270 research outputs found
A numerical method for computing time-periodic solutions in dissipative wave systems
A numerical method is proposed for computing time-periodic and relative
time-periodic solutions in dissipative wave systems. In such solutions, the
temporal period, and possibly other additional internal parameters such as the
propagation constant, are unknown priori and need to be determined along with
the solution itself. The main idea of the method is to first express those
unknown parameters in terms of the solution through quasi-Rayleigh quotients,
so that the resulting integro-differential equation is for the time-periodic
solution only. Then this equation is computed in the combined spatiotemporal
domain as a boundary value problem by Newton-conjugate-gradient iterations. The
proposed method applies to both stable and unstable time-periodic solutions;
its numerical accuracy is spectral; it is fast-converging; and its coding is
short and simple. As numerical examples, this method is applied to the
Kuramoto-Sivashinsky equation and the cubic-quintic Ginzburg-Landau equation,
whose time-periodic or relative time-periodic solutions with spatially-periodic
or spatially-localized profiles are computed. This method also applies to
systems of ordinary differential equations, as is illustrated by its simple
computation of periodic orbits in the Lorenz equations. MATLAB codes for all
numerical examples are provided in appendices to illustrate the simple
implementation of the proposed method.Comment: 24 pages, 5 figure
General -solitons and their dynamics in several nonlocal nonlinear Schr\"odinger equations
General -solitons in three recently-proposed nonlocal nonlinear
Schr\"odinger equations are presented. These nonlocal equations include the
reverse-space, reverse-time, and reverse-space-time nonlinear Schr\"odinger
equations, which are nonlocal reductions of the Ablowitz-Kaup-Newell-Segur
(AKNS) hierarchy. It is shown that general -solitons in these different
equations can be derived from the same Riemann-Hilbert solutions of the AKNS
hierarchy, except that symmetry relations on the scattering data are different
for these equations. This Riemann-Hilbert framework allows us to identify new
types of solitons with novel eigenvalue configurations in the spectral plane.
Dynamics of -solitons in these equations is also explored. In all the three
nonlocal equations, a generic feature of their solutions is repeated
collapsing. In addition, multi-solitons can behave very differently from
fundamental solitons and may not correspond to a nonlinear superposition of
fundamental solitons.Comment: 11 pages, 6 figure
Complete eigenfunctions of linearized integrable equations expanded around an arbitrary solution
Complete eigenfunctions of linearized integrable equations expanded around an
arbitrary solution are obtained for the Ablowitz-Kaup-Newell-Segur (AKNS)
hierarchy and the Korteweg-de Vries (KdV) hierarchy. It is shown that the
linearization operators and the recursion operator which generates the
hierarchy are commutable. Consequently, eigenfunctions of the linearization
operators are precisely squared eigenfunctions of the associated eigenvalue
problem. Similar results are obtained for the adjoint linearization operators
as well. These results make a simple connection between the direct
soliton/multi-soliton perturbation theory and the inverse-scattering based
perturbation theory for these hierarchy equations
Multiple permanent-wave trains in nonlinear systems
Multiple permanent-wave trains in nonlinear systems are constructed by the
asymptotic tail-matching method. Under some general assumptions, simple
criteria for the construction are presented. Applications to fourth-order
systems and coupled nonlinear Schr\"odinger equations are discussed
Partially-PT-symmetric optical potentials with all-real spectra and soliton families in multi-dimensions
Multi-dimensional complex optical potentials with partial parity-time (PT)
symmetry are proposed. The usual PT symmetry requires that the potential is
invariant under complex conjugation and simultaneous reflection in all spatial
directions. However, we show that if the potential is only partially
PT-symmetric, i.e., it is invariant under complex conjugation and reflection in
a single spatial direction, then it can also possess all-real spectra and
continuous families of solitons. These results are established analytically and
corroborated numerically.Comment: 4 pages, 3 figure
New classes of non-parity-time-symmetric optical potentials with all-real spectra and exceptional-point-free phase transition
Paraxial linear propagation of light in an optical waveguide with material
gain and loss is governed by a Schr\"odinger equation with a complex potential.
Properties of parity-time-symmetric complex potentials have been heavily
studied before. In this article, new classes of non-parity-time-symmetric
complex potentials featuring conjugate-pair eigenvalue symmetry in its spectrum
are constructed by operator symmetry methods. Due to this eigenvalue symmetry,
it is shown that the spectrum of these complex potentials is often all-real.
Under parameter tuning in these potentials, phase transition can also occur,
where pairs of complex eigenvalues appear in the spectrum. A peculiar feature
of the phase transition here is that, the complex eigenvalues may bifurcate out
from an interior continuous eigenvalue inside the continuous spectrum, in which
case a phase transition takes place without going through an exceptional point.
In one spatial dimension, this class of non-parity-time-symmetric complex
potentials is of the form , where is an arbitrary
parity-time-symmetric complex function. These potentials in two spatial
dimensions are also derived. Diffraction patterns in these complex potentials
are further examined, and unidirectional propagation behaviors are
demonstrated.Comment: 5 pages, 3 figure
Nonlinear behaviors of parity-time-symmetric lasers
We propose a time-dependent partial differential equation model to
investigate the dynamical behavior of the parity-time (PT) symmetric laser
during the nonlinear stage of its operation. This model incorporates physical
effects such as the refractive index distribution, dispersion, material loss,
nonlinear gain saturation and self-phase modulation. We show that when the loss
is weak, multiple stable steady states and time-periodic states of light exist
above the lasing threshold, rendering the laser multi-mode. However, when the
loss is strong, only a single stable steady state of broken PT symmetry exists
for a wide range of the gain amplitude, rendering the laser single-mode. These
results reveal the important role the loss plays in maintaining the single-mode
operation of PT lasers.Comment: 5 pages, 7 figure
Necessity of PT symmetry for soliton families in one-dimensional complex potentials
For the one-dimensional nonlinear Schroedinger equation with a complex
potential, it is shown that if this potential is not parity-time (PT)
symmetric, then no continuous families of solitons can bifurcate out from
linear guided modes, even if the linear spectrum of this potential is all real.
Both localized and periodic non-PT-symmetric potentials are considered, and the
analytical conclusion is corroborated by explicit examples. Based on this
result, it is argued that PT-symmetry of a one-dimensional complex potential is
a necessary condition for the existence of soliton families.Comment: 14 pages, 3 figure
A new nonlocal nonlinear Schroedinger equation and its soliton solutions
A new integrable nonlocal nonlinear Schroedinger (NLS) equation with clear
physical motivations is proposed. This equation is obtained from a special
reduction of the Manakov system, and it describes Manakov solutions whose two
components are related by a parity symmetry. Since the Manakov system governs
wave propagation in a wide variety of physical systems, this new nonlocal
equation has clear physical meanings. Solitons and multi-solitons in this
nonlocal equation are also investigated in the framework of Riemann-Hilbert
formulations. Surprisingly, symmetry relations of discrete scattering data for
this equation are found to be very complicated, where constraints between
eigenvectors in the scattering data depend on the number and locations of the
underlying discrete eigenvalues in a very complex manner. As a consequence,
general -solitons are difficult to obtain in the Riemann-Hilbert framework.
However, one- and two-solitons are derived, and their dynamics investigated. It
is found that two-solitons are generally not a nonlinear superposition of
one-solitons, and they exhibit interesting dynamics such as meandering and
sudden position shifts. As a generalization, other integrable and physically
meaningful nonlocal equations are also proposed, which include NLS equations of
reverse-time and reverse-space-time types as well as nonlocal Manakov equations
of reverse-space, reverse-time and reverse-space-time types.Comment: 11 pages, 3 figure
Transformations between nonlocal and local integrable equations
Recently, a number of nonlocal integrable equations, such as the PT-symmetric
nonlinear Schrodinger (NLS) equation and PT-symmetric Davey-Stewartson
equations, were proposed and studied. Here we show that many of such nonlocal
integrable equations can be converted to local integrable equations through
simple variable transformations. Examples include these nonlocal NLS and
Davey-Stewartson equations, a nonlocal derivative NLS equation, the reverse
space-time complex modified Korteweg-de Vries (CMKdV) equation, and many
others. These transformations not only establish immediately the integrability
of these nonlocal equations, but also allow us to construct their analytical
solutions from solutions of the local equations. These transformations can also
be used to derive new nonlocal integrable equations. As applications of these
transformations, we use them to derive rogue wave solutions for the partially
PT-symmetric Davey-Stewartson equations and the nonlocal derivative NLS
equation. In addition, we use them to derive multi-soliton and quasi-periodic
solutions in the reverse space-time CMKdV equation. Furthermore, we use them to
construct many new nonlocal integrable equations such as nonlocal short pulse
equations, nonlocal nonlinear diffusion equations, and nonlocal Sasa-Satsuma
equations.Comment: 15 pages, 4 figure
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