Hamiltonian Systems Based on a Lie Algebra

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Lie Algebra ◽  
Euler Equations ◽  
Burgers Equation ◽  
Kdv Equation ◽  
Three Dimensional ◽  
Hamiltonian Formalism ◽  
Virasoro Algebra ◽  
Inner Product ◽  
The Kdv Equation ◽  
Limiting Case

There is a remarkable analogy between Euler’s equations for a rigid body and his equations for an ideal fluid. The unifying idea is that of a Lie algebra with an inner product, which is not invariant, on it. The concepts of a vector space, Lie algebra, and inner product are reviewed. A hamiltonian dynamical system is derived from each metric Lie algebra. The Virasoro algebra (famous in string theory) is shown to lead to the KdV equation; and in a limiting case, to the Burgers equation for shocks. A hamiltonian formalism for two-dimensional Euler equations is then developed in detail. A discretization of these equations (using a spectral method) is then developed using mathematical ideas from quantum mechanics. Then a hamiltonian formalism for the full three-dimensional Euler equations is developed. The Clebsch variables which provide canonical pairs for fluid dynamics are then explained, in analogy to angular momentum.

On the integrability properties of variable coefficient Korteweg – de Vries equations

1996 ◽  
Vol 74 (9-10) ◽  
pp. 676-684 ◽  
Author(s):  
F. Güngör ◽  
M. Sanielevici ◽  
P. Winternitz
Keyword(s):  
Differential Equations ◽  
Variable Coefficient ◽  
Kdv Equation ◽  
Three Dimensional ◽  
Symmetry Algebra ◽  
Time Symmetry ◽  
Kdv Equations ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
De Vries

All variable coefficient Korteweg – de Vries (KdV) equations with three-dimensional Lie point symmetry groups are investigated. For such an equation to have the Painlevé property, its coefficients must satisfy seven independent partial differential equations. All of them are satisfied only for equations equivalent to the KdV equation itself. However, most of them are satisfied in all cases. If the symmetry algebra is either simple, or nilpotent, then the equations have families of single-valued solutions depending on two arbitrary functions of time. Symmetry reduction is used to obtain particular solutions. The reduced ordinary differential equations are classified.

A dynamic equation for 2D surface waves on deep water

2020 ◽  
Author(s):  
Dmitry Kachulin ◽  
Alexander Dyachenko ◽  
Vladimir Zakharov
Keyword(s):  
Fourth Order ◽  
Canonical Transformation ◽  
Three Dimensional ◽  
Fourier Method ◽  
Hamiltonian Formalism ◽  
Two Dimensional ◽  
Russian Science ◽  
Wave Interactions ◽  
Zakharov Equation ◽  
Limiting Case

<p>Using the Hamiltonian formalism and the theory of canonical transformations, we have constructed a model of the dynamics of two-dimensional waves on the surface of a three-dimensional fluid. We find and apply a canonical transformation to a water wave equation to remove all nonresonant cubic and fourth-order nonlinear terms. The found canonical transformation also allows us to significantly simplify the fourth-order terms in the Hamiltonian by replacing the coefficient of four-wave Zakharov interactions with a new simpler one. As a result, unlike the Zakharov equation (written in k-space), this equation can be written in x-space, which greatly simplifies its numerical simulation. In addition, our chosen form of a new coefficient of four-wave interactions allows us to generalize this equation to describe two-dimensional waves on the surface of a three-dimensional fluid. An effective numerical algorithm based on the pseudospectral Fourier method for solving the new 2D equation is developed. In the limiting case of plane (one-dimensional) waves, we found solutions in the form of breathers propagating in one direction. The dynamics of such nonlinear traveling waves perturbed in the transverse direction is numerically investigated.</p><p>The work was supported by the Russian Science Foundation (Grant No. 19-72-30028).</p>

scholarly journals Solutions of the KdV Equation through Analysis of Regular Symmetries

2021 ◽  
Vol 20 ◽  
pp. 387-398
Author(s):  
S. Y. Jamal ◽  
J. M. Manale
Keyword(s):  
Burgers Equation ◽  
Kdv Equation ◽  
Lie Symmetries ◽  
Third Order ◽  
The Kdv Equation ◽  
De Vries ◽  
Topological Manifolds

We investigate a case of the generalized Korteweg – De Vries Burgers equation. Our aim is to demonstrate the need for the application of further methods in addition to using Lie Symmetries. The solution is found through differential topological manifolds. We apply Lie’s theory to take the PDE to an ODE. However, this ODE is of third order and not easily solvable. It is through differentiable topological manifolds that we are able to arrive at a solution

scholarly journals The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160585 ◽  
Author(s):  
Divya Venkataraman ◽  
Samriddhi Sankar Ray
Keyword(s):  
Energy Spectrum ◽  
Numerical Simulations ◽  
Euler Equations ◽  
Burgers Equation ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Infinite Dimensional ◽  
Localized Structures ◽  
Finite Dimensional ◽  
Incompressible Euler

Solutions to finite-dimensional (all spatial Fourier modes set to zero beyond a finite wavenumber K G ), inviscid equations of hydrodynamics at long times are known to be at variance with those obtained for the original infinite dimensional partial differential equations or their viscous counterparts. Surprisingly, the solutions to such Galerkin-truncated equations develop sharp localized structures, called tygers (Ray et al. 2011 Phys. Rev. E 84 , 016301 ( doi:10.1103/PhysRevE.84.016301 )), which eventually lead to completely thermalized states associated with an equipartition energy spectrum. We now obtain, by using the analytically tractable Burgers equation, precise estimates, theoretically and via direct numerical simulations, of the time τ c at which thermalization is triggered and show that τ c ∼ K G ξ , with ξ = − 4 9 . Our results have several implications, including for the analyticity strip method, to numerically obtain evidence for or against blow-ups of the three-dimensional incompressible Euler equations.

EULER–POINCARÉ FLOWS ON THE LOOP BOTT–VIRASORO GROUP AND SPACE OF TENSOR DENSITIES AND (2 + 1)-DIMENSIONAL INTEGRABLE SYSTEMS

2010 ◽  
Vol 22 (05) ◽  
pp. 485-505
Author(s):  
PARTHA GUHA
Keyword(s):  
Kdv Equation ◽  
Virasoro Algebra ◽  
Coadjoint Orbit ◽  
Type Systems ◽  
Inner Product ◽  
Field Equations ◽  
Virasoro Group ◽  
The Right ◽  
Tensor Densities

Following the work of Ovsienko and Roger ([54]), we study loop Virasoro algebra. Using this algebra, we formulate the Euler–Poincaré flows on the coadjoint orbit of loop Virasoro algebra. We show that the Calogero–Bogoyavlenskii–Schiff equation and various other (2 + 1)-dimensional Korteweg–deVries (KdV) type systems follow from this construction. Using the right invariant H1 inner product on the Lie algebra of loop Bott–Virasoro group, we formulate the Euler–Poincaré framework of the (2 + 1)-dimensional of the Camassa–Holm equation. This equation appears to be the Camassa–Holm analogue of the Calogero–Bogoyavlenskii–Schiff type (2 + 1)-dimensional KdV equation. We also derive the (2 + 1)-dimensional generalization of the Hunter–Saxton equation. Finally, we give an Euler–Poincaré formulation of one-parameter family of (1 + 1)-dimensional partial differential equations, known as the b-field equations. Later, we extend our construction to algebra of loop tensor densities to study the Euler–Poincaré framework of the (2 + 1)-dimensional extension of b-field equations.

scholarly journals WELL-POSEDNESS FOR A FAMILY OF PERTURBATIONS OF THE KdV EQUATION IN PERIODIC SOBOLEV SPACES OF NEGATIVE ORDER

2013 ◽  
Vol 15 (06) ◽  
pp. 1350005
Author(s):  
XAVIER CARVAJAL PAREDES ◽  
RICARDO A. PASTRAN
Keyword(s):  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Burgers Equation ◽  
Kdv Equation ◽  
Well Posedness ◽  
Initial Value ◽  
Negative Order ◽  
The Kdv Equation ◽  
De Vries ◽  
Sivashinsky Equation

We establish local well-posedness in Sobolev spaces Hs(𝕋), with s ≥ -1/2, for the initial value problem issues of the equation [Formula: see text] where η > 0, (Lu)∧(k) = -Φ(k)û(k), k ∈ ℤ and Φ ∈ ℝ is bounded above. Particular cases of this problem are the Korteweg–de Vries–Burgers equation for Φ(k) = -k2, the derivative Korteweg–de Vries–Kuramoto–Sivashinsky equation for Φ(k) = k2 - k4, and the Ostrovsky–Stepanyams–Tsimring equation for Φ(k) = |k| - |k|3.

Analysis of three-dimensional aerospace configurations using the Euler equations

1989 ◽  
Author(s):  
N. KROLL ◽  
C. ROSSOW ◽  
S. SCHERR ◽  
J. SCHOENE ◽  
G. WICHMANN
Keyword(s):  
Euler Equations ◽  
Three Dimensional
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