting it all in context and in a common notation for ease of
comparison