14,234 research outputs found
Nonlinear waves and related nonintegrable and integrable systems
Spectral method related to Lame equation with finite-gap potential is used to
study the optical cascading equations. These equations are known not to be
integrable by inverse scattering method. Due to "partial integrability" two-gap
solutions are obtained in terms of products of elliptic functions and are
classified in five different families related to eigenvalues of appropriate
spectral problem. In special cases, when periodic solutions reduce to localized
solitary waves, previously known phase-locked solutions are recovered, and
additional one solution is obtained. For vector nonlinear Schrodinger equation
n=3 we present exact solutions in a form of multicomponent cnoidal waves.Comment: 15p., No fig, In: Prof. G. Manev's Legacy in Contemporary Aspects of
Astronomy, Gravitational and Theoretical Physics", Eds.: V. Gerdjikov and M.
Tsvetkov, Heron Press Ltd, Sofia, 2005. pp. 291-30
A multigrid method for elliptic grid generation using compact schemes
Traditional iterative methods are stalling numerical processes, in which the error has relatively small changes from one iteration to the next. Multigrid methods overcome the limitations of iterative methods and are computationally efficient. Convergence of iterative methods for elliptic partial differential equations is extremely slow. In particular, the convergence of the non-linear elliptic Poisson grid generation equations used for elliptic grid generation is very slow. Multigrid methods are fast converging methods when applied to elliptic partial differential equations. In this dissertation, a non-linear multigrid algorithm is used to accelerate the convergence of the non-linear elliptic Poisson grid generation method. The non-linear multigrid algorithm alters the performance characteristics of the non-linear elliptic Poisson grid generation method making it robust and fast in convergence. The elliptic grid generation method is based on the use of a composite mapping. It consists of a nonlinear transfinite algebraic transformation and an elliptic transformation. The composite mapping is a differentiable one-to-one mapping from the computational space onto the domains. Compact finite difference schemes are used for the discretization of the grid generation equations. Compared to traditional schemes, compact finite difference schemes provide better representation of shorter length scales and this feature brings them closer to spectral methods
Immersed Boundary Smooth Extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods
The Immersed Boundary method is a simple, efficient, and robust numerical
scheme for solving PDE in general domains, yet it only achieves first-order
spatial accuracy near embedded boundaries. In this paper, we introduce a new
high-order numerical method which we call the Immersed Boundary Smooth
Extension (IBSE) method. The IBSE method achieves high-order accuracy by
smoothly extending the unknown solution of the PDE from a given smooth domain
to a larger computational domain, enabling the use of simple Cartesian-grid
discretizations (e.g. Fourier spectral methods). The method preserves much of
the flexibility and robustness of the original IB method. In particular, it
requires minimal geometric information to describe the boundary and relies only
on convolution with regularized delta-functions to communicate information
between the computational grid and the boundary. We present a fast algorithm
for solving elliptic equations, which forms the basis for simple, high-order
implicit-time methods for parabolic PDE and implicit-explicit methods for
related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat,
Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise
convergence for Dirichlet problems and third-order pointwise convergence for
Neumann problems
A multidomain spectral method for solving elliptic equations
We present a new solver for coupled nonlinear elliptic partial differential
equations (PDEs). The solver is based on pseudo-spectral collocation with
domain decomposition and can handle one- to three-dimensional problems. It has
three distinct features. First, the combined problem of solving the PDE,
satisfying the boundary conditions, and matching between different subdomains
is cast into one set of equations readily accessible to standard linear and
nonlinear solvers. Second, touching as well as overlapping subdomains are
supported; both rectangular blocks with Chebyshev basis functions as well as
spherical shells with an expansion in spherical harmonics are implemented.
Third, the code is very flexible: The domain decomposition as well as the
distribution of collocation points in each domain can be chosen at run time,
and the solver is easily adaptable to new PDEs. The code has been used to solve
the equations of the initial value problem of general relativity and should be
useful in many other problems. We compare the new method to finite difference
codes and find it superior in both runtime and accuracy, at least for the
smooth problems considered here.Comment: 31 pages, 8 figure
On a periodic solution of the focusing nonlinear Schr\"odinger equation
A periodic two-phase algebro-geometric solution of the focusing nonlinear
Schr\"odinger equation is constructed in terms of elliptic Jacobi
theta-functions. A dependence of this solution on the parameters of a spectral
curve is investigated. An existence of a real smooth finite-gap solution of NLS
equation with complex initial phase is proven. Degenerations of the constructed
solution to one-phase traveling wave solution and solutions in the form of the
plane waves are carried.Comment: 24 pages, 9 figure
Numerical study of the transverse stability of NLS soliton solutions in several classes of NLS type equations
Dispersive PDEs are important both in applications (wave phenomena e.g. in
hy- drodynamics, nonlinear optics, plasma physics, Bose-Einstein
condensates,...) and a mathematically very challenging class of partial
differential equations, especially in the time dependent case. An important
point with respect to applications is the stability of exact solutions like
solitons. Whereas the linear or spectral stability can be addressed
analytically in some situations, the proof of full nonlinear (in-)stability
remains mostly an open question. In this paper, we numerically investi- gate
the transverse (in-)stability of the solitonic solution to the one-dimensional
cubic NLS equation, the well known isolated soliton, under the time evolution
of several higher dimensional models, being admissible as a tranverse
perturbation of the 1d cubic NLS. One of the recent work in this context [42]
allowed to prove the instability of the soliton, under the flow of the
classical (elliptic) 2d cubic NLS equation, for both localized or periodic
perturbations. The characteristics of this instability stay however unknown. Is
there a blow-up, dispersion..? We first illustrate how this instability occurs
for the elliptic 2d cubic NLS equation and then show that the elliptic-elliptic
Davey Stewartson system (a (2+1)-dimensional generalization of the cubic NLS
equation) behaves as the former in this context. Then we investigate hyperbolic
variants of the above models, for which no theory in this context is available.
Namely we consider the hyperbolic 2d cubic NLS equation and the
Davey-Stewartson II equations. For localized perturbations, the isolated
soliton appears to be unstable for the former case, but seems to be orbitally
stable for the latter. For periodic perturbations the soliton is found to be
unstable for all transversally perturbed models considered.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1401.474
On the variational structure of breather solutions
In this paper we give a systematic and simple account that put in evidence
that many breather solutions of integrable equations satisfy suitable
variational elliptic equations, which also implies that the stability problem
reduces in some sense to the study of the spectrum of explicit linear
systems (\emph{spectral stability}), and the understanding of how bad
directions (if any) can be controlled using low regularity conservation laws.
We exemplify this idea in the case of the modified Korteweg-de Vries (mKdV),
Gardner, and sine-Gordon (SG) equations. Then we perform numerical simulations
that confirm, at the level of the spectral problem, our previous rigorous
results, where we showed that mKdV breathers are and stable,
respectively. In a second step, we also discuss the Gardner and the Sine-Gordon
cases, where the spectral study of a fourth-order linear matrix system is the
key element to show stability. Using numerical methods, we confirm that all
spectral assumptions leading to the stability of SG breathers
are numerically satisfied, even in the ultra-relativistic, singular regime. In
a second part, we study the periodic mKdV case, where a periodic breather is
known from the work of Kevrekidis et al. We rigorously show that these
breathers satisfy a suitable elliptic equation, and we also show numerical
spectral stability. However, we also identify the source of nonlinear
instability in the case described in Kevrekidis et al. Finally, we present a
new class of breather solution for mKdV, believed to exist from geometric
considerations, and which is periodic in time and space, but has nonzero mean,
unlike standard breathers.Comment: 55 pages; This paper is an improved version of our previous paper
1309.0625 and hence we replace i
Spectral Methods for Numerical Relativity. The Initial Data Problem
Numerical relativity has traditionally been pursued via finite differencing.
Here we explore pseudospectral collocation (PSC) as an alternative to finite
differencing, focusing particularly on the solution of the Hamiltonian
constraint (an elliptic partial differential equation) for a black hole
spacetime with angular momentum and for a black hole spacetime superposed with
gravitational radiation. In PSC, an approximate solution, generally expressed
as a sum over a set of orthogonal basis functions (e.g., Chebyshev
polynomials), is substituted into the exact system of equations and the
residual minimized. For systems with analytic solutions the approximate
solutions converge upon the exact solution exponentially as the number of basis
functions is increased. Consequently, PSC has a high computational efficiency:
for solutions of even modest accuracy we find that PSC is substantially more
efficient, as measured by either execution time or memory required, than finite
differencing; furthermore, these savings increase rapidly with increasing
accuracy. The solution provided by PSC is an analytic function given
everywhere; consequently, no interpolation operators need to be defined to
determine the function values at intermediate points and no special
arrangements need to be made to evaluate the solution or its derivatives on the
boundaries. Since the practice of numerical relativity by finite differencing
has been, and continues to be, hampered by both high computational resource
demands and the difficulty of formulating acceptable finite difference
alternatives to the analytic boundary conditions, PSC should be further pursued
as an alternative way of formulating the computational problem of finding
numerical solutions to the field equations of general relativity.Comment: 15 pages, 5 figures, revtex, submitted to PR
Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations
This paper concerns with numerical approximations of solutions of second
order fully nonlinear partial differential equations (PDEs). A new notion of
weak solutions, called moment solutions, is introduced for second order fully
nonlinear PDEs. Unlike viscosity solutions, moment solutions are defined by a
constructive method, called vanishing moment method, hence, they can be readily
computed by existing numerical methods such as finite difference, finite
element, spectral Galerkin, and discontinuous Galerkin methods with
"guaranteed" convergence. The main idea of the proposed vanishing moment method
is to approximate a second order fully nonlinear PDE by a higher order, in
particular, a fourth order quasilinear PDE. We show by various numerical
experiments the viability of the proposed vanishing moment method. All our
numerical experiments show the convergence of the vanishing moment method, and
they also show that moment solutions coincide with viscosity solutions whenever
the latter exist.Comment: 24 pages and 30 figure
Rogue periodic waves of the mKdV equation
Traveling periodic waves of the modified Korteweg-de Vries (mKdV) equation
are considered in the focusing case. By using one-fold and two-fold Darboux
transformations, we construct explicitly the rogue periodic waves of the mKdV
equation expressed by the Jacobian elliptic functions dn and cn respectively.
The rogue dn-periodic wave describes propagation of an algebraically decaying
soliton over the dn-periodic wave, the latter wave is modulationally stable
with respect to long-wave perturbations. The rogue cn-periodic wave represents
the outcome of the modulation instability of the cn-periodic wave with respect
to long-wave perturbations and serves for the same purpose as the rogue wave of
the nonlinear Schrodinger equation (NLS), where it is expressed by the rational
function. We compute the magnification factor for the cn-periodic wave of the
mKdV equation and show that it remains the same as in the small-amplitude NLS
limit for all amplitudes. As a by-product of our work, we find explicit
expressions for the periodic eigenfunctions of the AKNS spectral problem
associated with the dn- and cn-periodic waves of the mKdV equation.Comment: 24 pages, 3 figure
- …