Asymptotic behavior of solutions to the coagulation-fragmentation equations. II. Weak fragmentation

The discrete coagulation-fragmentation equations are a model for the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones. The assumptions made on the fragmentation coefficients have the physical interpretation that surface effects are important. Our results on the asymptotic behavior of solutions generalize the corresponding results of Ball, Carr, and Penrose for the Becker-Doring equation.Dedicated to Oliver Penrose on the occasion of his 65th birthday     

Existence and uniqueness of solutions to superdifferential equations

We state and prove the theorem of existence and uniqueness of solutions to ordinary superdifferential equations on supermanifolds. It is shown that any supervector field, X = X₀ + X₁, has a unique integral flow, Г: ^1¦1 x (M, A_M) → (M, A_M), satisfying a given initial condition. A necessary and sufficient condition for this integral flow to yield an ^1¦1-action is obtained: the homogeneous components, X₀, and, X₁, of the given field must define a Lie superalgebra of dimension (1, 1). The supergroup structure on ^1¦1, however, has to be specified: there are three non-isomorphic Lie supergroup structures on ^1¦1, all of which have addition as the group operation in the underlying Lie group . On the other extreme, even if X₀, and X₁ do not close to form a Lie superalgebra, the integral flow of X is uniquely determined and is independent of the Lie supergroup structure imposed on ^1¦1. This fact makes it possible to establish an unambiguous relationship between the algebraic Lie derivative of supergeometric objects (e.g., superforms), and its geometrical definition in terms of integral flows. It is shown by means of examples that if a supergroup structure in ^1¦1 is fixed, some flows obtained from left-invariant supervector fields on Lie supergroups may fail to define an ^1¦1-action of the chosen structure. Finally, necessary and sufficient conditions for the integral flows of two supervector fields to commute are given.     

Existence,uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations

We study positive solutions of the Dirichlet problem: u(x)+f(u(x))=0,xD ⁿ ,u(x)=0,xD ⁿ , whereD ⁿ is ann-ball. We find necessary and sufficient conditions for solutions to be nondegenerate. We also give some new existence and uniqueness theorems.Research supported in part by NSF Contract Number MCS 80-02337     

Existence, uniqueness and limit property of solutions to quadratic Erdélyi-Kober type integral equations of fractional order

In this paper, we apply certain measure of noncompactness and fixed point theorem of Darbo type to derive the existence and limit property of solutions to quadratic Erdélyi-Kober type integral equations of fractional order with three parameters. Moreover, we also present the uniqueness and another existence results of the solutions to the above equations. Finally, two examples are given to illustrate the obtained results.     

Boltzmann equation on a lattice: Existence and uniqueness of solutions

Cercignani, Greenberg, and Zweifel proved the existence and uniqueness of solutions of the Boltzmann equation on a toroidal lattice under the assumption that the collision kernel is bounded. We give an alternative, considerably simpler, proof which is based on a fixed point argument.     

Generalized internal long waves equations: Construction,Hamiltonian structure,and conservation laws

Communications in Mathematical Physics - A general class of the ILW type equations is constructed. We introduce a Hamiltonian structure and construct an infinite number of conservation laws.     

Discretizing constant curvature surfaces via loop group factorizations: The discrete sine- and sinh-Gordon equations

The sine- and sinh-Gordon equations are the harmonic map equations for maps of the (Lorentz) plane into the 2-sphere. Geometrically they correspond to the integrability equations for surfaces of constant Gauss and constant mean curvature. There is a well-known dressing action of a loop group on the space of harmonic maps. By discretizing the vacuum solutions we obtain via the dressing action completely integrable discretizations (in both variables) of the sine- and sinh-Gordon equations. For the sine-Gordon equation we get Hirota's discretization. Since we work in a geometric context we also obtain discrete models for harmonic maps into the 2-sphere and discrete models of constant Gauss and mean curvature surfaces.     

The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit

A short-time existence theorem is proven for the Euler equations for nonisentropic compressible fluid flow in a bounded domain, and solutions with low Mach number and almost incompressible initial data are shown to be close to corresponding solutions of the equations for incompressible flow.Partially supported by Department of Energy contracts DEAC0276ER03077-III, V, Current address: Mathematics Department, Princeton University, Princeton, NJ 08544, USA     

Two integrable generalizations of WKI and FL equations:Positive and negative flows,and conservation laws

With the aid of Lenard recursion equations, an integrable hierarchy of nonlinear evolution equations associated with a 2 × 2 matrix spectral problem is proposed, in which the first nontrivial member in the positive flows can be reduced to a new generalization of the Wadati–Konno–Ichikawa(WKI) equation. Further, a new generalization of the Fokas–Lenells(FL) equation is derived from the negative flows. Resorting to these two Lax pairs and Riccati-type equations, the infinite conservation laws of these two corresponding equations are obtained.     

Travelling solitary waves in the discrete Schrödinger equation with saturable nonlinearity: Existence, stability and dynamics

The present work examines in detail the existence, stability and dynamics of travelling solitary waves in a Schrödinger lattice with saturable nonlinearity. After analysing the linear spectrum of the problem in the travelling wave frame, a pseudo-spectral numerical method is used to identify weakly nonlocal solitary waves. By finding zeros of an appropriately crafted tail condition, we can obtain the genuinely localized pulse-like solutions. Subsequent use of continuation methods allows us to obtain the relevant branches of solutions as a function of the system parameters, such as the frequency and intersite coupling strength. We examine the stability of the solutions in two ways: both by imposing numerical perturbations and observing the solution dynamics, as well as by considering the solutions as fixed points of an appropriate map and computing the corresponding Floquet matrix and its eigenvalues. Both methods indicate that our solutions are robustly localized. Finally, the interactions of these solutions are examined in collision type phenomena, observing that relevant collisions are near-elastic, although they may, under appropriate conditions, lead to the generation of an additional pulse.     

The dye laser and laser with a saturable absorber: A comparison of the time-dependent density matrix equations

It is shown that the general time-dependent equation for the matrix elements of the dye laser field is identical in form to that for the laser with a saturable absorber to order $\text{1}\text{n}{}_0$, where n₀ is the mean number of photons at threshold. The parameters of the saturable absorber have, of course, to be replaced by the appropriate parameters of the dye molecule.     

Delay-coupled discrete maps: Synchronization, bistability, and quasiperiodicity

The synchronization transition is studied in delay-coupled logistic maps. For low coupling, in-phase and out-of-phase synchronous dynamics coexist, and with increasing coupling there is a regime of quasiperiodicity before eventual attraction to a fixed point at a critical value of coupling that depends on the nonlinearity. The presence of a region of asynchrony separating two synchronized regimes—termed anomalous behaviour—has been observed earlier in continuous systems and is shown here to occur in delay mappings as well. There are regions of in-phase, anti-phase, and out-of-phase dynamics of periodic as well as chaotic attractors.     

Lattice equations,hierarchies and hamiltonian structures: The Kadomtsev-Petviashvili equation

By applying a suitable continuum limit to a lattice version of the Kadomtsev-Petviashvili (KP) equation we obtain an infinite hierarchy of integrable partial differential-difference equations defined on a two-dimensional lattice, together with the associated hamiltonian structure.     

一类高阶非线性波方程的李群分析、最优系统、精确解和守恒律

本文运用李群分析的方法研究了一类高阶非线性波方程,得到了五阶非线性波方程的对称以及方程的最优系统,进而运用幂级数的方法,求得了方程的精确幂级数解.最后,给出了五阶非线性波方程的一些守恒律.     

Underlying formation mechanisms of ultrasound-assisted brined porcine meat: The role of physicochemical modification,myofiber fragmentation and histological organization

Ultrasound treatment has been a good hurdle technique for meat curing processing, where both physical and chemical consequences can be involved towards final quality of obtained products. However, the specific correlation between ultrasound parameters and muscle fiber fragmentation and myofibrillar microstructural changes during curing deserve further evaluation. In present study, we comparatively studied the effect of ultrasound-assisted brining (UAB) and static brining (SB) on the muscle proteolysis events and microstructural/morphological variation of porcine meat as well as the physicochemical indices and histological characteristics. The results showed that UAB (20 kHz, 315 W for 1 h) could markedly enhance the muscle proteolysis with higher free-/peptide-bound alpha-amino-nitrogen (α-NH2-N) content (P < 0.05) than SB treatment and greatly improved the fragmentation of muscle fiber tissues of cured meat. Meanwhile, UAB processing favored more opening structures of myofibrillar proteins with more hydrophobic groups being exposed. The quantitative histological analysis revealed that, compared with SB treatment, UAB could significantly increase the gap between muscle fibers and the swelling of the perimysium (P < 0.01), proving an efficient curing process with better textural and water holding properties.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

A theorem allowing to derive deterministic evolution equations from stochastic evolution equations, II: The non-Markovian extension

Deterministic evolution equations of classical as well as quantum mechanical models are derived from a set of non-Markovian stochastic evolution equations after an average over realization using a theorem. Examples are given, show that deterministic differential equations that contain derivatives with respect to time higher than or equal to two can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations that increase in number after increasing the number of previous time steps in the updating rules that define a given model. Two explicit examples, the first containing updating rules that depend on two previous time steps and the second on three, are worked in some detail in order to show some features of the linear transformation that allow one to obtain the deterministic differential equations.     

The laws of relativistic thermodynamics: II. The density,momentum and energy laws

The conservation laws of particle number, momentum and energy are derived from relativistic kinetic theory. The first law of relativistic thermodynamics is formulated.