A new non-perturbative approach in quantum mechanics for time-independent Schr\"{o}dinger equations

A new non-perturbative approach is proposed to solve time-independent Schr\"{o}dinger equations in quantum mechanics and chromodynamics (QCD). It is based on the homotopy analysis method (HAM), which was developed by the author for highly nonlinear equations since 1992 and has been widely applied in many fields. Unlike perturbative methods, this HAM-based approach has nothing to do with small/large physical parameters. Besides, convergent series solution can be obtained even if the disturbance is far from the known status. A nonlinear harmonic oscillator is used as an example to illustrate the validity of this approach for disturbances that might be more than hundreds larger than the possible superior limit of the perturbative approach. This HAM-based approach could provide us rigorous theoretical results in quantum mechanics and chromodynamics (QCD), which can be directly compared with experimental data. Obviously, this is of great benefit not only for improving the accuracy of experimental measurements but also for validating physical theories.Comment: 26 pages, 12 figures, 9 table

Two new standing solitary waves in shallow water

In this paper, the closed-form analytic solutions of two new Faraday's standing solitary waves due to the parametric resonance of liquid in a vessel vibrating vertically with a constant frequency are given for the first time. Using a model based on the symmetry of wave elevation and the linearized Boussinesq equation, we gain the closed-form wave elevations of the two kinds of non-monotonically decaying standing solitary waves with smooth crest and the even or odd symmetry. All of them have never been reported, to the best of our knowledge. Besides, they can well explain some experimental phenomena. All of these are helpful to deepen and enrich our understandings about standing solitary waves and Faraday's wave.Comment: 12 pages, 4 figures, accepted by Wave Motio

On peaked solitary waves of Camassa-Holm equation

Unlike the Boussinesq, KdV and BBM equations, the celebrated Casamma-Holm (CH) equation can model both phenomena of soliton interaction and wave breaking. Especially, it has peaked solitary waves in case of omega=0. Besides, in case of omega > 0, its solitary wave "becomes $C^\infty$ and there is no derivative discontinuity at its peak", as mentioned by Camassa and Holm in 1993 (PRL). However, it is found in this article that the CH equation has peaked solitary waves even in case of omega > 0. Especially, all of these peaked solitary waves have an unusual property: their phase speeds have nothing to do with the height of peakons or anti-peakons. Therefore, in contrast to the traditional view-points, the peaked solitary waves are a common property of the CH equation: in fact, all mainstream models of shallow water waves admit such kind of peaked solitary wavesComment: 11 pages, 4 figures, 2 table

On cusped solitary waves in finite water depth

It is well-known that the Camassa-Holm (CH) equation admits both of the peaked and cusped solitary waves in shallow water. However, it was an open question whether or not the exact wave equations can admit them in finite water depth. Besides, it was traditionally believed that cusped solitary waves, whose 1st-derivative tends to infinity at crest, are essentially different from peaked solitary ones with finite 1st-derivative. Currently, based on the symmetry and the exact water wave equations, Liao [1] proposed a unified wave model (UWM) for progressive gravity waves in finite water depth. The UWM admits not only all traditional smooth progressive waves but also the peaked solitary waves in finite water depth: in other words, the peaked solitary progressive waves are consistent with the traditional smooth ones. In this paper, in the frame of the linearized UWM, we further give, for the first time, the cusped solitary waves in finite water depth, and besides reveal a close relationship between the cusped and peaked solitary waves: a cusped solitary wave is consist of an infinite number of peaked solitary ones with the same phase speed, so that it can be regarded as a special peaked solitary wave. This also well explains why and how a cuspon has an infinite 1st-derivative at crest. It is found that, like peaked solitary waves, the vertical velocity of a cusped solitary wave in finite water depth is also discontinuous at crest (x=0), and especially its phase speed has nothing to do with wave height, too. In addition, it is unnecessary to consider whether the peaked/cusped solitary waves given by the UWM are weak solution or not, since the governing equation is not necessary to be satisfied at crest. All of these would deepen and enrich our understandings about the cusped solitary waves.Comment: 9 pages, 2 figure

A comment on the arguments about the reliability and convergence of chaotic simulations

Yao and Hughes commented (Tellus-A, 60: 803 - 805, 2008) that "all chaotic responses are simply numerical noise and have nothing to do with the solutions of differential equations". However, using 1200 CPUs of the National Supercomputer TH-A1 and a parallel integral algorithm of the so-called "Clean Numerical Simulation" (CNS) based on the 3500th-order Taylor expansion and data in 4180-digit multiple precision, one can gain reliable, convergent chaotic solution of Lorenz equation in a rather long interval [0,10000]. This supports Lorenz's optimistic viewpoint (Tellus-A, 60: 806 - 807, 2008): "numerical approximations can converge to a chaotic true solution throughout any finite range of time".Comment: 2 page

Analytic Solutions of Von Karman Plate under Arbitrary Uniform Pressure --- Equations in Differential Form

The large deflection of a circular thin plate under uniform external pressure is a classic problem in solid mechanics, dated back to Von K{\'a}rm{\'a}n \cite{Karman}. {This problem is reconsidered in this paper using an analytic approximation method, namely the homotopy analysis method (HAM).} Convergent series solutions are obtained for four types of boundary conditions with rather high nonlinearity, even in the case of $w(0)/h>20$, where $w(0)/h$ denotes the ratio of central deflection to plate thickness. Especially, we prove that the previous perturbation methods for an arbitrary perturbation quantity (including the Vincent's [2] and Chien's [3] methods) and the modified iteration method [4] are only the special cases of the HAM. However, the HAM works well even when the perturbation methods become invalid. All of these demonstrate the validity and potential of the HAM for the Von K{\'a}rm{\'a}n's plate equations, and show the superiority of the HAM over perturbation methods for highly nonlinear problemsComment: 33 pages, 4 figure

On the mathematically reliable long-term simulation of chaos of Lorenz equation in the interval [0,10000]

Using 1200 CPUs of the National Supercomputer TH-A1 and a parallel integral algorithm based on the 3500th-order Taylor expansion and the 4180-digit multiple precision data, we have done a reliable simulation of chaotic solution of Lorenz equation in a rather long interval [0,10000] (Lorenz time unit). Such a kind of mathematically reliable chaotic simulation has never been reported. It provides us a numerical benchmark for mathematically reliable long-term prediction of chaos. Besides, it also proposes a safe method for mathematically reliable simulations of chaos in a finite but long enough interval. In addition, our very fine simulations suggest that such a kind of mathematically reliable long-term prediction of chaotic solution might have no physical meanings, because the inherent physical micro-level uncertainty due to thermal fluctuation might quickly transfer into macroscopic uncertainty so that trajectories for a long enough time would be essentially uncertain in physics.Comment: 11 pages, 1 figure, accepted by Science China - Physics, Mechanics & Astronomy (published Online: 2014-01-01

On the inherent self-excited macroscopic randomness of chaotic three-body system

What is the origin of macroscopic randomness (uncertainty)? This is one of the most fundamental open questions for human being. In this paper, 10000 samples of reliable (convergent), multiple-scale (from 1.0E-60 to 100) numerical simulations of a chaotic three-body system indicate that, without any external disturbance, the microscopic inherent uncertainty (in the level of 1.0E-60) due to physical fluctuation of initial positions of the three-body system enlarges exponentially into macroscopic randomness (at the level O(1)) until t=T*, the so-called physical limit time of prediction, but propagates algebraically thereafter when accurate prediction of orbit is impossible. Note that these 10000 samples use micro-level, inherent physical fluctuations of initial position, which have nothing to do with human being. Especially, the differences of these 10000 fluctuations are mathematically so small (in the level of 1.0E-60) that they are physically the SAME since a distance shorter than a Planck length does not make physical senses according to the spring theory. It indicates that the macroscopic randomness of the chaotic three-body system is self-excited, say, without any external force or disturbances, from the inherent micro-level uncertainty. This provides us the new concept "self-excited macroscopic randomness (uncertainty)". In addition, it is found that, without any external disturbance, the chaotic three-body system might randomly disrupt with the symmetry-breaking at t=1000 in about 25% probability, which provides us the new concepts "self-excited random disruption", "self-excited random escape" and "self-excited symmetry breaking" of the chaotic three-body system. It suggests that a chaotic three-body system might randomly evolve by itself, without any external forces or disturbance.Comment: 15 pages, 5 figures, accepted by Int. J. Bifurcation and Chaos, will be published via Open Acces

A commend on "Three Classes of Newtonian Three-Body Planar Periodic Orbits" by \v{S}uvakov and Dmitra\v{s}inovi\'{c} (PRL, 2013)

Currently, the fifteen new periodic solutions of Newtonian three-body problem with equal mass were reported by \v{S}uvakov and Dmitra\v{s}inovi\'{c} (PRL, 2013) [1]. However, using a reliable numerical approach (namely the Clean Numerical Simulation, CNS) that is based on the arbitrary-order Taylor series method and data in arbitrary-digit precision, it is found that at least seven of them greatly depart from the periodic orbits after a long enough interval of time. Therefore, the reported initial conditions of at least seven of the fifteen orbits reported by \v{S}uvakov and Dmitra\v{s}inovi\'{c} [1] are not accurate enough to predict periodic orbits. Besides, it is found that these seven orbits are unstable.Comment: 5 pages, 5 figure

Analytic Solutions of Von Karman Plate under Arbitrary Uniform Pressure (II): Equations in Integral Form

In this paper, the homotopy analysis method (HAM) is successfully applied to solve the Von Karman's plate equations in the integral form for a circular plate with the clamped boundary under an arbitrary uniform external pressure. Two HAM-based approaches are proposed. One is for a given external load Q, the other for a given central deflection. Both of them are valid for an arbitrary uniform external pressure by means of choosing a proper value of the so-called convergence-control parameters c_1 and c_2 in the frame of the HAM. Besides, it is found that iteration can greatly accelerate the convergence of solution series. In addition, we prove that the interpolation iterative method is a special case of the HAM-based 1st-order iteration approach for a given external load Q when c_1=-theta and c_2=-1, where theta denotes the interpolation parameter of the interpolation iterative method. Therefore, like Zheng and Zhou}, one can similarly prove that the HAM-based approaches are valid for an arbitrary uniform external pressure, at least in some special cases such as c_1=-theta and c_2=-1. Furthermore, it is found that the HAM-based iteration approaches converge much faster than the interpolation iterative method. All of these illustrate the validity and potential of the HAM for the famous Von Karman's plate equations, and show the superiority of the HAM over perturbation methods.Comment: 26 pages, 8 figure