scholarly journals Analytically Solvable Models and Physically Realizable Solutions to Some Problems in Nonlinear Wave Dynamics of Cylindrical Shells

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2227
Author(s):  
Andrey Bochkarev ◽  
Aleksandr Zemlyanukhin ◽  
Vladimir Erofeev ◽  
Aleksandr Ratushny
Keyword(s):  
Elastic Medium ◽  
Gordon Equation ◽  
Modulation Instability ◽  
Solvable Models ◽  
Axially Symmetric ◽  
Wave Dynamics ◽  
Periodic Traveling Waves ◽  
Bending Waves ◽  
Klein Gordon ◽  
Nonlinear Elastic

The axially symmetric propagation of bending waves in a thin Timoshenko-type cylindrical shell, interacting with a nonlinear elastic Winkler medium, is herein studied. With the help of asymptotic integration, two analytically solvable models were obtained that have no physically realizable solitary wave solutions. The possibility for the real existence of exact solutions, in the form of traveling periodic waves of the nonlinear inhomogeneous Klein–Gordon equation, was established. Two cases were identified, which enabled the development of the modulation instability of periodic traveling waves: (1) a shell preliminarily compressed along a generatrix, surrounded by an elastic medium with hard nonlinearity, and (2) a preliminarily stretched shell interacting with an elastic medium with soft nonlinearity.

scholarly journals GRAVITATIONAL FIELD OF A ROTATING GRAVITATIONAL DYON

2002 ◽  
Vol 17 (15n17) ◽  
pp. 1091-1096 ◽  
Author(s):  
N. DADHICH ◽  
Z. YA. TURAKULOV
Keyword(s):  
Gravitational Field ◽  
Gordon Equation ◽  
Equations Of Motion ◽  
Polar Angle ◽  
Vacuum Equation ◽  
Axially Symmetric ◽  
Einstein Vacuum Equation ◽  
Einstein Vacuum ◽  
Klein Gordon ◽  
Hamilton Jacobi Equation

We have obtained the general solution of the Einstein vacuum equation for the axially symmetric stationary metric in which both the Hamilton-Jacobi equation for particle motion and the Klein - Gordon equation are separable. It can be interpreted to describe the gravitational field of a rotating dyon, a particle endowed with both gravoelectric (mass) and gravomagnetic (NUT parameter) charges. Further, there also exists a duality relation between the two charges and the radial and the polar angle coordinates which keeps the solution invariant. The solution can however be transformed into the known Kerr - NUT solution indicating its uniqueness under the separability of equations of motion.

SEPARATING STATIONARY AXIALLY-SYMMETRIC ASYMPTOTICALLY FLAT METRICS

1989 ◽  
Vol 04 (12) ◽  
pp. 2953-2958 ◽  
Author(s):  
Z. YA. TURAKULOV
Keyword(s):  
Gordon Equation ◽  
Separation Of Variables ◽  
Vacuum Solution ◽  
Complete Separation ◽  
Kerr Metric ◽  
Axially Symmetric ◽  
Spherical System ◽  
Asymptotically Flat ◽  
Klein Gordon ◽  
Klein Gordon Equation

Stationary axially-symmetric asymptotically flat metrics allowing the complete separation of variables in the Klein-Gordon equation are considered. It is shown that if such metrics coincide at infinity with the metric of spherical system of coordinates, the variables for them in the Einstein equation are completely separable and the only vacuum solution is the Kerr metric.

scholarly journals Axisymmetric Longitudinal-Bending Waves in a Cylindrical Shell Interacting with a Nonlinear Elastic Medium

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Alexander I. Zemlyanukhin ◽  
Andrey V. Bochkarev ◽  
Lev I. Mogilevich ◽  
Ekaterina G. Tindova
Keyword(s):  
Cylindrical Shell ◽  
Elastic Medium ◽  
Numerical Solutions ◽  
Bending Waves ◽  
Nonlinear Elastic Medium ◽  
Longitudinal Bending ◽  
Infinite Cylindrical Shell ◽  
Nonlinear Elastic ◽  
Traveling Pulse ◽  
Good Agreement

A nonlinear differential equation is derived which describes the propagation of axisymmetric stationary longitudinal-bending waves in infinite cylindrical shell of Timoshenko type, interacting with the external nonlinear elastic medium. A modified perturbation method based on the use of diagonal Pade approximants was applied to build exact solitary-wave solutions of the derived equation in the form of traveling front and the traveling pulse. Numerical solutions of the equation, obtained by means of finite difference method, are in good agreement with the corresponding exact analytical ones.

scholarly journals Fractional Klein-Gordon equation with singular mass

2021 ◽  
Vol 143 ◽  
pp. 110579
Author(s):  
Arshyn Altybay ◽  
Michael Ruzhansky ◽  
Mohammed Elamine Sebih ◽  
Niyaz Tokmagambetov
Keyword(s):  
Gordon Equation ◽  
Klein Gordon ◽  
Klein Gordon Equation
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