Hydrodynamic Equations for the Toda Lattice

Hydrodynamic equations for the Ablowitz-Ladik discretization of the nonlinear Schroedinger equation

“…A numerical simulation of the integrable chain is reported in [20]. As we will see the AL lattice equations are structurally rather similar to the classical Toda lattice, for which fairly detailed notes are available [21]. When pointing out such similarity we merely refer to these notes.…”

“…In the case of the Toda lattice, see [21], Section 2, the Lax matrix is self-adjoint and its eigenvalues are real. Under the Toda dynamics the Lax matrix satisfies d dt L N " rB N , L N s with B N a related antisymmetric matrix.…”

“…For V " 0, one readily obtains F AL pP, 0q " logpP {πq with log intensity νpP q " P ´1 ą 0. Hence there is no high pressure phase as known for the Toda lattice, see [21], Section 8, and [30]. Thermal equilibrium corresponds to V pwq " β cos w with β the inverse temperature.…”

Soliton gas in integrable dispersive hydrodynamics

“…Finally, the recently emerged intriguing parallels between the spectral theory of soliton gas and generalised hydrodynamics [26], [118] provide yet another avenue for the novel crossdisciplinary research that could be of significant benefit for both dispersive hydrodynamics and quantum theory of many-body systems.…”

“…We note that in a different context, the collision rate assumption is at heart of the generalised hydrodynamics (GHD), a large-scale theory for the dynamics of quantum manybody integrable systems. In GHD the pointlike quasiparticles are subject to the instantaneous velocity-dependent spatial shifts upon colliding (see [19], [27], [28], [26], [118] and references therein). It is important to stress, however, that the validity of (2.12) in the context of classical soliton gases, although intuitively suggestive, is far from being obvious.…”