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Siu A. Chin

Publications and source records attributed to Siu A. Chin.

At least 37 records · Page 2Linked to original sources
arXiv2016

Peak-height formula for higher-order breathers of the nonlinear Schrödinger equation on non-uniform backgrounds

Given any background (or seed) solution of the nonlinear Schrödinger equation, the Darboux transformation can be used to generate higher-order breathers with much greater peak intensities. The Darboux transformation is generic in iterating a pair of generating solutions of the Lax-pair equation which preserves the solution of the nonlinear Schrödinger equation as a consistency condition. The Darboux transformation itself knows nothing about the background solution except through the initial pair of Lax solutions. Because of this, in this work, we can prove in an unified manner, and without knowing the analytical form of the background solution, that the peak-height of a high-order breather, is just a sum of peak-heights of first-order breathers plus that of the background, {\it irrespective} of the specific choice of the background. Detailed results are verified for breathers on a cnoidal background. Generalizations to more extended nonlinear Schrödinger equations are also indicated.

nlin.PS↗
arXiv2016

Maximal intensity higher-order Akhmediev breathers of the nonlinear Schrodinger equation and their systematic generation

It is well known that Akhmediev breathers of the nonlinear cubic Schrodinger equation can be superposed nonlinearly via the Darboux transformation to yield breathers of higher order. Surprisingly, we find that the peak height of each Akhmediev breather only adds {\it linearly} to form the peak height of the final breather. Using this new peak-height formula, we show that at any given periodicity, there exist a unique high-order breather of maximal intensity. Moreover, these high-order breathers form a continuous hierarchy, growing in intensity with increasing periodicity. For any such higher-order breather, a simple initial wave function can be extracted from the Darboux transformation to dynamically generate that breather from the nonlinear Schrodinger equation.

nlin.PS↗
arXiv2015

Anatomy of the Akhmediev breather: cascading instability, first formation time and Fermi-Pasta-Ulam recurrence

By invoking Bogoliubov's spectrum, we show that for the nonlinear Schrodinger equation, the modulation instability (MI) of its n = 1 Fourier mode on a finite background automatically triggers a further cascading instability, forcing all the higher modes to grow exponentially in locked-step with the n = 1 mode. This fundamental insight, the enslavement of all higher modes to the n = 1 mode, explains the formation of a triangular-shaped spectrum which generates the Akhmediev breather, predicts its formation time analytically from the initial modulation amplitude, and shows that the Fermi-Pasta-Ulam (FPU) recurrence is just a matter of energy conservation with a period twice the breather's formation time. For higher order MI with more than one initial unstable modes, while most evolutions are expected to be chaotic, we show that it is possible to have isolated cases of "super-recurrence", where the FPU period is much longer than that of a single unstable mode.

physics.optics↗
arXiv2015

High-order Path Integral Monte Carlo methods for solving quantum dot problems

The conventional second-order Path Integral Monte Carlo method is plagued with the sign problem in solving many-fermion systems. This is due to the large number of anti-symmetric free fermion propagators that are needed to extract the ground state wave function at large imaginary time. In this work, we show that optimized fourth-order Path Integral Monte Carlo methods, which use no more than 5 free-fermion propagators, can yield accurate quantum dot energies for up to 20 polarized electrons with the use of the Hamiltonian energy estimator.

physics.comp-ph↗
arXiv2013

A theory of explicit finite-difference schemes

Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as systematic ways of matching up to the operator solution of the partial differential equation. By completely abandon the idea of approximating derivatives directly, the theory provides a unified description of explicit finite-difference schemes for solving a general linear partial differential equation with constant coefficients to any time-marching order. As a result, the stability of the first-order algorithm for an entire class of linear equations can be determined all at once. Because the method is based on solution-matching, it can also be used to derive any order schemes for solving the general nonlinear advection equation.

math.NA↗
arXiv2012

Symplectic finite-difference methods for solving partial differential equations

The usual explicit finite-difference method of solving partial differential equations is limited in stability because it approximates the exact amplification factor by power-series. By adapting the same exponential-splitting method of deriving symplectic integrators, explicit symplectic finite-difference methods produce Saul'yev-type schemes which approximate the exact amplification factor by rational-functions. As with conventional symplectic integrators, these symplectic finite-difference algorithms preserve important qualitative features of the exact solution. Thus the symplectic diffusing algorithm is {\it unconditionally stable} and the symplectic advection algorithm is {\it unitary}. There is a one-to-one correspondence between symplectic integrators and symplectic finite-difference methods, including the key idea that one can systematically improve an algorithm by matching its modified Hamiltonian more closely to the original Hamiltonian. Consequently, the entire arsenal of symplectic integrators can be used to produce arbitrary high order time-marching algorithms for solving the diffusion and the advection equation.

math.NA↗
arXiv2010

Multi-product operator splitting as a general method of solving autonomous and non-autonomous equations

Prior to the recent development of symplectic integrators, the time-stepping operator $\e^{h(A+B)}$ was routinely decomposed into a sum of products of $\e^{h A}$ and $\e^{hB}$ in the study of hyperbolic partial differential equations. In the context of solving Hamiltonian dynamics, we show that such a decomposition give rises to both {\it even} and {\it odd} order Runge-Kutta and Nyström integrators. By use of Suzuki's forward-time derivative operator to enforce the time-ordered exponential, we show that the same decomposition can be used to solve non-autonomous equations. In particular, odd order algorithms are derived on the basis of a highly non-trivial {\it time-asymmetric} kernel. Such an operator approach provides a general and unified basis for understanding structure non-preserving algorithms and is especially useful in deriving very high-order algorithms via {\it analytical} extrapolations. In this work, algorithms up to the 100th order are tested by integrating the ground state wave function of the hydrogen atom. For such a singular Coulomb problem, the multi-product expansion showed uniform convergence and is free of poles usually associated with structure-preserving methods. Other examples are also discussed.

math.NA↗
arXiv2009

Explicit symplectic integrators for solving non-separable Hamiltonians

By exploiting the error functions of explicit symplectic integrators for solving separable Hamiltonians, I show that it is possible to develop explicit, time-reversible symplectic integrators for solving non-separable Hamiltonians of the product form. The algorithms are unusual in that they of fractional order.

physics.comp-ph↗
arXiv2009

Multi-product splitting and Runge-Kutta-Nystrom integrators

The splitting of $\e^{h(A+B)}$ into a single product of $\e^{h A}$ and $\e^{hB}$ results in symplectic integrators when $A$ and $B$ are classical Lie operators. However, at high orders, a single product splitting, with exponentially growing number of operators, is very difficult to derive. This work shows that, if the splitting is generalized to a sum of products, then a simple choice of the basis product reduces the problem to that of extrapolation, with analytically known coefficients and only quadratically growing number of operators. When a multi-product splitting is applied to classical Hamiltonian systems, the resulting algorithm is no longer symplectic but is of the Runge-Kutta-Nyström (RKN) type. Multi-product splitting, in conjunction with a special force-reduction process,explains why at orders $p=4$ and 6, RKN integrators only need $p-1$ force evaluations.

math.NA↗
arXiv2009

Extrapolated High-Order Propagators for Path Integral Monte Carlo Simulations

We present a new class of high-order imaginary time propagators for path-integral Monte Carlo simulations by subtracting lower order propagators. By requiring all terms of the extrapolated propagator be sampled uniformly, the subtraction only affects the potential part of the path integral. The negligible violation of positivity of the resulting path integral at small time steps has no discernable affect on the accuracy of our method. Thus in principle arbitrarily high order algorithms can be devised for path-integral Monte Carlo simulations. We verify this claim is by showing that fourth, sixth, and eighth order convergence can indeed be achieved in solving for the ground state of strongly interacting quantum many-body systems such as bulk liquid $^4$He.

physics.comp-ph↗
arXiv2008

Any order imaginary time propagation method for solving the Schrodinger equation

The eigenvalue-function pair of the 3D Schrödinger equation can be efficiently computed by use of high order, imaginary time propagators. Due to the diffusion character of the kinetic energy operator in imaginary time, algorithms developed so far are at most fourth-order. In this work, we show that for a grid based algorithm, imaginary time propagation of any even order can be devised on the basis of multi-product splitting. The effectiveness of these algorithms, up to the 12$^{\rm th}$ order, is demonstrated by computing all 120 eigenstates of a model C$_{60}$ molecule to very high precisions. The algorithms are particularly useful when implemented on parallel computer architectures.

cond-mat.mtrl-sci↗
arXiv2008

Relativistic Motion in a Constant Electromagnetic Field

For a relativistic charged particle moving in a constant electromagnetic field, its velocity 4-vector has been well studied. However, despite the fact that both the electromagnetic field and the equations of motion are purely real, the resulting 4-velocity is seemingly due to a complex electromagnetic field. This work shows that this is not due to some complex formalism used (such as Clifford algebra) but is intrinsically due to the fact that the $o(3,1)$ Lie algebra of the Lorentz group is equivalent to two commuting complex $su(2)$ algebras. Expressing the complex $su(2)$ generators in terms of the boost and rotation operators then naturally introduces a complex electromagnetic field. This work solves the equation of motion not as a matrix equation, but as an operator evolution equation in terms of the generators of the Lorentz group. The factorization of the real evolution operator into two commuting complex evolution operators then directly gives the time evolution of the velocity 4-vector without any reference to an intermediate field.

math-ph↗
arXiv2007

Higher-order splitting algorithms for solving the nonlinear Schrödinger equation and their instabilities

Since the kinetic and the potential energy term of the real time nonlinear Schrödinger equation can each be solved exactly, the entire equation can be solved to any order via splitting algorithms. We verified the fourth-order convergence of some well known algorithms by solving the Gross-Pitaevskii equation numerically. All such splitting algorithms suffer from a latent numerical instability even when the total energy is very well conserved. A detail error analysis reveals that the noise, or elementary excitations of the nonlinear Schrödinger, obeys the Bogoliubov spectrum and the instability is due to the exponential growth of high wave number noises caused by the splitting process. For a continuum wave function, this instability is unavoidable no matter how small the time step. For a discrete wave function, the instability can be avoided only for $\dt k_{max}^2{<\atop\sim}2 π$, where $k_{max}=π/Δx$.

physics.comp-ph↗
arXiv2007

Forward and non-forward symplectic integrators in solving classical dynamics problems

Forward time step integrators are splitting algorithms with only positive splitting coefficients. When used in solving physical evolution equations, these positive coefficients correspond to positive time steps. Forward algorithms are essential for solving time-irreversible equations that cannot be evolved using backward time steps. However, forward integrators are also better in solving time-reversible equations of classical dynamics by tracking as closely as possible the physical trajectory. This work compares in detail various forward and non-forward fourth-order integrators using three, fourth, five and six force evaluations. In the case of solving the 2D Kepler orbit, all non-forward integrators are optimized by simply minimizing the size of their backward time steps.

physics.comp-ph↗
arXiv2006

The physics of symplectic integrators: perihelion advances and symplectic corrector algorithms

Symplectic integrators evolve dynamical systems according to modified Hamiltonians whose error terms are also well-defined Hamiltonians. The error of the algorithm is the sum of each error Hamiltonian's perturbation on the exact solution. When symplectic integrators are applied to the Kepler problem, these error terms cause the orbit to precess. In this work, by developing a general method of computing the perihelion advance via the Laplace-Runge-Lenz vector even for non-separable Hamiltonians, I show that the precession error in symplectic integrators can be computed analytically. It is found that at each order, each paired error Hamiltonians cause the orbit to precess oppositely by exactly the same amount after each period. Hence, symplectic corrector, or process integrators, which have equal coefficients for these paired error terms, will have their precession errors exactly cancel after each period. Thus the physics of symplectic integrators determines the optimal algorithm for integrating long time periodic motions.

math-ph↗
arXiv2006

Model of molecular bonding based on the Bohr-Sommerfeld picture of atoms

We develop a model of molecular binding based on the Bohr-Sommerfeld description of atoms together with a constraint taken from conventional quantum mechanics. The model can describe the binding energy curves of H2, H3 and other molecules with striking accuracy. Our approach treats electrons as point particles with positions determined by extrema of an algebraic energy function. Our constrained model provides a physically appealing, accurate description of multi-electron chemical bonds.

physics.chem-ph↗
arXiv2005

A Fundamental Theorem on the Structure of Symplectic Integrators

I show that the basic structure of symplectic integrators is governed by a theorem which states {\it precisely}, how symplectic integrators with positive coefficients cannot be corrected beyond second order. All previous known results can now be derived quantitatively from this theorem. The theorem provided sharp bounds on second-order error coefficients explicitly in terms of factorization coefficients. By saturating these bounds, one can derive fourth-order algorithms analytically with arbitrary numbers of operators.

math-ph↗