412 research outputs found
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
KdV soliton interactions: a tropical view
Via a "tropical limit" (Maslov dequantization), Korteweg-deVries (KdV)
solitons correspond to piecewise linear graphs in two-dimensional space-time.
We explore this limit.Comment: 10 pages, 4 figures, conference "Physics and Mathematics of Nonlinear
Phenomena 2013
On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions
In this paper, we complement recent results of Bronski and Johnson and of
Johnson and Zumbrun concerning the modulational stability of spatially periodic
traveling wave solutions of the generalized Korteweg-de Vries equation. In this
previous work it was shown by rigorous Evans function calculations that the
formal slow modulation approximation resulting in the Whitham system accurately
describes the spectral stability to long wavelength perturbations. Here, we
reproduce this result without reference to the Evans function by using direct
Bloch-expansion methods and spectral perturbation analysis. This approach has
the advantage of applying also in the more general multi-periodic setting where
no conveniently computable Evans function is yet devised. In particular, we
complement the picture of modulational stability described by Bronski and
Johnson by analyzing the projectors onto the total eigenspace bifurcating from
the origin in a neighborhood of the origin and zero Floquet parameter. We show
the resulting linear system is equivalent, to leading order and up to
conjugation, to the Whitham system and that, consequently, the characteristic
polynomial of this system agrees (to leading order) with the linearized
dispersion relation derived through Evans function calculation.Comment: 19 pages
Time–space integrable decompositions of nonlinear evolution equations
AbstractSeparation of the time and space variables of evolution equations is analyzed, without using any structure associated with evolution equations. The resulting theory provides techniques for constructing time–space integrable decompositions of evolution equations, which transform an evolution equation into two compatible Liouville integrable ordinary differential equations in the time and space variables. The techniques are applied to the KdV, MKdV and diffusion equations, thereby yielding several new time–space integrable decompositions of these equations
Generalized Miura Transformations, Two-Boson KP Hierarchies and their Reduction to KDV Hierarchies
Bracket preserving gauge equivalence is established between several two-boson
generated KP type of hierarchies. These KP hierarchies reduce under symplectic
reduction (via Dirac constraints) to KdV, mKdV and Schwarzian KdV hierarchies.
Under this reduction the gauge equivalence is taking form of the conventional
Miura maps between the above KdV type of hierarchies.Comment: 12 pgs., LaTeX, IFT-P/011/93, UICHEP-TH/93-
Semi-stability of embedded solitons in the general fifth-order KdV equation
Evolution of perturbed embedded solitons in the general Hamiltonian
fifth-order Korteweg--de Vries (KdV) equation is studied. When an embedded
soliton is perturbed, it sheds a one-directional continuous-wave radiation. It
is shown that the radiation amplitude is not minimal in general. A dynamical
equation for velocity of the perturbed embedded soliton is derived. This
equation shows that a neutrally stable embedded soliton is in fact semi-stable.
When the perturbation increases the momentum of the embedded soliton, the
perturbed state approaches asymptotically the embedded soliton, while when the
perturbation reduces the momentum of the embedded soliton, the perturbed state
decays into radiation. Classes of initial conditions to induce soliton decay or
persistence are also determined. Our analytical results are confirmed by direct
numerical simulations of the fifth-order KdV equation
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