Periodic waves travelling along an unsmooth boundary via the fractal variational theory

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

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Fermat Metrics

Symmetry ◽

10.3390/sym13081422 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1422

Author(s):

Antonio Masiello

Keyword(s):

Variational Methods ◽

Morse Theory ◽

Sagnac Effect ◽

Variational Theory ◽

Global Hyperbolicity ◽

Fermat Principle ◽

Multiple Image ◽

Light Rays ◽

Stationary Spacetimes

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered.

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Quasi-periodic and non-periodic waves in (2+1) dimensions

The European Physical Journal D ◽

10.1140/epjd/e2008-00038-x ◽

2008 ◽

Vol 47 (2) ◽

pp. 221-225 ◽

Cited By ~ 1

Author(s):

C. L. Bai ◽

H. J. Niu

Keyword(s):

Periodic Waves

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Bifurcation Phenomena of Nonlinear Waves in a Generalized Zakharov-Kuznetsov Equation

Advances in Mathematical Physics ◽

10.1155/2013/812120 ◽

2013 ◽

Vol 2013 ◽

pp. 1-14

Author(s):

Yun Wu ◽

Zhengrong Liu

Keyword(s):

Solitary Waves ◽

Nonlinear Waves ◽

Blow Up ◽

Periodic Waves ◽

The Third ◽

Bifurcation Phenomena ◽

Kink Waves ◽

Fourth Kind

We study the bifurcation phenomena of nonlinear waves described by a generalized Zakharov-Kuznetsov equationut+au2+bu4ux+γuxxx+δuxyy=0. We reveal four kinds of interesting bifurcation phenomena. The first kind is that the low-kink waves can be bifurcated from the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The second kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves, the symmetric solitary waves, and the 2-blow-up waves. The third kind is that the periodic-blow-up waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves.

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On periodic water-waves and their convergence to solitary waves in the long-wave limit

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1981.0231 ◽

1981 ◽

Vol 303 (1481) ◽

pp. 633-669 ◽

Cited By ~ 59

Keyword(s):

Integral Equation ◽

Solitary Waves ◽

Water Waves ◽

Periodic Problem ◽

Existence Theory ◽

Periodic Waves ◽

Equation Formulation ◽

Long Wave ◽

Integral Equation Formulation ◽

Wave Limit

A detailed discussion of Nekrasov’s approach to the steady water-wave problems leads to a new integral equation formulation of the periodic problem. This development allows the adaptation of the methods of Amick & Toland (1981) to show the convergence of periodic waves to solitary waves in the long-wave limit. In addition, it is shown how the classical integral equation formulation due to Nekrasov leads, via the Maximum Principle, to new results about qualitative features of periodic waves for which there has long been a global existence theory (Krasovskii 1961, Keady & Norbury 1978).

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