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Dec 1998

Volume 39, Issue 12, pp. 6247-6756

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Chaos in the Gyldén problem

Florin Diacu and Dan Şelaru

J. Math. Phys. 39, 6537 (1998); http://dx.doi.org/10.1063/1.532663 (10 pages) | Cited 7 times

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We consider the Gyldén problem—a perturbation of the Kepler problem via an explicit function of time. For certain general classes of planar periodic perturbations, after proving a Poincaré–Melnikov-type criterion, we find a manifold of orbits in which the dynamics is given by the shift automorphism on the set of bi-infinite sequences with infinitely many symbols. We achieve the main result by computing the Melnikov integral explicitly. © 1998 American Institute of Physics.
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95.10.Ce Celestial mechanics (including n-body problems)
05.45.-a Nonlinear dynamics and chaos
45.05.+x General theory of classical mechanics of discrete systems

A generalized Hirota equation in 2+1 dimensions

Attilio Maccari

J. Math. Phys. 39, 6547 (1998); http://dx.doi.org/10.1063/1.532664 (5 pages) | Cited 21 times

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Using an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling, a new integrable and nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained starting from the Kadomtsev–Petviashvili equation. We apply the reduction technique to the Lax pair of the Kadomtsev–Petviashvili equation and demonstrate the integrability property of the new equation, because we obtain the corresponding Lax pair. The new equation reduces to the Hirota equation in the 1+1-dimensional limit. © 1998 American Institute of Physics.
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02.30.Jr Partial differential equations
02.30.Cj Measure and integration
41.20.Jb Electromagnetic wave propagation; radiowave propagation

Bifurcations of new eigenvalues for the Benjamin–Ono equation

Dmitry E. Pelinovsky and Catherine Sulem

J. Math. Phys. 39, 6552 (1998); http://dx.doi.org/10.1063/1.532665 (21 pages) | Cited 6 times

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A criterion for the emergence of new eigenvalues is found for the linear scattering problem associated with the Benjamin–Ono (BO) equation. This bifurcation occurs due to perturbations of nongeneric potentials which include the soliton solutions of the BO equation. The asymptotic approximation of an exponentially small new eigenvalue is derived. The method is based on the expansion of a localized function through a complete set of unperturbed eigenfunctions. Explicit expressions are obtained for the soliton potentials. © 1998 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos
02.30.-f Function theory, analysis
41.20.Jb Electromagnetic wave propagation; radiowave propagation
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
11.80.-m Relativistic scattering theory
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