THE N = 3 SUPERSYMMETRIC KdV HIERARCHY

1993 ◽  
Vol 08 (12) ◽  
pp. 1161-1169 ◽  
Author(s):  
C. M. YUNG
Keyword(s):  
Conservation Law ◽  
Hamiltonian Structure ◽  
Kdv Equation ◽  
Superconformal Algebra ◽  
Supersymmetric Extension ◽  
Kdv Hierarchy ◽  
The Kdv Equation

An N = 3 supersymmetric extension of the KdV equation, whose Hamiltonian structure is the classical O(3)-extended superconformal algebra is presented. Integrability of the equation is argued based on the existence of a non-trivial conservation law which reduces to the first non-trivial conservation law of the KdV equation.

scholarly journals ON GAUGE-EQUIVALENT FORMULATIONS OF N=4 SKdV HIERARCHY

1998 ◽  
Vol 13 (35) ◽  
pp. 2855-2862 ◽  
Author(s):  
E. IVANOV
Keyword(s):  
Hamiltonian Structure ◽  
Superconformal Algebra ◽  
Kdv Hierarchy ◽  
Small N

We point out that the N=4 supersymmetric KdV hierarchy, when written through the prepotentials of the bosonic chiral and antichiral N=2 supercurrents, exhibits a freedom related to the possibility to choose different gauges for the prepotentials. Some implications of this property are presented. In particular, we give the prepotential form of the "small" N=4 superconformal algebra, the second Hamiltonian structure algebra of the N=4 SKdV hierarchy, for two choices of gauge.

Asymptotic lower bound for the radius of spatial analyticity to solutions of KdV equation

2019 ◽  
Vol 21 (08) ◽  
pp. 1850061 ◽  
Author(s):  
Achenef Tesfahun
Keyword(s):  
Lower Bound ◽  
Initial Data ◽  
Recent Result ◽  
Analytic Functions ◽  
Conservation Law ◽  
Kdv Equation ◽  
Space Of Analytic Functions ◽  
Positive Formula ◽  
The Kdv Equation ◽  
Fixed Radius

It is shown that the uniform radius of spatial analyticity [Formula: see text] of solutions at time [Formula: see text] to the KdV equation cannot decay faster than [Formula: see text] as [Formula: see text] given initial data that is analytic with fixed radius [Formula: see text]. This improves a recent result of Selberg and da Silva, where they proved a decay rate of [Formula: see text] for arbitrarily small positive [Formula: see text]. The main ingredients in the proof are almost conservation law for the solution to the KdV equation in space of analytic functions and space-time dyadic bilinear [Formula: see text] estimates associated with the KdV equation.

scholarly journals Evolution equations having conservation laws with flux characteristics

2007 ◽  
Vol 49 (1) ◽  
pp. 39-52
Author(s):  
B. Van Brunt ◽  
M. Vlieg Hulstman
Keyword(s):  
Conservation Laws ◽  
Conservation Law ◽  
Evolution Equations ◽  
Kdv Equation ◽  
Divergence Form ◽  
Subject Classification ◽  
The Kdv Equation

A class of evolution equations in divergence form is studied in this paper. Specifically, we develop conditions under which the spatial divergence term, the flux, corresponds to the characteristic of a conservation law. The KdV equation is a prominent example of an equation having a flux term that is also a characteristic for a conservation law. We show that the flux term must be self-adjoint. General equations for the corresponding conservation laws and Hamiltonian densities are derived and supplemented with examples. 2000 Mathematics subject classification: primary 35K.

scholarly journals ON THE CLASSICAL WN(l) ALGEBRAS

1992 ◽  
Vol 07 (24) ◽  
pp. 6053-6080 ◽  
Author(s):  
DIDIER A. DEPIREUX ◽  
PIERRE MATHIEU
Keyword(s):  
Hamiltonian Structure ◽  
Superconformal Algebra ◽  
Hamiltonian Reduction ◽  
Integrable Hierarchy ◽  
Kdv Hierarchy ◽  
General Aspects

We analyze the WN(l) algebras according to their conjectured realization as the second Hamiltonian structure of the integrable hierarchy resulting from the interchange of x and t in the l th flow of the sl (N) KdV hierarchy. The W4(3) algebra is derived explicitly along these lines, thus providing further support for the conjecture. This algebra is found to be equivalent to that obtained by the method of Hamiltonian reduction. Furthermore, its twisted version reproduces the algebra associated to a certain nonprincipal embedding of sl(2) into sl(4), or equivalently, the u(2) quasi-superconformal algebra. General aspects of the WN(l) algebras are also presented. We point out in particular that the x↔t interchange approach of the WN(l) algebra appears straightforward only when N and l are coprime.

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 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.

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.

scholarly journals REDUCTION OF TODA LATTICE HIERARCHY TO GENERALIZED KdV HIERARCHIES AND THE TWO-MATRIX MODEL

1995 ◽  
Vol 10 (17) ◽  
pp. 2537-2577 ◽  
Author(s):  
H. ARATYN ◽  
E. NISSIMOV ◽  
S. PACHEVA ◽  
A.H. ZIMERMAN
Keyword(s):  
Poisson Bracket ◽  
Hamiltonian Structure ◽  
Toda Lattice ◽  
Free Field ◽  
String Model ◽  
Matrix Formulation ◽  
Hamiltonian Action ◽  
Kdv Hierarchy ◽  
Associated Matrix ◽  
Toda Lattice Hierarchy

Toda lattice hierarchy and the associated matrix formulation of the 2M-boson KP hierarchies provide a framework for the Drinfeld-Sokolov reduction scheme realized through Hamiltonian action within the second KP Poisson bracket. By working with free currents, which Abelianize the second KP Hamiltonian structure, we are able to obtain a unified formalism for the reduced SL (M+1, M−k) KdV hierarchies interpolating between the ordinary KP and KdV hierarchies. The corresponding Lax operators are given as superdeterminants of graded SL (M+1, M−k) matrices in the diagonal gauge and we describe their bracket structure and field content. In particular, we provide explicit free field representations of the associated W(M, M−k) Poisson bracket algebras generalizing the familiar nonlinear WM+1 algebra. Discrete Bäcklund transformations for SL (M+1, M−k) KdV are generated naturally from lattice translations in the underlying Toda-like hierarchy. As an application we demonstrate the equivalence of the two-matrix string model to the SL (M+1, 1) KdV hierarchy.

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