scholarly journals Hydrodynamic reductions of multidimensional dispersionless PDEs: The test for integrability

2004 ◽  
Vol 45 (6) ◽  
pp. 2365-2377 ◽  
Author(s):  
E. V. Ferapontov ◽  
K. R. Khusnutdinova
Keyword(s):  
Hydrodynamic Reductions

scholarly journals Hydrodynamic Reductions and Solutions of a Universal Hierarchy

2004 ◽  
Vol 140 (2) ◽  
pp. 1073-1085 ◽  
Author(s):  
L. Martínez Alonso ◽  
A. B. Shabat
Keyword(s):  
Hydrodynamic Reductions

scholarly journals Generalized hydrodynamic reductions of the kinetic equation for a soliton gas

2012 ◽  
Vol 171 (2) ◽  
pp. 675-682 ◽  
Author(s):  
M. V. Pavlov ◽  
V. B. Taranov ◽  
G. A. El
Keyword(s):  
Kinetic Equation ◽  
Hydrodynamic Reductions

scholarly journals Hamiltonian formalism of two-dimensional Vlasov kinetic equation

2014 ◽  
Vol 470 (2172) ◽  
pp. 20140343
Author(s):  
Maxim V. Pavlov
Keyword(s):  
Kinetic Equation ◽  
Hamiltonian Structure ◽  
Bubbly Flow ◽  
Fluid Motion ◽  
Hamiltonian Formalism ◽  
Finite Depth ◽  
Dimensional Case ◽  
Two Dimensional ◽  
The One ◽  
Hydrodynamic Reductions

In this paper, the two-dimensional Benney system describing long wave propagation of a finite depth fluid motion and the multi-dimensional Russo–Smereka kinetic equation describing a bubbly flow are considered. The Hamiltonian approach established by J. Gibbons for the one-dimensional Vlasov kinetic equation is extended to a multi-dimensional case. A local Hamiltonian structure associated with the hydrodynamic lattice of moments derived by D. J. Benney is constructed. A relationship between this hydrodynamic lattice of moments and the two-dimensional Vlasov kinetic equation is found. In the two-dimensional case, a Hamiltonian hydrodynamic lattice for the Russo–Smereka kinetic model is constructed. Simple hydrodynamic reductions are presented.

scholarly journals On a class of integrable Hamiltonian equations in 2+1 dimensions

2021 ◽  
Vol 477 (2249) ◽  
pp. 20210047
Author(s):  
Ben Gormley ◽  
Eugene V. Ferapontov ◽  
Vladimir S. Novikov
Keyword(s):  
Lax Pairs ◽  
Hamiltonian Density ◽  
Hamiltonian Equations ◽  
Commuting Flows ◽  
Integrability Conditions ◽  
Function Of Two Variables ◽  
Hydrodynamic Reductions ◽  
Non Locality

We classify integrable Hamiltonian equations of the form u t = ∂ x ( δ H δ u ) , H = ∫ h ( u , w )   d x d y , where the Hamiltonian density h ( u , w ) is a function of two variables: dependent variable u and the non-locality w = ∂ x − 1 ∂ y u . Based on the method of hydrodynamic reductions, the integrability conditions are derived (in the form of an involutive PDE system for the Hamiltonian density h ). We show that the generic integrable density is expressed in terms of the Weierstrass σ -function: h ( u , w ) =  σ ( u ) e w . Dispersionless Lax pairs, commuting flows and dispersive deformations of the resulting equations are also discussed.

scholarly journals Hydrodynamic reductions of the heavenly equation

2003 ◽  
Vol 20 (11) ◽  
pp. 2429-2441 ◽  
Author(s):  
E V Ferapontov ◽  
M V Pavlov
Keyword(s):  
Hydrodynamic Reductions

scholarly journals Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

2010 ◽  
Vol 21 (2) ◽  
pp. 151-191 ◽  
Author(s):  
G. A. El ◽  
A. M. Kamchatnov ◽  
M. V. Pavlov ◽  
S. A. Zykov
Keyword(s):  
Kinetic Equation ◽  
Hydrodynamic Reductions

scholarly journals Symmetric matrix ensemble and integrable hydrodynamic chains

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Costanza Benassi ◽  
Marta Dell’Atti ◽  
Antonio Moro
Keyword(s):  
Partition Function ◽  
Thermodynamic Limit ◽  
Arbitrary Number ◽  
Symmetric Matrix ◽  
Hydrodynamic Type ◽  
Number Of Components ◽  
Matrix Ensemble ◽  
Particular Solution ◽  
Hydrodynamic Reductions ◽  
Hydrodynamic Chain

AbstractThe partition function of the Symmetric Matrix Ensemble is identified with the $$\tau $$ τ -function of a particular solution of the Pfaff Lattice. We show that, in the case of even power interactions, in the thermodynamic limit, the $$\tau $$ τ -function corresponds to the solution of an integrable chain of hydrodynamic type. We prove that the hydrodynamic chain so obtained is diagonalisable and admits hydrodynamic reductions in Riemann invariants in an arbitrary number of components.

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