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Two kinds of peaked solitary waves of the KdV,BBM and Boussinesq equations

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Science China(Physics,Mechanics & Astronomy) 2012年第12期55卷 2469-2475页

作者： liao shijun Department of Mathematics&State Key Laboratory of Ocean Engineering School of Naval ArchitectureOcean and Civil EngineeringShanghai Jiao Tong UniversityShanghai 200240China

It is well-known that the celebrated Camassa-Holm equation has the peaked solitary waves,which have been not reported for other mainstream models of shallow water *** this letter,the closed-form solutions of peaked solitary waves of the KdV equation,the BBM equation and the Boussinesq equation are given for the first *** of them have either a peakon or an *** of them exactly satisfies the corresponding Rankine-Hogoniot jump condition and could be understood as weak ***,the peaked solitary waves might be common for most of shallow water wave models,no matter whether or not they are integrable and/or admit breaking-wave solutions.

关键词： solitary waves peaked crest progressive discontinuity weak solution

NUMERICALLY SOLVING STRONGLY NONLINEAR PROBLEMS BY MEANS OF NO ITERATIONS

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Journal of Hydrodynamics 1998年第1期10卷 102-108页

作者： liao, shijun Shanghai Jiaotong Univ Shanghai China

Based on the Homotopy Analysis Method, a direct numerical method for strongly nonlinear problems was proposed. The 2-D laminar flow over semi-infinite plate was used. The method can give the accurate enough approximations of a strongly nonlinear problem by means of no iteration and can provide a family of iterative formulas with traditional approaches.

关键词： Iterative methods Laminar flow Nonlinear equations Numerical methods Viscous flow

Clean numerical simulation:a new strategy to obtain reliable solutions of chaotic dynamic systems

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Applied Mathematics and Mechanics(English Edition) 2018年第11期39卷 1529-1546页

作者： Xiaoming LI shijun liao School of Naval Architecture Ocean and Civil EngineeringShanghai Jiao Tong UniversityShanghai 200240China State Key Laboratory of Ocean Engineering Shanghai 200240China Collaborative Innovative Center for Advanced Ship and Deep-Sea Exploration Shanghai 200240China Ministry-of-Education Key Laboratory of Scientific and Engineering Computing Shanghai 200240China

It is well-known that chaotic dynamic systems, e.g., three-body system and turbulent flow, have sensitive dependence on the initial conditions(SDIC). Unfortunately,numerical noises, i.e., truncation error and round-off error, always exist in practice. Thus,due to the SDIC, the long-term accurate prediction of chaotic dynamic systems is practically impossible. In this paper, a new strategy for chaotic dynamic systems, i.e., the clean numerical simulation(CNS), is briefly described, and applied to a few Hamiltonian chaotic systems. With negligible numerical noises, the CNS can provide convergent(reliable) chaotic trajectories in a long enough interval of time. This is very important for Hamiltonian systems, and thus should have many applications in various fields. It is found that the traditional numerical methods in double precision cannot give not only reliable trajectories but also reliable Fourier power spectra and autocorrelation functions(ACFs). In addition, even the statistic properties of chaotic systems cannot be correctly obtained by means of traditional numerical algorithms in double precision, as long as these statistics are time-dependent. The CNS results strongly suggest that one had better be very careful on the direct numerical simulation(DNS) results of statistically unsteady turbulent flows, although DNS results often agree well with experimental data when the turbulent flow is in a statistical stationary state.

关键词： chaos numerical noise clean numerical simulation(CNS) reliability of computation

More than six hundred new families of Newtonian periodic planar collisionless three-body orbits

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Science China(Physics,Mechanics & Astronomy) 2017年第12期60卷 63-69页

作者： XiaoMing Li shijun liao School of Naval Architecture Ocean and Civil Engineering Shanghai Jiaotong University Shanghai 200240 China MoE Key Laboratory in Scientific and Engineering Computing Shanghai 200240 China

The famous three-body problem can be traced back to Isaac Newton in the 1680 s. In the 300 years since this "three-body problem"was first recognized, only three families of periodic solutions had been found, until 2013 when ˇSuvakov and Dmitraˇsinovi′c [***. Lett. 110, 114301(2013)] made a breakthrough to numerically find 13 new distinct periodic orbits, which belong to 11 new families of Newtonian planar three-body problem with equal mass and zero angular momentum. In this paper, we numerically obtain 695 families of Newtonian periodic planar collisionless orbits of three-body system with equal mass and zero angular momentum in case of initial conditions with isosceles collinear configuration, including the well-known figure-eight family found by Moore in 1993, the 11 families found by ˇSuvakov and Dmitraˇsinovi′c in 2013, and more than 600 new families that have never been reported, to the best of our knowledge. With the definition of the average period T = T=Lf, where Lf is the length of the so-called "free group element", these 695 families suggest that there should exist the quasi Kepler's third law T* ≈ 2:433 ± 0:075 for the considered case, where T*= T|E|^(3/2) is the scale-invariant average period and E is its total kinetic and potential energy,respectively. The movies of these 695 periodic orbits in the real space and the corresponding close curves on the "shape sphere"can be found via the website: http://***/three-body/***.

关键词： three-body problem periodic orbits clean numerical simulation(CNS)

Analytic approximations of Von Krmn plate under arbitrary uniform pressure—equations in integral form

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Science China(Physics,Mechanics & Astronomy) 2018年第1期61卷 57-67页

作者： XiaoXu Zhong shijun liao School of Naval Architecture Ocean and Civil Engineering Shanghai Jiao Tong University Collaborative Innovative Center for Advanced Ship and Deep-Sea Exploration State Key Laboratory of Ocean Engineering Ministry-of-Education Key Laboratory of Scientific and Engineering Computing

Analytic approximations of the Von Karman＇s plate equations in integral form for a circular plate under external uniform pressure to arbitrary magnitude are successfully obtained by means of the homotopy analysis method （HAM）, an analytic approximation technique for highly nonlinear problems. Two HAM-based approaches are proposed for either a given external uniform pressure Q or a given central deflection, respectively. Both of them are valid for uniform pressure to arbitrary magnitude by choosing proper values of the so-called convergence-control parameters c1 and c2 in the frame of the HAM. Besides, it is found that the HAM- based iteration approaches generally converge much faster than the interpolation iterative method. Furthermore, we prove that the interpolation iterative method is a special case of the first-order HAM iteration approach for a given external uniform pressure Q when c1= -0 and c2 = -1, where 0 denotes the interpolation iterative parameter. Therefore, according to the convergence theorem of Zheng and Zhou about the interpolation iterative method, the HAM-based approaches are valid for uniform pressure to arbitrary magnitude at least in the special case c1 = -0 and c2= -1. In addition, we prove that the HAM approach for the Von karman＇s plate equations in differential form is just a special case of the HAM for the Von karman＇s plate equations in integral form mentioned in this paper. All of these illustrate the validity and great potential of the HAM for highly nonlinear problems, and its superiority over perturbation techniques.

关键词： circular plate high nonlinearity homotopy analysis method

On peaked solitary waves of the Degasperis-Procesi equation

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Science China(Physics,Mechanics & Astronomy) 2013年第2期56卷 418-422页

作者： liao shijun State Key Laboratory of Ocean Engineering Dept.of MathematicsSchool of Naval ArchitectureOcean and Civil EngineeringShanghai Jiao Tong University

The Degasperis-Procesi （DP） equation describing the propagation of shallow water waves contains a physical parameter co, and it is well-known that the DP equation admits solitary waves with a peaked crest when ω = 0. In this article, we illustrate, for the first time, that the DP equation admits peaked solitary waves even when ω≠ 0. This is helpful to enrich our knowledge and deepen our understandings about peaked solitary waves of the DP equation.

关键词： peaked solitary waves discontinuity Degasperis-Procesi equation

Avoiding Small Denominator Problems by Means of the Homotopy Analysis Method

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Advances in Applied Mathematics and Mechanics 2023年第2期15卷 267-299页

作者： shijun liao Center of Marine Numerical Experiment State Key Laboratory of Ocean EngineeringShanghai 200240China School of Naval Architecture Ocean and Civil EngineeringShanghai Jiaotong UniversityShanghai 200240China

The so-called“small denominator problem”was a fundamental problem of dynamics,as pointed out by Poincar´*** denominators appear most commonly in perturbative *** Duffing equation is the simplest example of a non-integrable system exhibiting all problems due to small *** this paper,using the forced Duffing equation as an example,we illustrate that the famous“small denominator problems”never appear if a non-perturbative approach based on the homotopy analysis method(HAM),namely“the method of directly defining inverse mapping”(MDDiM),is *** HAM-based MDDiM provides us great freedom to directly define the inverse operator of an undetermined linear operator so that all small denominators can be completely avoided and besides the convergent series of multiple limit-cycles of the forced Duffing equation with high nonlinearity are successfully ***,from the viewpoint of the HAM,the famous“small denominator problems”are only artifacts of perturbation ***,completely abandoning perturbation methods but using the HAM-based MDDiM,one would be never troubled by“small denominators”.The HAM-based MDDiM has general meanings in mathematics and thus can be used to attack many open problems related to the so-called“small denominators”.

关键词： Small denominator problem Duffing equation limit cycle homotopy analysis method(HAM) MDDiM

On the periodic solutions of the three-body problem

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National Science Review 2019年第6期6卷 1070-1071页

作者： shijun liao Xiaoming Li Advanced Center of Computing School of Naval Architecture Ocean and Civil EngineeringShanghai Jiao Tong University School of Physics and Astronomy Shanghai Jiao Tong University MOE Key Laboratory of Disaster Forecast and Control in Engineering School of Mechanics and Construction Engineering Jinan University

The motion of three bodies that attract each other by universal gravitation was first investigated by Newton [1] and was reconsidered by many great mathematicians and scientists, including Euler, Lagrange, Poincaré [2] and Hilbert. The famous three-body problem has had a great influence on physics, mathematics and non-linear dynamics. It has led to a new

关键词： On the periodic solutions of the three-body problem

One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems

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Science China(Physics,Mechanics & Astronomy) 2021年第1期64卷 108-113页

作者： XiaoMing Li XiaoChen Li shijun liao MOE Key Laboratory of Disaster Forecast and Control in Engineering School of Mechanics and Construction EngineeringJinan UniversityGuangzhou510632China Department of Earth Atmospheric and Planetary SciencesMassachusetts Institute of TechnologyCambridge02139USA School of Civil Engineering and Transportation South China University of TechnologyGuangzhou510641China Department of Engineering Science University of OxfordOxfordOX13PJUK Center of Advanced Computing School of Naval ArchitectureOcean and Civil EngineeringShanghai Jiao Tong UniversityShanghai200240China School of Physics and Astronomy Shanghai Jiao Tong UniversityShanghai200240China

The three-body problem can be traced back to Newton in 1687,but it is still an open question *** that only a few periodic orbits of three-body systems were found in 300 years after Newton mentioned this famous *** triple systems are common in astronomy,practically all observed periodic triple systems are hierarchical(similar to the Sun,Earth and Moon).It has traditionally been believed that non-hierarchical triple systems would be unstable and thus should disintegrate into a stable binary system and a single star,and consequently stable periodic orbits of non-hierarchical triple systems have been expected to be rather ***,we report here one family of 135445 periodic orbits of non-hierarchical triple systems with unequal masses;13315 among them are *** with the narrow mass range(only 10-5)in which stable"Figure-eight"periodic orbits of three-body systems exist,our newly found stable periodic orbits have fairly large mass *** find that many of these numerically found stable non-hierarchical periodic orbits have mass ratios close to those of hierarchical triple systems that have been measured with astronomical *** implies that these stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses quite possibly can be observed in *** investigation also suggests that there should exist an infinite number of stable periodic orbits of non-hierarchical triple systems with distinctly unequal *** that our approach has general meaning:in a similar way,every known family of periodic orbits of three-body systems with two or three equal masses can be used as a starting point to generate thousands of new periodic orbits of triple systems with distinctly unequal masses.

关键词： three-body problem non-hierarchical system periodic orbits