Optical solitons and closed form solutions to the (3+1)-dimensional resonant Schrödinger dynamical wave equation

2020 ◽  
Vol 34 (30) ◽  
pp. 2050291 ◽  
Author(s):  
Usman Younas ◽  
Aly R. Seadawy ◽  
M. Younis ◽  
S. T. R. Rizvi
Keyword(s):  
Closed Form ◽  
Gordon Equation ◽  
Velocity Dispersion ◽  
Group Velocity Dispersion ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Closed Form Solutions ◽  
Contour Plots ◽  
Complex Phenomena ◽  
Selection Of

This paper investigates the new solitons and closed form solutions to [Formula: see text] dimensional resonant nonlinear Schrödinger equation (RNLSE) that explains the behavior of waves with the effect of group velocity dispersion and resonant nonlinearities in the optical fiber. The soliton solutions in single and combined forms like dark, singular, and dark-singular in mixed form are extracted by means of two innovative integration norms namely extended sinh-Gordon equation expansion and [Formula: see text]-expansion function methods. Moreover, kink and closed form solutions are also observed under different constraint conditions. By choosing the suitable selection of the parameters, three dimensional, two dimensional, and contour plots are sketched. The obtained outcomes show that the applied computational strategies are direct, efficient, concise and can be implemented in more complex phenomena with the assistant of symbolic computations.

Dynamics of exact closed-form solutions to the Schamel Burgers and Schamel equations with constant coefficients using a novel analytical approach

2021 ◽  
Author(s):  
Sanjaya K. Mohanty ◽  
Sachin Kumar ◽  
Manoj K. Deka ◽  
Apul N. Dev
Keyword(s):  
Closed Form ◽  
Acoustic Waves ◽  
Evolution Equations ◽  
Kdv Equation ◽  
Three Dimensional ◽  
Nonlinear Evolution Equations ◽  
Closed Form Solutions ◽  
Extended Technique ◽  
Contour Plots ◽  
Constant Coefficients

In this paper, we investigate two different constant-coefficient nonlinear evolution equations, namely the Schamel Burgers equation and the Schamel equation. These models also have a great deal of potential for studying ion-acoustic waves in plasma physics and fluid dynamics. The primary goal of this paper is to establish closed-form solutions and dynamics of analytical solutions to the Schamel Burgers and the Schamel equations, which are special examples of the well-known Schamel–Korteweg-de Vries (S-KdV) equation. We derive completely novel solutions to the considered models using a variety of computation programmes and a newly proposed extended generalized [Formula: see text] expansion approach. The newly formed solutions, which include hyperbolic and trigonometric functions as well as rational function solutions, have been produced. The annihilation of three-dimensional shock waves, periodic waves, single soliton, singular soliton, and combo soliton, multisoliton as well as their three-dimensional and contour plots are used to show the dynamical representations of the acquired solutions. These results demonstrate that the proposed extended technique is efficient, reliable and simple.

Solutions to the Vector Tetrahedron Equation

1965 ◽  
Vol 87 (2) ◽  
pp. 228-234 ◽  
Author(s):  
Milton A. Chace
Keyword(s):  
Closed Form ◽  
Digital Computer ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Spherical Coordinates ◽  
Tetrahedron Equation ◽  
Position Determination ◽  
Closed Form Solutions

A set of nine closed-form solutions are presented to the single, three-dimensional vector tetrahedron equation, sum of vectors equals zero. The set represents all possible combinations of unknown spherical coordinates among the vectors, assuming the coordinates are functionally independent. Optimum use is made of symmetry. The solutions are interpretable and can be evaluated reliably by digital computer in milliseconds. They have been successfully applied to position determination of many three-dimensional mechanisms.

scholarly journals Reconstruction of three-dimensional maps based on closed-form solutions of the variational problem of multisensor data registration

2019 ◽  
Vol 484 (6) ◽  
pp. 672-677
Author(s):  
A. V. Vokhmintcev ◽  
A. V. Melnikov ◽  
K. V. Mironov ◽  
V. V. Burlutskiy
Keyword(s):  
Closed Form ◽  
Large Scale ◽  
Three Dimensional ◽  
Dynamic Environment ◽  
Closed Form Solution ◽  
Point Clouds ◽  
Accurate Method ◽  
Form Solution ◽  
Closed Form Solutions ◽  
Reconstruction Methods

A closed-form solution is proposed for the problem of minimizing a functional consisting of two terms measuring mean-square distances for visually associated characteristic points on an image and meansquare distances for point clouds in terms of a point-to-plane metric. An accurate method for reconstructing three-dimensional dynamic environment is presented, and the properties of closed-form solutions are described. The proposed approach improves the accuracy and convergence of reconstruction methods for complex and large-scale scenes.

Kinematics of a Novel Single-Loop Under-Actuated Wrist

2012 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Path Planning ◽  
Closed Form ◽  
Nonholonomic Constraint ◽  
Three Dimensional ◽  
Single Loop ◽  
Closed Form Solutions ◽  
Position Analysis ◽  
Tracking Tasks ◽  
Planning Algorithm ◽  
Path Planning Algorithm

A novel type of parallel wrist (PW) is proposed which, differently from previously presented PWs, features a single-loop architecture and only one nonholonomic constraint. Due to the presence of a nonholonomic constraint, the proposed PW type is under-actuated, that is, it is able to control the platform orientation in a three-dimensional workspace by employing only two actuated pairs, one prismatic (P) and the other revolute (R); and it cannot perform tracking tasks. Position analysis and path planning of this novel PW are studied. In particular, all the relevant position analysis problems are solved in closed form, and, based on these closed-form solutions, a path-planning algorithm is built.

Nonautonomous Solitons for the Coupled Variable-Coefficient Cubic-Quintic Nonlinear Schrödinger Equations with External Potentials in the Non-Kerr Fibre

2015 ◽  
Vol 70 (12) ◽  
pp. 985-994
Author(s):  
Bing-Qing Mao ◽  
Yi-Tian Gao ◽  
Yu-Jie Feng ◽  
Xin Yu
Keyword(s):  
Variable Coefficient ◽  
Velocity Dispersion ◽  
Group Velocity Dispersion ◽  
Soliton Solutions ◽  
Einstein Condensation ◽  
Bose Einstein Condensation ◽  
Nonlinear Schrödinger ◽  
Similarity Transformations ◽  
Constant Coefficients ◽  
Bose Einstein

AbstractVariable-coefficient nonlinear Schrödinger (NLS)-type models are used to describe certain phenomena in plasma physics, nonlinear optics, arterial mechanics, and Bose–Einstein condensation. In this article, the coupled variable-coefficient cubic-quintic NLS equations with external potentials in the non-Kerr fibre are investigated. Via symbolic computation, similarity transformations and relevant constraints on the coefficient functions are obtained. Based on those transformations, such equations are transformed into the coupled cubic-quintic NLS equations with constant coefficients. Nonautonomous soliton solutions are derived, and propagation and interaction of the nonautonomous solitons in the non-Kerr fibre are investigated analytically and graphically. Those soliton solutions are related to the group velocity dispersion r(x) and external potentials h1(x) and h2(x, t). With the different choices of r(x), parabolic, cubic, and periodically oscillating solitons are obtained; with the different choices of h2(x, t), we can see the dromion-like structures and nonautonomous solitons with smooth step-like oscillator frequency profiles, to name a few.

Nonlinear Spatial Equilibria and Stability of Cables Under Uni-axial Torque and Thrust

1994 ◽  
Vol 61 (4) ◽  
pp. 879-886 ◽  
Author(s):  
C.-L. Lu ◽  
N. C. Perkins
Keyword(s):  
Closed Form ◽  
Elliptic Integral ◽  
Broad Class ◽  
Local Equilibrium ◽  
Three Dimensional ◽  
Equilibrium States ◽  
Form Complex ◽  
Closed Form Solutions ◽  
The Stability ◽  
Low Tension

Low tension cables subject to torque may form complex three-dimensional (spatial) equilibria. The resulting nonlinear static deformations, which are dominated by cable flexure and torsion, may produce interior loops or kinks that can seriously degrade the performance of the cable. Using Kirchhoffrod assumptions, a theoretical model governing cable flexure and torsion is derived herein and used to analyze (1) globally large equilibrium states, and (2) local equilibrium stability. For the broad class of problems described by pure boundary loading, the equilibrium boundary value problem is integrable and admits closed-form elliptic integral solutions. Attention is focused on the example problem of a cable subject to uni-axial torque and thrust. Closed-form solutions are presented for the complex three-dimensional equilibrium states which, heretofore, were analyzed using purely numerical methods. Moreover, the stability of these equilibrium states is assessed and new and important stability conclusions are drawn.

On the Efficiency of Analyzing 3D Anisotropic, Transversely Isotropic, and Isotropic Bodies in BEM

2010 ◽  
Vol 26 (4) ◽  
pp. 483-491
Author(s):  
Y. C. Shiah ◽  
W. X. Sun
Keyword(s):  
Closed Form ◽  
Computational Efficiency ◽  
Transverse Isotropy ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Fundamental Solutions ◽  
Engineering Practice ◽  
Closed Form Solutions ◽  
Anisotropic Bodies ◽  
Isotropic Bodies

ABSTRACTDue to a lack of closed-form solutions for three dimensional anisotropic bodies, the computational burden of evaluating the fundamental solutions in the boundary element method (BEM) has been a research focus over the years. In engineering practice, transversely isotropic material has gained popularity in the use of composites. As a degenerate case of the generally anisotropic material, transverse isotropy still needs to be treated separately to ease the computations. This paper aims to investigate the computational efficiency of the BEM implementations for 3D anisotropic, transversely isotropic, and isotropic bodies. For evaluating the fundamental solutions of 3D anisotropy, the explicit formulations reported in [1,2] are implemented. For treating transversely isotropic materials, numerous closed form solutions have been reported in the literature. For the present study, the formulations presented by Pan and Chou [3] are particularly employed. At the end, a numerical example is presented to compare the computational efficiency of the three cases and to demonstrate how the CPU time varies with the number of meshes.

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