The generalized Kupershmidt deformation for constructing new integrable systems from integrable bi-Hamiltonian systems

Based on the Kupershmidt deformation for any integrable bi-Hamiltonian systems presented in [4], we propose the generalized Kupershmidt deformation to construct new systems from integrable bi-Hamiltonian systems, which provides a nonholonomic perturbation of the bi-Hamiltonian systems. The generalized Kupershmidt deformation is conjectured to preserve integrability. The conjecture is verified in a few representative cases: KdV equation, Boussinesq equation, Jaulent-Miodek equation and Camassa-Holm equation. For these specific cases, we present a general procedure to convert the generalized Kupershmidt deformation into the integrable Rosochatius deformation of soliton equation with self-consistent sources, then to transform it into a $t$-type bi-Hamiltonian system. By using this generalized Kupershmidt deformation some new integrable systems are derived. In fact, this generalized Kupershmidt deformation also provides a new method to construct the integrable Rosochatius deformation of soliton equation with self-consistent sources.Comment: 21 pages, to appear in Journal of Mathematical Physic

Tunable coupled-mode dispersion compensation and its application to on-chip resonant four-wave mixing

We propose and demonstrate localized mode coupling as a viable dispersion engineering technique for phase-matched resonant four-wave mixing (FWM). We demonstrate a dual-cavity resonant structure that employs coupling-induced frequency splitting at one of three resonances to compensate for cavity dispersion, enabling phase-matching. Coupling strength is controlled by thermal tuning of one cavity enabling active control of the resonant frequency-matching. In a fabricated silicon microresonator, we show an 8 dB enhancement of seeded FWM efficiency over the non-compensated state. The measured four-wave mixing has a peak wavelength conversion efficiency of -37.9 dB across a free spectral range (FSR) of 3.334 THz ($\sim$27 nm). Enabled by strong counteraction of dispersion, this FSR is, to our knowledge, the largest in silicon to demonstrate FWM to date. This form of mode-coupling-based, active dispersion compensation can be beneficial for many FWM-based devices including wavelength converters, parametric amplifiers, and widely detuned correlated photon-pair sources. Apart from compensating intrinsic dispersion, the proposed mechanism can alternatively be utilized in an otherwise dispersionless resonator to counteract the detuning effect of self- and cross-phase modulation on the pump resonance during FWM, thereby addressing a fundamental issue in the performance of light sources such as broadband optical frequency combs

Deriving N-soliton solutions via constrained flows

The soliton equations can be factorized by two commuting x- and t-constrained flows. We propose a method to derive N-soliton solutions of soliton equations directly from the x- and t-constrained flows.Comment: 8 pages, AmsTex, no figures, to be published in Journal of Physics

Constructing N-soliton solution for the mKdV equation through constrained flows

Based on the factorization of soliton equations into two commuting integrable x- and t-constrained flows, we derive N-soliton solutions for mKdV equation via its x- and t-constrained flows. It shows that soliton solution for soliton equations can be constructed directly from the constrained flows.Comment: 10 pages, Latex, to be published in "J. Phys. A: Math. Gen.

Ultra-low-loss CMOS-Compatible Waveguide Crossing Arrays Based on Multimode Bloch Waves and Imaginary Coupling

We experimentally demonstrate broadband waveguide crossing arrays showing ultra low loss down to $0.04\,$dB/crossing ($0.9\%$), matching theory, and crosstalk suppression over $35\,$dB, in a CMOS-compatible geometry. The principle of operation is the tailored excitation of a low-loss spatial Bloch wave formed by matching the periodicity of the crossing array to the difference in propagation constants of the 1$^\text{st}$- and 3$^\text{rd}$-order TE-like modes of a multimode silicon waveguide. Radiative scattering at the crossing points acts like a periodic imaginary-permittivity perturbation that couples two supermodes, which results in imaginary (radiative) propagation-constant splitting and gives rise to a low-loss, unidirectional breathing Bloch wave. This type of crossing array provides a robust implementation of a key component enabling dense photonic integration

BsB_s Semileptonic Decays to DsD_s and Ds∗D_s^* in Bethe-Salpeter Method

Using the relativistic Bethe-Salpeter method, the electron energy spectrum and the semileptonic decay widths of $B^0_s\to D^-_s \ell^+{\nu_\ell}$ and $B^0_s\to D_s^{*-}\ell^+{\nu_\ell}$ are calculated. We obtained large branching ratios, $Br(B_s\to D_se\nu_e)=(2.85\pm0.35)$ and $Br (B_s\to D_s^*e\nu_e)=(7.09\pm0.88)$, which can be easily detected in the future experiment.Comment: 3 pages, 3 figures

Higher Order Potential Expansion for the Continuous Limits of the Toda Hierarchy

A method for introducing the higher order terms in the potential expansion to study the continuous limits of the Toda hierarchy is proposed in this paper. The method ensures that the higher order terms are differential polynomials of the lower ones and can be continued to be performed indefinitly. By introducing the higher order terms, the fewer equations in the Toda hierarchy are needed in the so-called recombination method to recover the KdV hierarchy. It is shown that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda hierarchy tend towards the corresponding ones of the KdV hierarchy in continuous limit.Comment: 20 pages, Latex, to be published in Journal of Physics

B\"{a}cklund transformations for high-order constrained flows of the AKNS hierarchy: canonicity and spectrality property

New infinite number of one- and two-point B\"{a}cklund transformations (BTs) with explicit expressions are constructed for the high-order constrained flows of the AKNS hierarchy. It is shown that these BTs are canonical transformations including B\"{a}cklund parameter $\eta$ and a spectrality property holds with respect to $\eta$ and the 'conjugated' variable $\mu$ for which the point $(\eta, \mu)$ belongs to the spectral curve. Also the formulas of m-times repeated Darboux transformations for the high-order constrained flows of the AKNS hierarchy are presented.Comment: 21 pages, Latex, to be published in J. Phys.