On solutions to the Degasperis-Procesi equation

In this paper we consider a new integrable equation (the Degasperis-Procesi equation) derived recently by Degasperis and Procesi (1999) [3]. Analogous to the Camassa-Holm equation, this new equation admits blow-up phenomenon and infinite propagation speed. First, we give a proof for the blow-up criterion established by Zhou (2004) in [12]. Then, infinite propagation speed for the Degasperis-Procesi equation is proved in the following sense: the corresponding solution u(x,t) with compactly supported initial datum u₀(x) does not have compact x-support any longer in its lifespan. Moreover, we show that for any fixed time t>0 in its lifespan, the corresponding solution u(x,t) behaves as: u(x,t)=L(t)e−x for x?1, and u(x,t)=l(t)ex for x?−1, with a strictly increasing function L(t)>0 and a strictly decreasing function l(t)<0 respectively.     

Necessary and sufficient conditions for the existence and ɛ-uniqueness of bounded solutions of the equation x′ = f(x) − h(t)

We introduce the notion of ?-unique bounded solution to the nonlinear differential equation x′ = f(x) ? h(t), where f: ? → ? is a continuous function and h(t) is an arbitrary continuous function bounded on ?. We derive necessary and sufficient conditions for the existence and ?-uniqueness of bounded solutions to this equation.     

Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain

In this paper, we study the long-time behavior of solutions for the parabolic equation with non-linear Laplacian principal part in Rn. We prove the existence of a global (L₂(Rn), L_∞(Rn))-attractor when n?p and the existence of a global (L₂(Rn), )-attractor when n>p.     

Regularity and long time behavior of the Boltzmann equation for granular gases

This paper investigates regularity of solutions of the Boltzmann equation with dissipative collisions in a thermal bath. In the case of pseudo-Maxwellian approximation, we prove that for any initial datum f₀(ξ) in the set of probability density with zero bulk velocity and finite temperature, the unique solution of the equation satisfies f(ξ,t)∈H^∞(R³) for all t>0. Furthermore, for any t₀>0 and s?0 the Hs norm of f(ξ,t) is bounded for t?t₀. As a consequence, the exponential convergence to the unique steady state is also established under the same initial condition.     

Gain of regularity for the KP-I equation

In this paper we study the smoothness properties of solutions to the KP-I equation. We show that the equation's dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data ? possesses certain regularity and sufficient decay as x→∞, then the solution u(t) will be smoother than ? for 0<t?T where T is the existence time of the solution.     

Analysis of the backward-euler/langevin method for molecular dynamics

This paper develops the theory of a recently introduced computational method for molecular dynamics. The method in question uses the backward-Euler method to solve the classical Langevin equations of a molecular system. Parameters are chosen to produce a cutoff frequency ω_c, which may be set equal to kT/h to simulate quantum-mechanical effects. In the present paper, an ensemble of identical Hamiltonian systems modeled by the backward-Euler/Langevin method is considered, an integral equation for the equilibrium phase-space density is derived, and an asymptotic analysis of that integral equation in the limit Δt → 0 is performed. The result of this asymptotic analysis is a second-order partial differential equation for the equilibrium phase-space density expressed as a function of the constants of the motion. This equation is solved in two special cases: a system of coupled harmonic oscillators and a diatomic molecule with a stiff bond.     

On the Hermitian positive definite solutions of nonlinear matrix equation X + A∗XA = Q

Nonlinear matrix equation X^s + A∗X^−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ? 1, 0 < t ? 1 and 0 < s ? 1, t ? 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.     

Upper semicontinuity of global attractors for a viscoelastic equations with nonlinear density and memory effects

This paper is devoted to showing the upper semicontinuity of global attractors associated with the family of nonlinear viscoelastic equations in a three‐dimensional space, for f growing up to the critical exponent and dependent on ρ ∈ [0,4), as ρ→0⁺. This equation models extensional vibrations of thin rods with nonlinear material density ?(?_tu) = |?_tu|^ρ and presence of memory effects. This type of problems has been extensively studied by several authors; the existence of a global attractor with optimal regularity for each ρ ∈ [0,4) were established only recently. The proof involves the optimal regularity of the attractors combined with Hausdorff's measure.     

On the Cauchy problem for some abstract nonlinear differential equations

In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A ₁(t)u,A ₂(t)u,?,A _?(t)u), (A _i (t),i = 1, 2, ?,?), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E ^? intoE and ?A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (A _i(t),i = 1, 2, ?,?), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations.     

Some well-posed functional equations which generate semigroups

Consider the abstract linear functional equation (FE) (Dx)(t) = f(t) (t ? 0), x(t) = ?(t) (t ? 0) in a Banach space B. A theorem is proven which contains the following result as a special case. Let Y(R; B; η) be a L^p-space or C₀-space on R = (?t8, ∞), with a suitable weight function η, and with values in B. Let D be a closed (unbounded) causal linear operator in Y(R; B; η), which commutes with translations. Suppose that D + λI has a continuous causal inverse for some complex λ, and that D restricted to those functions in Y(R;B;η) which vanish on R^? = (?∞, 0] has a continuous causal inverse. Then (FE) generates a strongly continuous semigroup of translation type on a Banach space, which is essentially the cross product of the restriction of the domain of D to R^? and Y(R⁺; B; η). Examples with B = Cⁿ on how the theory applies to a neutral functional differential equation, a difference equation, a Volterra integrodifferential equation (with nonintegrable kernel but integrable resolvent), and a fractional order functional differential equation are given. Also, an abstract neutral functional differential equation in a Hilbert space is studied and applications to an abstract Volterra integrodifferential equation in a Banach space are indicated.     

Finite‐dimensional attractors and exponential attractors for degenerate doubly nonlinear equations

We consider the following doubly nonlinear parabolic equation in a bounded domain Ω??³: where the nonlinearity f is allowed to have a degeneracy with respect to ?_tu of the form ?_tu|?_tu|^p at some points x∈Ω. Under some natural assumptions on the nonlinearities f and g, we prove the existence and uniqueness of a solution of that problem and establish the finite‐dimensionality of global and exponential attractors of the semigroup associated with this equation in the appropriate phase space. Copyright © 2009 John Wiley & Sons, Ltd.     

Hyperbolic–parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation

We consider the hyperbolic?Cparabolic singular perturbation problem for a nondegenerate quasilinear equation of Kirchhoff type with weak dissipation. This means that the dissipative term is multiplied by a coefficient b(t) which tends to 0 as t ?? +???. The case where b(t) ~ (1?+?t)^?p with p?<?1 has recently been considered. The result is that the hyperbolic problem has a unique global solution, and the difference between solutions of the hyperbolic problem and the corresponding solutions of the parabolic problem converges to zero both as t ???+??? and as ${\varepsilon \to 0^{+}}$ . In this paper we show that these results cannot be true for p?> 1, but they remain true in the critical case p?=?1.     

Rationale of the chakravāla process of Jayadeva and Bhāskara II

The old Indian chakravāla method for solving the Bhāskara-Pell equation or varga-prak$\text{r}\text{?}$ti x² ? Dy² = 1 is investigated and explained in detail. Previous misconceptions are corrected, for example that chakravāla, the short cut method bhāvanā included, corresponds to the quick-method of Fermat. The chakravāla process corresponds to a half-regular best approximating algorithm of minimal length which has several deep minimization properties. The periodically appearing quantities (jye$\text{s}\text{?}$$\text{t}\text{?}$ha-mūla, kani$\text{s}\text{?}$$\text{t}\text{?}$ha-mūla, k$\text{s}\text{?}$epaka, ku$\text{t}\text{?}$$\text{t}\text{?}$akāra, etc.) are correctly understood only with the new theory.     

Homogenization of transport equations: A simple PDE approach to the Kubo formula

We are interested in the behavior with respect to the small parameter ?>0 of solutions ρ? of the conservative transport(-diffusion) equation t_∂ρ?+∇x(ρ?u?)=ηΔxρ?, with η?0, driven by a large random velocity field: |u?|=O(1/?). Assuming that the velocity does not have long-time memory we justify the convergence of the expectation Eρ? to the solution of a diffusion equation. This question has been widely investigated; here we present a simple proof which only relies on PDE tools.     

On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI

Painlevé's transcendental differential equation PVI may be expressed as the consistency condition for a pair of linear differential equations with 2×2 matrix coefficients with rational entries. By a construction due to Tracy and Widom, this linear system is associated with certain kernels which give trace class operators on Hilbert space. This paper expresses such operators in terms of Hankel operators Γ? of linear systems which are realised in terms of the Laurent coefficients of the solutions of the differential equations. Let P(t,∞):L²(0,∞)→L²(t,∞) be the orthogonal projection; then the Fredholm determinant τ(t)=det(I−P(t,∞)Γ?) defines the τ function, which is here expressed in terms of the solution of a matrix Gelfand-Levitan equation. For suitable values of the parameters, solutions of the hypergeometric equation give a linear system with similar properties. For meromorphic transfer functions that have poles on an arithmetic progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in L²(0,∞); so det(I−Γ?P(t,∞)) can be expressed as a series of finite determinants. This applies to elliptic functions of the second kind, such as satisfy Lamé's equation with ?=1.     

Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line

Consider an evolution family U=(U(t,s))_t?s?0 on a half-line R₊ and a semi-linear integral equation . We prove the existence of stable manifolds of solutions to this equation in the case that (U(t,s))_t?s?0 has an exponential dichotomy and the nonlinear forcing term f(t,x) satisfies the non-uniform Lipschitz conditions: ‖f(t,x₁)−f(t,x₂)‖?φ(t)‖x₁−x₂‖ for φ being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory.     

Numerical analytical method of studying some linear functional differential equations

This paper presents the results of studying a scalar linear functional differential equation of a delay type ?(t) = a(t)x(t ? 1) + b(t)x(t/q) + f(t), q > 1. Primary attention is given to the original problem with the initial point, when the initial condition is specified at the initial point, and the classical solution, whose substitution into the original equation transforms it into an identity, is sought. The method of polynomial quasi-solutions, based on representation of an unknown function x(t) as a polynomial of degree N, is applied as the method of investigation. Substitution of this function into the original equation yields a residual Δ(t) = O(t ^N ), for which an accurate analytical representation is obtained. In this case, the polynomial quasi-solution is understood as an exact solution in the form of a polynomial of degree N, disturbed because of the residual of the original initial problem. Theorems of existence of polynomial quasi-solutions for the considered linear functional differential equation and exact polynomial solutions have been proved. Results of a numerical experiment are presented.     

Global Dynamics of Froude-Type Oscillators with Superlinear Damping Terms

This paper deals with the damped superlinear oscillator $$x'' + a(t)\phi_p\bigl(x' \bigr) + b(t)\phi_q\bigl(x'\bigr)+ \omega^2x = 0, $$ where a(t) and b(t) are continuous and nonnegative for t≥0; p and q are real numbers greater than or equal to 2; ? _r (x′)=|x′| ^r?2 x′. This equation is a generalization of nonlinear ship rolling motion with Froude’s expression, which is very familiar in marine engineering, ocean engineering and so on. Our concern is to establish a necessary and sufficient condition for the equilibrium to be globally asymptotically stable. In particular, the effect of the damping coefficients a(t), b(t) and the nonlinear functions ? _p (x′), ? _q (x′) on the global asymptotic stability is discussed. The obtained criterion is judged by whether the integral of a particular solution of the first-order nonlinear differential equation $$u' + \omega^{p-2}a(t)\phi_p(u) + \omega^{q-2}b(t)\phi_q(u) + 1 = 0 $$ is divergent or convergent. In addition, explicit sufficient conditions and explicit necessary conditions are given for the equilibrium of the damped superlinear oscillator to be globally attractive. Moreover, some examples are included to illustrate our results. Finally, our results are extended to be applied to a more complicated model.     

On the behavior of the solution of the Burgers equation as the viscosity goes to zero

The solution of the initial boundary-value problem u_?′ ? ?D²u_? + u_?Du_? = f on (a, b) x(0, T), u_?(a, t) = u_?(b, t) = 0 and u_?(x, 0) = 0 on (a, b), is shown to converge to the solution of the limiting equation as the viscosity tends to zero. Estimates on the rate of convergence are given.     

A nonlinear problem having a continuous locus of singular points and no multiple solutions

The boundary-value problem ?z″ = (z² ? t²)z′, ? > 0, z(? 1) = α, z(0) = β, t? [?1, 0], has been shown to have a solution, and moreover, depending on the choice of α and β, multiple solutions to it exist. We consider the more general equation f(z, t)z″ = (z^r ? t^s)z′ for a particular non-negative function f(z, t), and integrate the equation exactly. Depending on α and β, we find that either there are no solutions, or that only unique solutions exist. The conclusion is that the presence of a continuous locus of singular points, given by z^r = t^s, does not necessarily produce multiple solutions.