We consider the cubic nonlinear Schrödinger (NLS) equation as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension. We prove that for each s > -1/2 there exists a conserved energy which is equivalent to the Hs-norm of the solution. For the Korteweg-de Vries (KdV) equation, there is a similar conserved energy for every s ≥-1.

We present results for the moments of observed spectra in inclusive semileptonic B-meson decays to charm hadrons B̄→Xc -ν̄. Moments of the hadronic-mass and the combined mass-and-energy spectra for different minimum electron or muon momenta between 0.8 and 1.9GeV/c are obtained from a sample of 232×106 Υ(4S)→BB̄ events, collected with the BABAR detector at the PEP-II asymmetric-energy B-meson factory at SLAC. We also present a reevaluation of the moments of electron-energy spectra and partial decay fractions B(B̄→Xce-ν̄) for minimum electron momenta between 0.6 and 1.5GeV/c based on a sample of 51×106 Υ(4S)→BB̄ events. The measurements are used for the extraction of the total decay fraction, the Cabibbo-Kobayashi-Maskawa (CKM) matrix element |Vcb|, the quark masses mb and mc, and four heavy-quark QCD parameters in the framework of a Heavy-Quark Expansion (HQE). We find B(B̄→Xc -ν̄)= (10.64±0.17±0.06)% and |Vcb|=(42.05±0.45±0.70) ×10-3. © 2010 The American Physical Society.

We have performed a search for the rare leptonic decay B+ --> mu+ nu_mu with data collected at the Upsilon(4S) resonance by the BABAR experiment at the PEP-II storage ring. In a sample of 88.4 million BBbar pairs, we find no significant evidence for a signal and set an upper limit on the branching fraction BR(B+ --> mu+ nu_mu) < 6.6 x 10^{-6} at the 90% confidence level.

We report a measurement of the first four moments of the hadronic mass distribution in [Formula Presented] decays. The measurements are based on 89 million [Formula Presented] events where the hadronic decay of one of the B mesons is fully reconstructed and a charged lepton from the decay of the other B meson is identified. The moments are presented for minimum lepton momenta ranging from 0.9 to 1.6 GeV in the B rest frame. It is expected that such measurements will lead to improved determinations of [Formula Presented] and [Formula Presented] © 2004 The American Physical Society.

We study the rare B meson decays B±,0→J/ψK+K-K±,0, B±,0→J/ψφK±,0, and search for B0→J/ψφ, using 469×106 BB¯ events collected at the γ(4S) resonance with the BABAR detector at the PEP-II e+e- asymmetric energy collider. We present new measurements of branching fractions and a study of the J/ψφ mass distribution in search of new charmonium-like states. In addition, we search for the decay B0→J/ψφ and find no evidence of a signal.

We perform a time-dependent Dalitz plot analysis of B0 → K0 S π+π- in order to extract the CP violation parameters of f0(980)K0 S and ρ0(770)K0 S and direct CP-asymmetries of Kz.ast;+(892)π-. The results are obtained from a data sample of (383 ± 3) × 106 BB decays, collected with the BABAR detector at the PEP-II asymmetric-energy B factory at SLAC. The measured values of 2βeff in B0 decays to f0(980)K0S and ρ0(770)K0 S are (89+22-20 ± 5 ±8) and (37+19 -17 ± 5 ±6), respectively, where the first quoted uncertainty is statistical, the second is systematic and the third is Dalitz plot signal model uncertainty. We measure the significance of 2βeff (f0(980)K0S) 6= 0 to be 4.3 σ. In decays to Kz.ast;(892)π we find ACP = -0.18 ± 0.10 ± 0.03 ± 0.03. The measured phase difference between the decay amplitudes of B0 → Kz.ast;+(892)π- and B0 → Kz.ast;-(892)π+ is (-164 ± 24 ± 12 ± 15). All results are preliminary.

We report the result of a search for the rare decay B0→γγ in 426fb-1 of data, corresponding to 226×106 B0B ̄0 pairs, collected on the Υ(4S) resonance at the PEP-II asymmetric-energy e+e- collider using the BABAR detector. We use a maximum likelihood fit to extract the signal yield and observe 21-12+13 signal events with a statistical significance of 1.8σ. This corresponds to a branching fraction B(B0→γγ)=(1.7±1.1(stat.) ±0.2(syst.))×10-7. Based on this result, we set a 90% confidence level upper limit of B(B0→γγ)<3.2×10 -7. © 2011 American Physical Society.

We present a measurement of the B0→Λ̄pπ- branching fraction performed using the BABAR detector at the PEP-II asymmetric e+e- collider. Based on a sample of 467×106 BB̄ pairs we measure B(B0→Λ̄pπ-)=[3.07±0.31(stat)±0.23(syst)] ×10-6. The measured differential spectrum as a function of the dibaryon invariant mass m(Λ̄p) shows a near-threshold enhancement similar to that observed in other baryonic B decays. We study the Λ̄ polarization as a function of Λ̄ energy in the B0 rest frame (EΛ̄*) and compare it with theoretical expectations of fully longitudinally right-polarized Λ̄ at large EΛ̄*. © 2009 The American Physical Society.

We report on the inclusive branching fractions of B- and of B̄0 mesons decaying to D0X, D̄0X, D+X, D-X, Ds+X, Ds-X, Λc+X, Λc-X, based on a sample of 88.9 × 106 BB̄ events recorded with the BABAR detector at the Υ (4S) resonance. Events are selected by completely reconstructing one B and searching for a reconstructed charmed particle in the rest of the event. We measure the number of charmed and of anticharmed particles per B decay and derive the total charm yield per B - decay nc- = 1.313 ± 0.037 ± 0.062-0.042+0.063 and per B̄0 decay n c0 = 1.276 ± 0.062 ± 0.058 -0.046+0.066, where the first uncertainty is statistical, the second is systematic, and the third reflects the charm branching-fraction uncertainties.