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.

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.

scholarly journals Bäcklund Transformations between the KdV Equation and a New Nonlinear Evolution Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Xifang Cao
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
The Other ◽  
Bäcklund Transformations ◽  
Backlund Transformations ◽  
The Kdv Equation ◽  
New Equation

We first give a Bäcklund transformation from the KdV equation to a new nonlinear evolution equation. We then derive two Bäcklund transformations with two pseudopotentials, one of which is from the KdV equation to the new equation and the other from the new equation to itself. As applications, by applying our Bäcklund transformations to known solutions, we construct some novel solutions to the new equation.

scholarly journals STRINGS, WAVES, DRUMS: SPECTRA AND INVERSE PROBLEMS

2009 ◽  
Vol 07 (02) ◽  
pp. 131-183
Author(s):  
RICHARD BEALS ◽  
PETER C. GREINER
Keyword(s):  
Inverse Problems ◽  
Integrable Systems ◽  
Wave Motion ◽  
Kdv Equation ◽  
Spectral Geometry ◽  
Quantum Inverse ◽  
The Kdv Equation ◽  
Vibrating String ◽  
Famous Paper ◽  
Index Problem

This survey treats a number of interconnected topics related in one way or another to the famous paper of Mark Kac, "Can one hear the shape of a drum?": wave motion, classical and quantum inverse problems, integrable systems, and the relations between spectra and geometry. We sketch the history and some of the principal developments from the vibrating string to quantum inverse problems, the KdV equation and integrable systems, spectral geometry and the index problem.

scholarly journals A QUANTUM BRST–ANTI-BRST APPROACH TO CLASSICAL INTEGRABLE SYSTEMS

2004 ◽  
Vol 19 (09) ◽  
pp. 693-702 ◽  
Author(s):  
MICHAEL CHESTERMAN ◽  
MARCELO B. SILKA
Keyword(s):  
Wave Function ◽  
Quantum Theory ◽  
Inverse Scattering ◽  
Integrable Systems ◽  
Classical Mechanics ◽  
Lax Pair ◽  
Liouville Integrability ◽  
Lax Equation ◽  
Scattering Wave ◽  
Classical Integrable

We reformulate the conditions of Liouville integrability in the language of Gozzi et al.'s quantum BRST–anti-BRST description of classical mechanics. The Das–Okubo geometrical Lax equation is particularly suited for this approach. We find that the Lax pair and inverse scattering wave function appear naturally in certain sectors of the quantum theory.

scholarly journals A family of wave equations with some remarkable properties

2018 ◽  
Vol 474 (2210) ◽  
pp. 20170763 ◽  
Author(s):  
Priscila Leal da Silva ◽  
Igor Leite Freire ◽  
Júlio Cesar Santos Sampaio
Keyword(s):  
Wave Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Lax Pair ◽  
Integrable Equation ◽  
Dispersive Equations ◽  
Nonlinear Dispersive Equations ◽  
Type Transformation ◽  
The Family ◽  
The Kdv Equation

We consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters. Conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions. Recursion operators are found for two members of the family investigated. For one of them, a Lax pair is also obtained, proving its complete integrability. From the Lax pair, we construct a Miura-type transformation relating the original equation to the Korteweg–de Vries (KdV) equation. This transformation, on the other hand, enables us to obtain solutions of the equation from the kernel of a Schrödinger operator with potential parametrized by the solutions of the KdV equation. In particular, this allows us to exhibit a kink solution to the completely integrable equation from the 1-soliton solution of the KdV equation. Finally, peakon-type solutions are also found for a certain choice of the parameters, although for this particular case the equation is reduced to a homogeneous second-order nonlinear evolution equation.

BILINEARIZATION AND SOLUTIONS OF THE KdV6 EQUATION

2008 ◽  
Vol 06 (04) ◽  
pp. 401-412 ◽  
Author(s):  
A. RAMANI ◽  
B. GRAMMATICOS ◽  
R. WILLOX
Keyword(s):  
Evolution Equation ◽  
Arbitrary Function ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Singularity Analysis ◽  
Backlund Transformation ◽  
Integrable Evolution Equation ◽  
The Kdv Equation ◽  
Integrable Evolution ◽  
Kdv6 Equation

We examine the recently proposed KdV6 integrable evolution equation. Starting from solutions suggested by singularity analysis and using the auto-Bäcklund transformation, we construct solutions of the KdV6 which involve one arbitrary function of time. Next, we proceed to bilinearize the equation and derive a new, simpler, auto-Bäcklund transformation. Starting from the solutions of the KdV equation we construct those of the KdV6 in the form of M kinks and N poles and which indeed involve an arbitrary function of time.

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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