WIGNER FUNCTION APPROACH TO OSCILLATING SOLUTIONS OF THE 1D-QUINTIC NONLINEAR SCHR ¨ ODINGER EQUATION ALEX MAHALOV AND SERGEI K. SUSLOV Abstract. We study oscillating solutions of the 1D-quintic nonlinear Schr¨ odinger equation with the help of Wigner’s quasiprobability distribution in quantum phase space. An “absolute squeezing property”, namely a periodic in time total localization of wave packets at some finite spatial points without violation of the Heisenberg uncertainty principle, is analyzed in this nonlinear model. As is widely known, mean-field theory is very successful in description of both static and dynamic properties of Bose-Einstein condensates [1], [2]. The macroscopic wave function obeys a 3D-cubic nonlinear Schr¨ odinger equation. At the same time, there are several reasons to consider higher order nonlinearity in the Gross–Pitaevskii model [2]. The quintic case is of particular importance because near Feshbach resonance one may turn the scattering length to zero when the dominant interaction among atoms is due to three-body effects (see [3], [4], [5], [6], [7], [8] and the references therein; in 7 Li-condensate, for example, the scattering length is reported as small as 0.01 Bohr radii [8]). Then the nonlinear term in the mean-field equation has the quintic form. Another examples include a 1D-Bose gas in the limit of impenetrable particles [9], [10], [11] and collapse of a plane Langmuir soliton in plasma [12], [13]. A finite time blow up of solutions of the unidimensional quintic nonlinear Schr¨ odinger equation is studied in many publications (see, for example, [13], [14], [15], [16], [17], [18], [19], [20]). This case is critical because any decrease of the power of nonlinearity results in the global existence of solutions [21], [22] (see also [10] and [23]). Related hidden symmetry, explicit oscillating and blow up solutions, the uncertainty relation and squeezing from the viewpoint of Wigner’s function approach are topics discussed in this Letter. 1. Symmetry Group The quintic derivative nonlinear Schr¨ odinger equation in a parabolic confinement, iψ t + ψ xx − x 2 ψ = ig ( |ψ| 2 ψ x + ψ 2 ψ * x ) + h |ψ| 4 ψ (1.1) (g and h are constants), is invariant under the following change of variables: ψ (x,t)= √ β (0) |z (t)| e i(α(t)x 2 +δ(t)x+κ(t)) χ (β (t) x + ε (t) , −γ (t)) (1.2) Date : January 1, 2013. 1991 Mathematics Subject Classification. Primary 81Q05, 35C05. Secondary 42A38. Key words and phrases. One dimensional quintic nonlinear Schr¨ odinger equation, traveling wave and blow up solutions, Wigner function, Heisenberg uncertainty relation, Tonks–Girardeau gas of impenetrable bosons. 1