Solitons in relativistic mean field models

Assuming that the nucleus can be treated as a perfect fluid we study the conditions for the formation and propagation of Korteweg–de Vries (KdV) solitons in nuclear matter. The KdV equation is obtained from the Euler and continuity equations in nonrelativistic hydrodynamics. The existence of these solitons depends on the nuclear equation of state, which, in our approach, comes from well-known relativistic mean field models. We reexamine early works on nuclear solitons, replacing the old equations of state by new ones, based on QHD and on its variants. Our analysis suggests that KdV solitons may indeed be formed in the nucleus with a width which, in some cases, can be smaller than one Fermi.     

Shock wave formation in hot nuclear matter

Assuming that nuclear matter can be treated as a perfect fluid, we study the propagation of perturbations in the baryon density at high temperature. The equation of state is derived from the non-linear Walecka model. The expansion of the Euler and continuity equations of relativistic hydrodynamics around equilibrium configurations lead to the breaking wave equation for the density perturbation. We solve it numerically for this perturbation and follow the propagation of the initial pulses.     

Anomalous hydrodynamics of fractional quantum Hall states

We propose a comprehensive framework for quantum hydrodynamics of the fractional quantum Hall (FQH) states. We suggest that the electronic fluid in the FQH regime can be phenomenologically described by the quantized hydrodynamics of vortices in an incompressible rotating liquid. We demonstrate that such hydrodynamics captures all major features of FQH states, including the subtle effect of the Lorentz shear stress. We present a consistent quantization of the hydrodynamics of an incompressible fluid, providing a powerful framework to study the FQH effect and superfluids. We obtain the quantum hydrodynamics of the vortex flow by quantizing the Kirchhoff equations for vortex dynamics.     

On the fluid approximation to a nonlinear Schrödinger equation

We present a heuristic proof that the nonlinear Schrödinger equation (NLS) - in 2 + 1 dimensions has a family of solutions which can be well approximated by a collection of point vortices for a planar incompressible fluid. The novelty of our approach is that we begin with a representation of the NLS as a compressible perturbation of Euler's equations for hydrodynamics.     

Vortex structures with complex points singularities in two-dimensional Euler equations. New exact solutions

In this work we present a new class of exact stationary solutions for two-dimensional (2D) Euler equations. Unlike already known solutions, the new ones contain complex singularities. We consider point singularities which have a vector field index greater than 1 as complex. For example, the dipole singularity is complex because its index is equal to 2. We present in explicit form a large class of exact localized stationary solutions for 2D Euler equations with a singularity whose index is equal to 3. The solutions obtained are expressed in terms of elementary functions. These solutions represent a complex singularity point surrounded by a vortex satellite structure. We also discuss the motion equation of singularities and conditions for singularity point stationarity which provide the stationarity of the complex vortex configuration.     

Statistical equilibrium solutions of the shallow water equations

A statistical method for calculating equilibrium solutions of the shallow water equations, a model of essentially 2D fluid flow with a free surface, is described. The model contains a competing acoustic turbulent direct energy cascade, and a 2D turbulent inverse energy cascade. It is shown, nonetheless that, just as in the corresponding theory of the inviscid Euler equation, the infinite number of conserved quantities constrains the flow sufficiently to produce nontrivial large-scale vortex structures which are solutions to a set of explicitly derived coupled nonlinear partial differential equations.     

Nonlinear waves in self-gravitating compressible fluid and generalized Cole-Hopf substitutions

A general representation of solutions is constructed for one-dimensional flows of viscous compressible fluid, which allows their exact solutions to be found. A general representation of solutions to Euler equations for three-dimensional compressible fluid flows is found. By analogy, a representation is constructed for flows in a uniformly rotating frame of reference.     

Hydrodynamic impulse in a compressible fluid

A suitable expression for hydrodynamic impulse in a compressible fluid is deduced. The development of appropriate impulse formulation for compressible Euler equations confirms the propriety of the hydrodynamic impulse expression for a compressible fluid given here. Implications of the application of this formulation to a compressible vortex ring are pointed out. Extension of Benjamin's variational characterization of a moving axisymmetric vortex system to a compressible fluid is explored.     

Differential-geometric structures associated with the Lagrangian and a nonvariational interpretation of the Euler equations and Nöther’s theorem

Differential-geometry structures associated with Lagrangians are studied. A relative invariant E embraced by an extension of fundamental object is constructed (in the paper, E is referred to as the Euler relative invariant) such that the equation E = 0 is an invariant representation of the Euler equation for the variational functional. For this reason, a nonvariational interpretation of the Euler equations becomes possible, because the Euler equations need not be connected with the variational problem, and one can regard the equations from the very beginning as an equation arising when equating the Euler relative invariant to zero. Local diffeomorphisms between two structures associated with Lagrangians are also discussed. The theorem concerning conditions under which the vanishing condition for the Euler relative invariant of one of these structures leads to vanishing of the Euler invariant relative of the other structure can be treated as a nonvariational interpretation of Nöther’s theorem.     

Numerical solution of under-resolved detonations

A new fractional-step method is proposed for the numerical solution of high speed reacting flows, where the chemical time scales are often much smaller than the fluid dynamical time scales. When the problem is stiff, because of insufficient spatial/temporal resolution, a well-known spurious numerical phenomenon occurs in standard finite volume schemes: the incorrect calculation of the speed of propagation of discontinuities. The new method is first illustrated considering a one-dimensional scalar hyperbolic advection/reaction equation with stiff source term, which may be considered as a model problem to under-resolved detonations. During the reaction step, the proposed scheme replaces the cell average representation with a two-value reconstruction, which allows us to locate the discontinuity position inside the cell during the computation of the source term. This results in the correct propagation of discontinuities even in the stiff case. The method is proved to be second-order accurate for smooth solutions of scalar equations and is applied successfully to the solution of the one-dimensional reactive Euler equations for Chapman–Jouguet detonations.     

Finite time analyticity for the two and three dimensional Kelvin-Helmholtz instability

The well-posed property for the finite time vortex sheet problem with analytic initial data was first conjectured by Birkhoff in two dimensions and is shown here to hold both in two and three dimensions. Incompressible, inviscid and irrotational flow with a velocity jump across an interface is assumed. In two dimensions, global existence of a weak solution to the Euler equation with such initial conditions is established. In three dimensions, a Lagrangian representation of the vortex sheet analogous to the Birkhoff equation in two dimensions is presented.This work was performed while C.B. was visiting the Dept. de Mathématiques, Nice     

Stochastic methods in quantum field theory and hydrodynamics

We give a review of recent work in quantum field theory and hydrodynamics, in which methods of stochastic analysis, in particular of stochastic equations, are used. The review includes the discussion of the canonical formalism for models of quantum fields as well as the discussion of statistical solutions of the Euler equations for the motion of an inviscid fluid.     

Topology-Preserving Diffusion of Divergence-Free Vector Fields and Magnetic Relaxation

The usual heat equation is not suitable to preserve the topology of divergence-free vector fields, because it destroys their integral line structure. On the contrary, in the fluid mechanics literature, one can find examples of topology-preserving diffusion equations for divergence-free vector fields. They are very degenerate since they admit all stationary solutions to the Euler equations of incompressible fluids as equilibrium points. For them, we provide a suitable concept of “dissipative solutions”, which shares common features with both P.-L. Lions’s dissipative solutions to the Euler equations and the concept of “curves of maximal slopes”, à la De Giorgi, recently used to study the scalar heat equation in very general metric spaces. We show that the initial value problem admits such global solutions, at least in the two space variable case, and they are unique whenever they are smooth.