The algebraic method is developed to obtain new exact solutions, including stationary wave solutions and traveling wave solutions, for the cubic-quintic nonlinear Schrödinger (NLS) equation. Specifically, we present two general solution formulae, which degenerate to the corresponding solution of the cubic NLS equation, when the quintic nonlinear term is absent. It is expected that they are useful in correlative physics fields.

We prove the existence of a weak solution to a two-dimensional resonant 3×3 system of conservation laws with $BV$ initial data. Due to possible resonance (coinciding eigenvalues), spatial $BV$ estimates are in general not available. Instead, we use an entropy dissipation bound combined with the time translation invariance property of the system to prove existence based on a two-dimensional compensated compactness argument adapted from [E. Tadmor, M. Rascle, and P. Bagnerini, J. Hyperbolic Differ. Equ., 2(3), 697-712, 2005]. Existence is proved under the assumption that the flux functions in the two directions are linearly independent.

We propose and analyze discretization methods for solving finite systems of nonlinearly coupled Schrödinger equations, which arise as an asymptotic limit of the three-dimensional Gross-Pitaevskii equation with strongly anisotropic potential. A pseudo-spectral method with Hermite basis functions combined with a Crank-Nicolson type method is introduced. Numerical experiments are presented, including a comparison with an alternative discretization approach.

The alignment algorithm of Zhang and Zha is an effective method recently proposed for nonlinear manifold learning (or dimensionality reduction). By first computing local coordinates of a data set, it constructs an alignment matrix from which a global coordinate is obtained from its null space. In practice, the local coordinates can only be constructed approximately and so is the alignment matrix. This together with roundoff errors requires that we compute the the eigenspace associated with a few smallest eigenvalues of an approximate alignment matrix. For this purpose, it is important to know the first nonzero eigenvalue of the alignment matrix or a lower bound in order to computationally separate the null space. This paper bounds the smallest nonzero eigenvalue, which serves as an indicator of how difficult it is to correctly compute the desired null space of the approximate alignment matrix.

This paper deals with the statistical modeling of turbulent two-phase flows consisting of particles or droplets immersed in a gas. The problem of gaseous turbulence alone being very complex, we concentrate here on the simpler case of an a priori given forced isotropic homogeneous turbulence acting on the particles, whose mean square velocity and integral Lagrangian time-scale are given constants. Our main objective is to derive a hierarchy of reduced models from the joint particle-gas pdf (probability density function). The latter equation may therefore be regarded as a master equation for our problem. The reduced models describe the dispersion of a cloud of particles observed at di®erent time scales compared to the dynamic response time of the particles and the characteristic time scale of the turbulence along their trajectories. These derivations rely on very classical Chapman-Enskog expansions. We recover in particular the result of Tchen [C. M. Tchen, /i>"Mean value and correlation problems connected with the motion of small particles suspended in a turbulent fluid, PhD thesis, Delft, The Hague, Martinus Nijhoff, 1947] stating that the diffusion rate is the same for small or large particles in homogeneous turbulence, under the assumption that the lagrangian statistical properties along their paths are the same. Moreover, our approach allows us to prove that the long-time limit of the joint particle-gas distribution function is a bi-maxwellian distribution, whatever the size of the particles. This is consistent with some usual assumptions made in the literature for the derivation of particle collision models [J. Laviéville, E. Deutsch and O. Simonin, Large eddy simulation of interactions between colliding particles and a homogeneous isotropic turbulence field, Gas-Solid Flows, ASME, 228, 347-358, 1995], [Leonid I. Zaichik, Olivier Simonin and Vladimir M. Alipchenkov, Two statistical models for predicting collision rates of inertia particles in homogeneous isotropic turbulence, Phys. Fluids, 15(10), 2995-3005, 2003].

We derive coupled mass and energy balance laws from a High-Field limit of thermostatted Boltzmann equations. The starting point is a Boltzmann equation for elastic collisions subjected to a large force field. By adding a thermostat correction, it is possible to expand the solutions about a High-Field equilibrium obtained when balancing the thermostatted field drift operator with the elastic collision operator. To this aim, a hydrodynamic type scaling of the thermostatted Boltzmann equation is used, considering that the leading 'collision operator' actually consists of the combination of the thermostatted field operator and of the elastic collision operator. At leading order in the Knudsen number, the resulting model consist of coupled nonlinear first order partial differential equations. We investigate two cases. The first one is based on a one-dimensional BGK-type operator. The second one is three dimensional and concerns a Fokker-Planck collision operator. In both cases, we show that the resulting models are hyperbolic, thereby indicating that they might be appropriate for a use in physically realistic situations.

Modular multiplication plays an important role to several public-key cryptosystems such as the RSA cryptosystem. This paper proposes an efficient modulo $p$ multiplication algorithm with moderate factors of $p$+1 and $p$-1. In order to improve the RSA decryption performance, users can utilize our proposed algorithm and the strong prime criterion. It will prove that the decryption method based on our proposed algorithm can run at a speed almost 6.5 times faster than that of the traditional method, or almost 2 times faster than that of the method based on the Chinese Remainder Theorem. Furthermore, the proposed algorithm can greatly enhance the performance of RSA encryption.

We prove well-posedness for the initial value problem for a vortex sheet in 3D fluids, in the presence of surface tension. We first reformulate the problem by making a favorable choice of variables and parameterizations. We then perform energy estimates for the evolution equations. It is important to note that the Kelvin-Helmholtz instability is present for the vortex sheet in the absence of surface tension. Accordingly, we must construct the energy functional carefully with an eye toward the regularization of this instability. Well-posedness follows from the estimates.

We consider a Brownian ratchet model where the particle on the ratchet is coupled to a cargo. We show that in a distinguished limit where the diffusion coefficient of the cargo is small, and the amplitude of thermal fluctuations is small, the system becomes completely coherent: the times at which the particle jumps across the teeth of the ratchet become deterministic. We also show that the dynamics of the ratchet-cargo system do not depend on the fine structure of the Brownian ratchet. These results are relevant in the context of molecular motors transporting a load, which are often modeled as a ratchet-cargo compound. They explain the regularity of the motor gait that has been observed in numerical experiments, as well as justify the coarsening into Markov jump processes which is commonly done in the literature.

Viscous stabilizations of the quantum hydrodynamic equations are studied. The quantum hydrodynamic model consists of the conservation laws for the particle density, momen- tum, and energy density, including quantum corrections from the Bohm potential. Two different stabilizations are analyzed. First, viscous terms are derived using a Fokker-Planck collision operator in the Wigner equation. The existence of solutions (with strictly positive particle density) to the isothermal, stationary, one-dimensional viscous model for general data and nonhomogeneous boundary conditions is shown. The estimates depend on the viscosity and do not allow to perform the inviscid limit. Second, the numerical viscosity of the second upwind finite-difference discretization of the inviscid quantum hydrodynamic model is computed. Finally, numerical simulations using the non-isothermal, stationary, one-dimensional model of a resonant tunnelling diode show the influence of the viscosity on the solution.

HMM-like multiscale integrators and projective integration methods are two different types of multiscale integrators which have been introduced to simulate efficiently systems with widely disparate time scales. The original philosophies of these methods, reviewed here, were quite different. Recently, however, projective integration methods seem to have evolved in a way that make them increasingly similar to HMM-integrators and quite different from what they were originally. Nevertheless, the strategy of extrapolation which was at the core of the original projective integration methods has its value and should be extended rather than abandoned. An attempt in this direction is made here and it is shown how the strategy of extrapolation can be generalized to stochastic dynamical systems with multiple time scales, in a way reminiscent of Chorin’s artificial compressibility method and the Car-Parrinello method used in molecular dynamics. The result is a seamless integration scheme, i.e. one that does not require knowing explicitly what the slow and fast variables are.