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Hopf solitons in modern mathematical physics

# Hydrodynamics

## The model description

The Euler equations of motion of an ideal liquid: (1.1) (1.2) (1.3) The potential of body forces (1.4) Consider a barotropic fluid flow (1.5) i.e. density of the liquid depends only on the pressure (1.6) We introduce the definition for vorticity (1.7) In 1961 it was shown [J.-J. Moreau, Comptes rendus de l'Académie des Sciences 252, 2810 (1961) copy] that for the equations (1.1)-(1.3) with condition (1.5), the integral named helicity (1.8) (for the flow with good boundary conditions at infinity) remains a constant (1.9) Indeed, it is easy to verify the identity (1.10) by substituting the time-derivatives of the velocity V from (1.1)-(1.3). From (1.8) and (1.10) it follows that if the motion is localized in space (the velocity decreases rapidly at infinity), then (1.9).

Later, the same conclusions were reached by Moffat [H.K. Moffat "The degree of knottedness of tangled vortex lines", J. Fluid Mech. 35 (1969), 117 ref]. He also described the meaning of the topological invariant (1.8) although apparently he did not know about the direct relationship with the Hopf fibration [H.Hopf "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche", Math. Annalen 104 (1931), 637 ref] and about Whitehead expression [J.H.C. Whitehead "An expression of Hopf’s invariant as an integral", Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 117 ref].

Direct indication that the flow with nonzero H are hopfions has been done in [E.A.Kuznetsov, A.V. Mikhailov "On the topological meaning of canonical Clebsch variables", Phys. Lett. A 77 (1980), 37 ref]. In this work has been defined a direct relationship between the vectors and the n-field and the Clebsch variables; (1.11) A topological invariant H follows directly from the Euler equations, but for the physically correct solution of the system, we need an additional relationship - continuity equation: (1.12) ## The results of recent years

In [E. A. Kuznetsov, V. P. Ruban "Collapse of vortex lines in hydrodynamics", JETP 91 (2000), 775 ref] reported that one of the simplest initially hopfion flow of an incompressible fluid (2.1) leads to the collapse of type (2.2) ## Still unsolved problems

1. Analytical localized solution in 3D for V(x,y,z,t) and p(x,y,z,t) for set of equations (1.1)-(1.3),(1.12), with realistic Π and ρ(p) functions; continuous and free from any singularities in full space in initial time. Especially hopfions, i.e. with non-zero H integral.
2. Good numerical simulation of the evolution from the initial hopfion configuration.
3. Problems of the collapse for the hopfion flows.

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