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  • 1
    Electronic Resource
    Electronic Resource
    Chaos in the one-dimensional gravitational three-body problem (1993)
    Woodbury, NY : American Institute of Physics (AIP)
    Chaos 3 (1993), S. 183-203 
    Details
    ISSN: 1089-7682
    Source: AIP Digital Archive
    Topics: Physics
    Notes: We have investigated the appearance of chaos in the one-dimensional Newtonian gravitational three-body system (three masses on a line with −1/r pairwise potential). In the center of mass coordinates this system has two degrees of freedom and can be conveniently studied using Poincaré sections. We have concentrated in particular on how the behavior changes when the relative masses of the three bodies change. We consider only the physically more interesting case of negative total energy. For two mass choices we have calculated 18 000 full orbits (with initial states on a 100×180 lattice on the Poincaré section) and obtained dwell time distributions. For 105 mass choices we have calculated Poincaré maps for 10×18 starting points. Our results show that the Poincaré section (and hence the phase space) divides into three well defined regions with orbits of different characteristics: (1) There is a region of fast scattering, with a minimum of pairwise collisions. This region consists of ‘scallops' bordering the E=0 line, within a scallop the orbits vary smoothly. The number of the scallops increases as the mass of the central particle decreases. (2) In the chaotic scattering region the interaction times are longer, and both the interaction time and the final state depend sensitively on the starting point on the Poincaré section. For both (1) and (2) the initial and final states consist of a binary + single particle. (3) The third region consists of quasiperiodic orbits where the three masses are bound together forever. At the center of the quasiperiodic region there is a periodic orbit discovered (numerically) by Schubart in 1956. The stability of the Schubart orbit turns out to correlate strongly with the global behavior.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.165984
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  • 2
    Electronic Resource
    Electronic Resource
    A search for bilinear equations passing Hirota's three-soliton condition. III. Sine–Gordon-type bilinear equations (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 2586-2592 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this paper the results of a search for pairs of bilinear equations of the type Ai(Dx,Dt)F⋅F +Bi(Dx,Dt)G⋅F +Ci(Dx,Dt)G⋅G=0, i=1,2, which have standard type three-soliton solutions, are presented. The freedom to rotate in (F,G) space is fixed by the one-soliton ansatz F=1, G=en, then the Bi determine the dispersion manifold while Ai and Ci are auxiliary functions. In this paper it is assumed that B1 and B2 are even and proportional, and that Ai and Ci are quadratic. As new results, B1=aD3x Dt +DtDy+b, A2=−C2=DxDt, and generalizations of the sine–Gordon model B1=DxDt+a with a family of auxiliary functions Ai and Ci are obtained.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527750
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  • 3
    Electronic Resource
    Electronic Resource
    A search for bilinear equations passing Hirota's three-soliton condition. I. KdV-type bilinear equations (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 1732-1742 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this paper the results of a search for bilinear equations of the type P(Dx, Dt)F ⋅ F=0, which have three-soliton solutions, are presented. Polynomials up to order 8 have been studied. In addition to the previously known cases of KP, BKP, and DKP equations and their reductions, a new polynomial P=DxDt(Dx2 +(square root of)3DxDt+Dt2) +aDx2+bDxDt +cDt2 has been found. Its complete integrability is not known, but it has three-soliton solutions. Infinite sequences of models with linear dispersion manifolds have also been found, e.g., P=DxMDtNDyP, if some powers are odd, and P=DxMDtN(Dx2 −1)P, if M and N are odd.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527815
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  • 4
    Electronic Resource
    Electronic Resource
    The eight tetrahedron equations (1997)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 3603-3615 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this paper we derive from arguments of string scattering a set of eight tetrahedron equations, with different index orderings. It is argued that this system of equations is the proper system that represents integrable structures in three dimensions generalizing the Yang–Baxter equation. Under additional restrictions this system reduces to the usual tetrahedron equation in the vertex form. Most known solutions fall under this class, but it is by no means necessary. Comparison is made with the work on braided monoidal 2-categories also leading to eight tetrahedron equations. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532055
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  • 5
    Electronic Resource
    Electronic Resource
    Solving the two-dimensional constant quantum Yang–Baxter equation (1993)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 34 (1993), S. 1725-1756 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A detailed analysis of the constant quantum Yang–Baxter equation Rk1k2j1j2 Rl1k3k1j3Rl2l3k2k3= Rk2k3j2j3 Rk1l3j1k3Rl1l2k1k2 in two dimensions is presented, leading to an exhaustive list of its solutions. The set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the Gröbner-basis methods. Each solution was then transformed into a canonical form (based on the various trace matrices of R) for final elimination of duplicates and subcases. If we use homogeneous parametrization the solutions can be combined into 23 distinct cases, modulo the well-known C, P, and T reflections, and rotations and scalings R˜=κ(Q⊗Q)R(Q⊗Q)−1.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530185
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  • 6
    Electronic Resource
    Electronic Resource
    New Integrable Systems (1988)
    Oxford, UK : Blackwell Publishing Ltd
    Annals of the New York Academy of Sciences 536 (1988), S. 0 
    Details
    ISSN: 1749-6632
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Natural Sciences in General
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1749-6632.1988.tb51560.x
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  • 7
    Electronic Resource
    Electronic Resource
    A search for bilinear equations passing Hirota's three-soliton condition. IV. Complex bilinear equations (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 628-635 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this paper the results of a search for complex bilinear equations with two-soliton solutions are presented. The following basic types are discussed: (a) the nonlinear Schrödinger equation B(Dx, ...)G⋅F=0, A(Dx,Dt) F⋅F=GG*, and (b) the Benjamin–Ono equation P(Dx, ...)F⋅F*=0. It is found that the existence of two-soliton solutions is not automatic, but introduces conditions that are like the usual three- and four-soliton conditions. The search was limited by the degree of A=2, and by degree of P≤4. The main results are the following: (1) (iaD3x+DxDt +iDy+b)G⋅F=0, D2xF⋅F=GG*; (2) (D2x+aD2y +iDt+b)G⋅F =0, DxDy F⋅F=GG*; (3) (iaD3x+D2x +iDt)F⋅F*=0; and (4) (DxDt+i(aDx +bDt))F⋅F*=0.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528002
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  • 8
    Electronic Resource
    Electronic Resource
    A search for bilinear equations passing Hirota's three-soliton condition. II. mKdV-type bilinear equations (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 2094-2101 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this paper (second in a series) [for part I, see J. Math. Phys. 30, 1732 (1987)] the search for bilinear equations having three-soliton solutions continues. This time pairs of bilinear equations of the type P1(Dx,Dt)F⋅G=0, P2(Dx,Dt)F⋅G=0, where P1 is an odd polynomial and P2 is quadratic, are considered. The main results are the following new bilinear systems: P1=aDx7+bDx5 +Dx2Dt+Dy, P2=Dx2; P1=aDx3+bDt3 +Dy, P2=DxDt; and P1=DxDtDy +aDx+bDt, P2=DxDt. In addition to these, several models with linear dispersion manifolds were obtained, as before.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527421
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  • 9
    Electronic Resource
    Electronic Resource
    A numerical investigation of the one-dimensional newtonian three-body problem II. Positive energies (1989)
    Springer
    Celestial mechanics and dynamical astronomy 47 (1989), S. 321-331 
    Details
    ISSN: 1572-9478
    Keywords: three-body problem ; collinear motion ; Newtonian potential
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Numerical orbit integrations have been conducted to characterize the types of trajectories in the one-dimensional Newtonian three-body problem with equal masses and positive energy. At positive energies the basic types of motions are “binary + single particle’ and ‘ionization’, and when time goes from −∞ to +∞ all possible transitions between these states can take place. Properties of individual orbits have been summarized in the form of graphical maps in a two-dimensional grid of initial values. The basic motion types exist at all positive energies, but the binary + single particle configuration is obtained only in a narrow region of initial values if the total energy is large. At very large energies the equations of motion can be solved approximately, and this asymptotic result, exact in the limit of infinite energy, is presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00051009
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  • 10
    Electronic Resource
    Electronic Resource
    A numerical investigation of the one-dimensional Newtonian three-body problem (1991)
    Springer
    Celestial mechanics and dynamical astronomy 51 (1991), S. 379-394 
    Details
    ISSN: 1572-9478
    Keywords: three-body problem ; collinear motion ; periodic orbits ; stability of motion
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The one-dimensional Newtonian three-body problem is known to have stable (quasi-)periodic orbits when the masses are equal. The existence and size of the stable region is discussed here in the case where the three masses are arbitrary. We consider only the stability of the periodic (generalized) Schubart's (1956) orbit. If this orbit is linearly stable it is almost always surrounded by a region of stable quasi-periodic orbits and the size and shape of this stable region depends on the masses. The three-dimensional linear stability of the periodic orbits is also determined. Final results show that the region of stability has a complicated shape and some of the stable regions in the mass-plane are quite narrow. The non-linear three-dimensional stability is studied independently by extensive numerical integrations and the results are found to be in agreement with the linear stability analysis. The boundaries of stable region in the mass-plane are given in terms of polynomial approximations. The results are compared with a similar work by Héenon (1977).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00052929
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