Infinitely many commuting symmetries and constants of motion in involution for explicitly time‐dependent evolution equations

1984 ◽  
Vol 25 (4) ◽  
pp. 918-922 ◽  
Author(s):  
W. Oevel ◽  
A. S. Fokas
Keyword(s):  
Evolution Equations ◽  
Time Dependent ◽  
Constants Of Motion

scholarly journals A Variational Formulation of Nonequilibrium Thermodynamics for Discrete Open Systems with Mass and Heat Transfer

Entropy ◽  
2018 ◽  
Vol 20 (3) ◽  
pp. 163 ◽  
Author(s):  
François Gay-Balmaz ◽  
Hiroaki Yoshimura
Keyword(s):  
Heat Transfer ◽  
Entropy Production ◽  
Variational Formulation ◽  
Evolution Equations ◽  
Nonequilibrium Thermodynamics ◽  
Open Systems ◽  
Internal Heat ◽  
Nonholonomic Constraints ◽  
Time Dependent ◽  
Nonlinear Constraint

We propose a variational formulation for the nonequilibrium thermodynamics of discrete open systems, i.e., discrete systems which can exchange mass and heat with the exterior. Our approach is based on a general variational formulation for systems with time-dependent nonlinear nonholonomic constraints and time-dependent Lagrangian. For discrete open systems, the time-dependent nonlinear constraint is associated with the rate of internal entropy production of the system. We show that this constraint on the solution curve systematically yields a constraint on the variations to be used in the action functional. The proposed variational formulation is intrinsic and provides the same structure for a wide class of discrete open systems. We illustrate our theory by presenting examples of open systems experiencing mechanical interactions, as well as internal diffusion, internal heat transfer, and their cross-effects. Our approach yields a systematic way to derive the complete evolution equations for the open systems, including the expression of the internal entropy production of the system, independently on its complexity. It might be especially useful for the study of the nonequilibrium thermodynamics of biophysical systems.

NONCOMMUTATIVE KdV HIERARCHY

2007 ◽  
Vol 19 (07) ◽  
pp. 677-724 ◽  
Author(s):  
FRANÇOIS TREVES
Keyword(s):  
Evolution Equations ◽  
Vries Equation ◽  
Kdv Equation ◽  
Completely Integrable Systems ◽  
Maximal Abelian Subalgebra ◽  
Abelian Subalgebra ◽  
Kdv Hierarchy ◽  
Noncommutative Version ◽  
Constants Of Motion ◽  
Hamiltonian Evolution

The noncommutative version of the Korteweg–de Vries equation studied here is shown to admit infinitely many constants of motion and to give rise to a hierarchy of higher-order Hamiltonian evolution equations, each one the noncommutative version of the commutative KdV equation of the same order. The noncommutative KdV polynomials span, topologically, a maximal Abelian subalgebra of the Lie algebra of noncommutative Bäcklund transformations. Two classes of examples of "completely integrable" systems of evolution equations to which the theory applies are described in the last two sections.

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