We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg--de Vries, equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg--de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.Comment: 25 pages, 24 figures, slightly revised and corrected versio

Atomic Bose-Einstein condensates confined to a dual-ring trap support Josephson vortices as topologically stable defects in the relative phase. We propose a test of the scaling laws for defect formation by quenching a Bose gas to degeneracy in this geometry. Stochastic Gross-Pitaevskii simulations reveal a -1/4 power-law scaling of defect number with quench time for fast quenches, consistent with the Kibble-Zurek mechanism. Slow quenches show stronger quench-time dependence that is explained by the stability properties of Josephson vortices, revealing the boundary of the Kibble-Zurek regime. Interference of the two atomic fields enables clear long-time measurement of stable defects and a direct test of the Kibble-Zurek mechanism in Bose-Einstein condensation.

We investigate the dynamics of turbulent flow in a two-dimensional trapped Bose-Einstein condensate by solving the Gross-Pitaevskii equation numerically. The development of the quantum turbulence is activated by the disruption of an initially embedded vortex quadrupole. By calculating the incompressible kinetic-energy spectrum of the superflow, we conclude that this quantum turbulent state is characterized by the Kolmogorov-Saffman scaling law in the wave-number space. Our study predicts the coexistence of two subinertial ranges responsible for the energy cascade and enstrophy cascade in this prototype of two-dimensional quantum turbulence.

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We propose a scheme for generating mesoscopic superpositions of two distinguishable pair coherent states of motion in a two-dimensional ion trap. In our scheme, the trapped ion is excited bichromatically by five laser beams along different directions in the X-Y plane of the ion trap. Four of these have the same frequency and can be derived from the same source, reducing the demands on the experimentalist. We show that if the initial vibrational state is given by a two-mode Fock state, as demonstrated in recent experiments, these highly correlated two-mode ''Schrödinger cat'' states are realized when the system reaches its steady state, which is indicated by the extinction of the fluorescence emitted by the ion. ͓S1050-2947͑96͒02611-X͔ PACS number͑s͒: 42.50. Vk, 42.50.Dv, 32.80.Pj Ever since Schrödinger suggested his famous cat experiment in 1935 ͓1͔, superpositions of macroscopically distinguishable quantum states ͑which are also known as Schrö-dinger cat states͒ have become a longstanding exemplar of the peculiarities which occur in the interpretation of quantum reality. To explore this subtlety and to gain insight into the fundamental issues of quantum theory, a number of schemes have been proposed for the realization of such states. In quantum optics, the Schrödinger cat states are usually described as superpositions of different coherent states ͓2͔, as coherent states are the closest quantum states to a classical description of the field of definite complex amplitude. Specifically, the archetype of a Schrödinger cat state is given by the superposition ͉⌿͘ cat ϭN͓͉␣͘ϩexp(i)͉Ϫ␣͘], where ͉␣͘ is a coherent state of the single-mode quantized field and N is a normalization coefficient. In particular, these states are referred to the even, odd, and Yurke-Stoler ͓3͔ coherent states when ϭ0,, and /2, respectively. They have been extensively studied and shown to exhibit nonclassical properties such as squeezing and sub-Poissonian statistics ͓2͔.In a recent paper Gerry and Grobe ͓4͔ have proposed a two-mode generalization of Schrödinger cat states which are defined as superpositions of two different pair coherent states ͑PCS͒. For two independent boson annihilation operators â and b , a pair coherent state ͉,q͘ is defined as an eigenstate of both the pair annihilation operator â b and the number difference operator Q ϭâwhere is a complex number and q is the ''charge'' which is a fixed integer. Without loss of generality, we may set qу0 and the PCS can be explicitly expanded as a superposition of the two-mode Fock states, i.e.,where N q ϭ͓͉͉ Ϫq I q (2͉͉)͔ Ϫ1/2 is the normalization constant (I q is the modified Bessel function of the first kind of order q). Pair coherent states are regarded as an important type of correlated two-mode state with prominent nonclassical properties such as sub-Poissonian statistics, strong intermode correlation in the number fluctuations, squeezing of quadrature variances, and violations of Cauchy-Schwarz inequalities ͓5͔.The correlated two-mode Schrödinger cat states ͉,q,͘ are defined as...