scholarly journals New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions

2017 ◽  
Vol 4 (5) ◽  
pp. 232-241 ◽  
Author(s):  
Aysegul Dascioglu ◽  
Sevil Culha ◽  
Dilek Varol Bayram
Keyword(s):  
Analytical Solutions ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Jacobi Elliptic Functions

scholarly journals Rigorous Asymptotics of a KdV Soliton Gas

2021 ◽  
Author(s):  
M. Girotti ◽  
T. Grava ◽  
R. Jenkins ◽  
K. D. T.-R. McLaughlin
Keyword(s):  
Hilbert Problem ◽  
Kdv Equation ◽  
Broad Class ◽  
Elliptic Functions ◽  
Jacobi Elliptic Functions ◽  
Cnoidal Wave ◽  
Large Space ◽  
The Kdv Equation ◽  
Long Time ◽  
Riemann Hilbert Problem

AbstractWe analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are characterized by a Riemann–Hilbert problem which we show arises as the limit $$N\rightarrow + \infty $$ N → + ∞ of a gas of N-solitons. We show that this gas of solitons in the limit $$N\rightarrow \infty $$ N → ∞ is slowly approaching a cnoidal wave solution for $$x \rightarrow - \infty $$ x → - ∞ up to terms of order $$\mathcal {O} (1/x)$$ O ( 1 / x ) , while approaching zero exponentially fast for $$x\rightarrow +\infty $$ x → + ∞ . We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.

scholarly journals Behavior of a Free Dual-Spin Gyrostat with Different Ratios of Inertia Moments

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Vladimir Aslanov
Keyword(s):  
Exact Solutions ◽  
Nonlinear Equations ◽  
Analytical Solutions ◽  
Elliptic Functions ◽  
Rigid Bodies ◽  
Attitude Motion ◽  
Jacobi Elliptic Functions ◽  
Spacecraft Dynamics ◽  
Exact Analytical Solutions ◽  
Canonical Variables

The attitude motion is studied of asymmetric dual-spin gyrostats which may be modeled as free systems of two rigid bodies, one asymmetric and one axisymmetric. Exact analytical solutions of the attitude motion are presented for all possible ratios of inertia moments of these bodies. The dynamics of free gyrostats with zero internal torque is considered. The dimensionless nonlinear equations of the gyrostat are written in Serret-Andoyer canonical variables. The previously known exact solutions are complemented by new several solutions in terms of Jacobi elliptic functions. The results of the study can be useful for the analysis of dual-spin spacecraft dynamics.

ON EXPLICIT SOLUTIONS OF A COUPLED KdV-mKdV EQUATION

1989 ◽  
Vol 03 (06) ◽  
pp. 871-875 ◽  
Author(s):  
Chanchal Guha-Roy
Keyword(s):  
Kdv Equation ◽  
Elliptic Functions ◽  
Mkdv Equation ◽  
Explicit Solutions ◽  
Jacobi Elliptic Functions ◽  
Modified Kdv Equation ◽  
De Vries

A coupled Korteweg de Vries (KdV)-modified KdV equation is presented, some explicit solutions of which are exhibited in terms of Jacobi elliptic functions.

scholarly journals A New Approach for Solving Fractional Partial Differential Equations in the Sense of the Modified Riemann-Liouville Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Jacobi Elliptic Function ◽  
Jacobi Elliptic Functions ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Liouville Derivative

Based on a fractional complex transformation, certain fractional partial differential equation in the sense of the modified Riemann-Liouville derivative is converted into another ordinary differential equation of integer order, and the exact solutions of the latter are assumed to be expressed in a polynomial in Jacobi elliptic functions including the Jacobi sine function, the Jacobi cosine function, and the Jacobi elliptic function of the third kind. The degree of the polynomial can be determined by the homogeneous balance principle. With the aid of mathematical software, a series of exact solutions for the fractional partial differential equation can be found. For demonstrating the validity of this approach, we apply it to solve the space fractional KdV equation and the space-time fractional Fokas equation. As a result, some Jacobi elliptic functions solutions for the two equations are obtained.

Nonlocal Symmetries, Explicit Solutions, and Wave Structures for the Korteweg–de Vries Equation

2016 ◽  
Vol 71 (8) ◽  
pp. 735-740
Author(s):  
Zheng-Yi Ma ◽  
Jin-Xi Fei
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Group Method ◽  
Physical Interaction ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
The Kdv Equation ◽  
De Vries ◽  
Exact Invariant

AbstractFrom the known Lax pair of the Korteweg–de Vries (KdV) equation, the Lie symmetry group method is successfully applied to find exact invariant solutions for the KdV equation with nonlocal symmetries by introducing two suitable auxiliary variables. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to the hyperbolic and Jacobi elliptic functions are derived. Figures show the physical interaction between the cnoidal waves and a solitary wave.

Minimum Energy Control of a Unicycle Model Robot

2021 ◽  
Author(s):  
Youngjin Kim ◽  
Tarunraj Singh
Keyword(s):  
Optimal Control ◽  
Minimum Energy ◽  
Kinematic Model ◽  
Elliptic Functions ◽  
Terminal Point ◽  
Jacobi Elliptic Functions ◽  
Differential Drive ◽  
Terminal Constraints ◽  
Differential Drive Robot ◽  
Point To Point

Abstract Point-to-point path planning for a kinematic model of a differential-drive wheeled mobile robot (WMR) with the goal of minimizing input energy is the focus of this work. An optimal control problem is formulated to determine the necessary conditions for optimality and the resulting two point boundary value problem is solved in closed form using Jacobi elliptic functions. The resulting nonlinear programming problem is solved for two variables and the results are compared to the traditional shooting method to illustrate that the Jacobi elliptic functions parameterize the exact profile of the optimal trajectory. A set of terminal constraints which lie on a circle in the first quadrant are used to generate a set of optimal solutions. It is noted that for maneuvers where the angle of the vector connecting the initial and terminal point is greater than a threshold, which is a function of the radius of the terminal constraint circle, the robot initially moves into the third quadrant before terminating in the first quadrant. The minimum energy solution is compared to two other optimal control formulations: (1) an extension of the Dubins vehicle model where the constant linear velocity of the robot is optimized for and (2) a simple turn and move solution, both of whose optimal paths lie entirely in the first quadrant. Experimental results are used to validate the optimal trajectories of the differential-drive robot.

CLASS NUMBER FORMULAS FOR CERTAIN BICYCLIC BIQUADRATIC NUMBER FIELDS AS FINITE SUMS INVOLVING JACOBI ELLIPTIC FUNCTIONS

2012 ◽  
Vol 08 (05) ◽  
pp. 1257-1270
Author(s):  
M. A. GÓMEZ-MOLLEDA ◽  
JOAN-C. LARIO
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Elliptic Functions ◽  
Number Fields ◽  
Jacobi Elliptic Functions ◽  
Classical Class ◽  
Class Number Formula ◽  
Number Formula ◽  
Finite Sums ◽  
Finite Sum

We give formulas for the class numbers of bicyclic biquadratic number fields containing an imaginary quadratic field of class number one. The class number is expressed as a finite sum in terms of the basic Jacobi elliptic functions, playing a similar role as the trigonometric sine in Dirichlet classical class number formula.

scholarly journals Optimal Control on the Rotation Group SO (3)

2012 ◽  
Vol 28 (2) ◽  
pp. 321-328
Author(s):  
CLAUDIU C. REMSING ◽  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Elliptic Functions ◽  
Rotation Group ◽  
Jacobi Elliptic Functions ◽  
Hamilton Equations ◽  
Extremal Curve

A typical left-invariant optimal control problem on the rotation group SO (3) is investigated. The reduced Hamilton equations associated with an extremal curve are derived in a simple and elegant manner. These equations are then explicitly integrated by Jacobi elliptic functions.

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