88 research outputs found
On asymptotically equivalent shallow water wave equations
The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at
linear order in the asymptotic expansion for unidirectional shallow water
waves. However, at quadratic order, this asymptotic expansion produces an
entire {\it family} of shallow water wave equations that are asymptotically
equivalent to each other, under a group of nonlinear, nonlocal, normal-form
transformations introduced by Kodama in combination with the application of the
Helmholtz-operator. These Kodama-Helmholtz transformations are used to present
connections between shallow water waves, the integrable 5th-order Korteweg-de
Vries equation, and a generalization of the Camassa-Holm (CH) equation that
contains an additional integrable case. The dispersion relation of the full
water wave problem and any equation in this family agree to 5th order. The
travelling wave solutions of the CH equation are shown to agree to 5th order
with the exact solution
Triplet Pairing in Neutron Matter
The separation method developed earlier by us [Nucl. Phys. {\bf A598} 390
(1996)] to calculate and analyze solutions of the BCS gap equation for
S pairing is extended and applied to P--F pairing in
pure neutron matter. The pairing matrix elements are written as a separable
part plus a remainder that vanishes when either momentum variable is on the
Fermi surface. This decomposition effects a separation of the problem of
determining the dependence of the gap components in a spin-angle representation
on the magnitude of the momentum (described by a set of functions independent
of magnetic quantum number) from the problem of determining the dependence of
the gap on angle or magnetic projection. The former problem is solved through a
set of nonsingular, quasilinear integral equations, providing inputs for
solution of the latter problem through a coupled system of algebraic equations
for a set of numerical coefficients. An incisive criterion is given for finding
the upper critical density for closure of the triplet gap. The separation
method and its development for triplet pairing exploit the existence of a small
parameter, given by a gap-amplitude measure divided by the Fermi energy. The
revised BCS equations admit analysis revealing universal properties of the full
set of solutions for P pairing in the absence of tensor coupling,
referring especially to the energy degeneracy and energetic order of these
solutions. The angle-average approximation introduced by Baldo et al. is
illuminated in terms of the separation-transformed BCS problem and the small
parameter expansion..
Sasa--Satsuma equation: Soliton on a background and its limiting cases
We present a multi-parameter family of a soliton on a background solutions to the Sasa-Satsuma equation. The solution is controlled by a set of several free parameters that control the background amplitude as well as the soliton itself. This family of solutions admits a few nontrivial limiting cases that are considered in detail. Among these special cases is the NLSE limit and the limit of rogue wave solutions
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