Evolution of geometric properties on solution curve of KdV equation

2017 ◽  
Vol 43 (1) ◽  
pp. 20-27
Author(s):  
Ling XU ◽  
Keyword(s):  
Kdv Equation ◽  
Solution Curve ◽  
Geometric Properties

scholarly journals Diffeomorphisms on S1, projective structures and integrable systems

2002 ◽  
Vol 44 (1) ◽  
pp. 169-180 ◽  
Author(s):  
Partha Guha
Keyword(s):  
Evolution Equation ◽  
Integrable Systems ◽  
Kdv Equation ◽  
Lax Pair ◽  
Projective Connection ◽  
Solution Curve ◽  
Projective Structures ◽  
Lax Equation ◽  
Homogeneous Coordinates ◽  
The Kdv Equation

AbstractIn this paper we consider a projective connection as defined by the nth-order Adler-Gelfand-Dikii (AGD) operator on the circle. It is well-known that the Korteweg-de Vries (KdV) equation is the archetypal example of a scalar Lax equation defined by a Lax pair of scalar nth-order differential (AGD) operators. In this paper we derive (formally) the KdV equation as an evolution equation of the AGD operator (at least for n ≤ 4) under the action of Vect(S1). The solutions of the AGD operator define an immersion R → RPn−1 in homogeneous coordinates. In this paper we derive the Schwarzian KdV equation as an evolution of the solution curve associated with Δ(n), for n ≤ 4.

Some Properties of Para-Sasakian Manifolds

2017 ◽  
Vol 5 (2) ◽  
pp. 73-78
Author(s):  
Jay Prakash Singh ◽  
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Sasakian Manifold ◽  
Subject Classification ◽  
Killing Vector Field ◽  
Sasakian Manifolds ◽  
Geometric Properties ◽  
Paper Author

In this paper author present an investigation of some differential geometric properties of Para-Sasakian manifolds. Condition for a vector field to be Killing vector field in Para-Sasakian manifold is obtained. Mathematics Subject Classification (2010). 53B20, 53C15.

Integrable Models

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Soliton Solution ◽  
Water Waves ◽  
Heisenberg Model ◽  
Kdv Equation ◽  
Short Review ◽  
Lax Pair ◽  
Shallow Water Waves ◽  
Initial Value ◽  
The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

Effect of 12 months of creatine supplementation and whole-body resistance training on measures of bone, muscle and strength in older males

2020 ◽  
pp. 026010602097524
Author(s):  
Darren G Candow ◽  
Philip D Chilibeck ◽  
Julianne Gordon ◽  
Emelie Vogt ◽  
Tim Landeryou ◽  
...  
Keyword(s):  
Bone Mineral Density ◽  
Resistance Training ◽  
Bone Mineral ◽  
Creatine Supplementation ◽  
Whole Body ◽  
Mineral Density ◽  
Geometric Properties ◽  
Section Modulus ◽  
Body Resistance ◽  
Older Males

Background: The combination of creatine supplementation and resistance training (10–12 weeks) has been shown to increase bone mineral content and reduce a urinary indicator of bone resorption in older males compared with placebo. However, the longer-term effects (12 months) of creatine and resistance training on bone mineral density and bone geometric properties in older males is unknown. Aim: To assess the effects of 12 months of creatine supplementation and supervised, whole-body resistance training on bone mineral density, bone geometric properties, muscle accretion, and strength in older males. Methods: Participants were randomized to supplement with creatine ( n = 18, 49–69 years, 0.1 g·kg-1·d-1) or placebo ( n = 20, 49–67 years, 0.1 g·kg-1·d-1) during 12 months of supervised, whole-body resistance training. Results: After 12 months of training, both groups experienced similar changes in bone mineral density and geometry, bone speed of sound, lean tissue and fat mass, muscle thickness, and muscle strength. There was a trend ( p = 0.061) for creatine to increase the section modulus of the narrow part of the femoral neck, an indicator of bone bending strength, compared with placebo. Adverse events did not differ between creatine and placebo. Conclusions: Twelve months of creatine supplementation and supervised, whole-body resistance training had no greater effect on measures of bone, muscle, or strength in older males compared with placebo.

scholarly journals Remarkable Classes of Almost 3-Contact Metric Manifolds

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 8
Author(s):  
Giulia Dileo
Keyword(s):  
Geometric Properties ◽  
New Class ◽  
Contact Metric Manifolds ◽  
Cosymplectic Manifolds ◽  
Sasaki Manifolds

We introduce a new class of almost 3-contact metric manifolds, called 3-(0,δ)-Sasaki manifolds. We show fundamental geometric properties of these manifolds, analyzing analogies and differences with the known classes of 3-(α,δ)-Sasaki (α≠0) and 3-δ-cosymplectic manifolds.

scholarly journals Wave Breaking in Undular Bores with Shear Flows

Water Waves ◽  
2021 ◽  
Author(s):  
Maria Bjørnestad ◽  
Henrik Kalisch ◽  
Malek Abid ◽  
Christian Kharif ◽  
Mats Brun
Keyword(s):  
Turbulent Flows ◽  
Wave Breaking ◽  
Vries Equation ◽  
Kdv Equation ◽  
Flow Depth ◽  
Present Contribution ◽  
Undular Bore ◽  
Wave Flume ◽  
The Kdv Equation ◽  
The Waves

AbstractIt is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg–de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent.

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