3,427 research outputs found
Lie symmetry analysis and group invariant solutions of the nonlinear Helmholtz equation
We consider the nonlinear Helmholtz (NLH) equation describing the beam
propagation in a planar waveguide with Kerr-like nonlinearity under
non-paraxial approximation. By applying the Lie symmetry analysis, we determine
the Lie point symmetries and the corresponding symmetry reductions in the form
of ordinary differential equations (ODEs) with the help of the optimal systems
of one-dimensional subalgebras. Our investigation reveals an important fact
that in spite of the original NLH equation being non-integrable, its symmetry
reductions are of Painlev\'e integrable. We study the resulting sets of
nonlinear ODEs analytically either by constructing the integrals of motion
using the modified Prelle-Singer method or by obtaining explicit travelling
wave-like solutions including solitary and symbiotic solitary wave solutions.
Also, we carry out a detailed numerical analysis of the reduced equations and
obtain multi-peak nonlinear wave trains. As a special case of the NLH equation,
we also make a comparison between the symmetries of the present NLH system and
that of the standard nonlinear Schr\"odinger equation for which symmetries are
long available in the literature.Comment: Accepted for publication in "Applied Mathematics and Computation". 18
pages, 15 figure
Vlasov moments, integrable systems and singular solutions
The Vlasov equation for the collisionless evolution of the single-particle
probability distribution function (PDF) is a well-known Lie-Poisson Hamiltonian
system. Remarkably, the operation of taking the moments of the Vlasov PDF
preserves the Lie-Poisson structure. The individual particle motions correspond
to singular solutions of the Vlasov equation. The paper focuses on singular
solutions of the problem of geodesic motion of the Vlasov moments. These
singular solutions recover geodesic motion of the individual particles.Comment: 16 pages, no figures. Submitted to Phys. Lett.
Momentum Maps and Measure-valued Solutions (Peakons, Filaments and Sheets) for the EPDiff Equation
We study the dynamics of measure-valued solutions of what we call the EPDiff
equations, standing for the {\it Euler-Poincar\'e equations associated with the
diffeomorphism group (of or an -dimensional manifold )}.
Our main focus will be on the case of quadratic Lagrangians; that is, on
geodesic motion on the diffeomorphism group with respect to the right invariant
Sobolev metric. The corresponding Euler-Poincar\'e (EP) equations are the
EPDiff equations, which coincide with the averaged template matching equations
(ATME) from computer vision and agree with the Camassa-Holm (CH) equations in
one dimension. The corresponding equations for the volume preserving
diffeomorphism group are the well-known LAE (Lagrangian averaged Euler)
equations for incompressible fluids. We first show that the EPDiff equations
are generated by a smooth vector field on the diffeomorphism group for
sufficiently smooth solutions. This is analogous to known results for
incompressible fluids--both the Euler equations and the LAE equations--and it
shows that for sufficiently smooth solutions, the equations are well-posed for
short time. In fact, numerical evidence suggests that, as time progresses,
these smooth solutions break up into singular solutions which, at least in one
dimension, exhibit soliton behavior. With regard to these non-smooth solutions,
we study measure-valued solutions that generalize to higher dimensions the
peakon solutions of the (CH) equation in one dimension. One of the main
purposes of this paper is to show that many of the properties of these
measure-valued solutions may be understood through the fact that their solution
ansatz is a momentum map. Some additional geometry is also pointed out, for
example, that this momentum map is one leg of a natural dual pair.Comment: 27 pages, 2 figures, To Alan Weinstein on the occasion of his 60th
Birthda
Singular solutions of a modified two-component Camassa-Holm equation
The Camassa-Holm equation (CH) is a well known integrable equation describing
the velocity dynamics of shallow water waves. This equation exhibits
spontaneous emergence of singular solutions (peakons) from smooth initial
conditions. The CH equation has been recently extended to a two-component
integrable system (CH2), which includes both velocity and density variables in
the dynamics. Although possessing peakon solutions in the velocity, the CH2
equation does not admit singular solutions in the density profile. We modify
the CH2 system to allow dependence on average density as well as pointwise
density. The modified CH2 system (MCH2) does admit peakon solutions in velocity
and average density. We analytically identify the steepening mechanism that
allows the singular solutions to emerge from smooth spatially-confined initial
data. Numerical results for MCH2 are given and compared with the pure CH2 case.
These numerics show that the modification in MCH2 to introduce average density
has little short-time effect on the emergent dynamical properties. However, an
analytical and numerical study of pairwise peakon interactions for MCH2 shows a
new asymptotic feature. Namely, besides the expected soliton scattering
behavior seen in overtaking and head-on peakon collisions, MCH2 also allows the
phase shift of the peakon collision to diverge in certain parameter regimes.Comment: 25 pages, 11 figure
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