Left Half-line Research Articles

Around strongly operator convex functions

We establish the subadditivity of strongly operator convex functions on (0,∞) and (−∞,0). By utilizing the properties of strongly operator convex functions, we derive the subadditivity property of operator monotone functions on (−∞,0). We introduce new operator inequalities involving strongly operator convex functions and weighted operator means. In addition, we explore the relationship between strongly operator convex and Kwong functions on (0,∞). Moreover, we study strongly operator convex functions on (a,∞) with −∞<a and on the left half-line (−∞,b) with b<∞. We demonstrate that any nonconstant strongly operator convex function on (a,∞) is strictly operator decreasing, and any nonconstant strongly operator convex function on (−∞,b) is strictly operator monotone. Consequently, for a strongly operator convex function g on (a,∞) or (−∞,b), we provide lower bounds for |g(A)−g(B)| whenever A−B>0.

Stability of mKdV breathers on the half-line

In this paper we study the stability problem for mKdV breathers on the left half-line. We are able to show that leftwards moving breathers, initially located far away from the origin, are strongly stable for the problem posed on the left half-line, when assuming homogeneous boundary conditions. The proof involves a Lyapunov functional which is almost conserved by the mKdV flow once we control some boundary terms which naturally arise.

Analytic smoothing estimates for the Korteweg–de Vries equation with steplike data

In this paper, we prove analytic smoothing estimates for the Korteweg–de Vries equation. In the first result, we obtain explicit analytic smoothing estimates for initial data in Faddeev class which decays exponentially in the positive direction. In the second result, we go beyond Faddeev class and generalize the result to non-decaying initial data. Particularly, step functions supported in the left half line and their perturbations by Faddeev class potentials decaying exponentially in positive direction are involved by the second result. Finally, we discuss some of its applications to control problems such as observability inequalities for the KdV equation.

The initial-boundary value problem for the Kawahara equation on the half-line

This paper concerns the initial-boundary value problem of the Kawahara equation posed on the right and left half-lines. We prove the local well-posedness in the low regularity Sobolev space. We introduce the Duhamel boundary forcing operator, which is introduced by Colliander and Kenig (Commun Partial Differ Equ 27:2187–2266, 2002) in the context of Airy group operators, to construct solutions on the whole line. We also give the bilinear estimate in $$X^{s,b}$$ space for $$b < \frac{1}{2}$$ , which is almost sharp compared to IVP of Kawahara equation (Chen et al. in J Anal Math 107:221–238, 2009; Jia and Huo in J Differ Equ 246:2448–2467, 2009).

The initial-boundary value problem for the Schrödinger–Korteweg–de Vries system on the half-line

We prove local well-posedness for the initial-boundary value problem (IBVP) associated to the Schrödinger–Korteweg–de Vries system on right and left half-lines. The results are obtained in the low regularity setting by using two analytic families of boundary forcing operators, one of these families being developed by Holmer to study the IBVP associated to the Korteweg–de Vries equation [The initial-boundary value problem for the Korteweg–de Vries equation, Comm. Partial Differential Equations 31 (2006) 1151–1190] and the other one was recently introduced by Cavalcante [The initial-boundary value problem for some quadratic nonlinear Schrödinger equations on the half-line, Differential Integral Equations 30(7–8) (2017) 521–554] in the context of nonlinear Schrödinger with quadratic nonlinearities.

Stability of KdV solitons on the half-line

In this paper we study the stability problem for KdV solitons on the left and right half-lines. Unlike standard KdV, these are not exact solutions to the equations posed on the half-line, and, contrary to NLS, no exact soliton solution seems to exist. However, we are able to show that solitons posed initially far away from the origin are strongly stable for the problem posed on the right half-line, assuming homogeneous boundary conditions. For the problem posed on the left half-line, the positive infinite-time stability problem makes no sense for the case of KdV solitons, but in this setting we prove a result of stability for all negative times. The proof involves finding and using two almost conserved quantities adapted to the evolution of the KdV soliton in the particular case of the half-line. Adaptations to other boundary conditions or star graphs are also discussed.

Local well-posedness of the fifth-order KdV-type equations on the half-line

This paper is a continuation of authors' previous work \cite{CK2018-1}. We extend the argument \cite{CK2018-1} to fifth-order KdV-type equations with different nonlinearities, in specific, where the scaling argument does not hold. We establish the $X^{s,b}$ nonlinear estimates for $b < \frac12$, which is almost optimal compared to the standard $X^{s,b}$ nonlinear estimates for $b > \frac12$ \cite{CGL2010, JH2009}. As an immediate conclusion, we prove the local well-posedness of the initial-boundary value problem (IBVP) for fifth-order KdV-type equations on the right half-line and the left half-line.

Front evolution of the Fredrickson-Andersen one spin facilitated model

The Fredrickson-Andersen one spin facilitated model (FA-1f) on Z belongs to the class of kinetically constrained spin models (KCM). Each site refreshes with rate one its occupation variable to empty (respectively occupied) with probability q (respectively p = 1 − q), provided at least one nearest neighbor is empty. Here, we study the non equilibrium dynamics of FA-1f started from a configuration entirely occupied on the left half-line and focus on the evolution of the front, namely the position of the leftmost zero. We prove, for q larger than a threshold ¯ q < 1, a law of large numbers and a central limit theorem for the front, as well as the convergence to an invariant measure of the law of the process seen from the front.

Dynamics of the spin- 12 Heisenberg chain initialized in a domain-wall state

We study the dynamics of an isotropic spin-1/2 Heisenberg chain starting in a domain-wall initial condition, where the spins are initially up on the left half-line and down on the right half-line. We focus on the long-time behavior of the magnetization profile. We perform extensive time-dependent density-matrix renormalization group simulations (up to t=350) and find that the data are compatible with a diffusive behavior. Subleading corrections decay slowly blurring the emergence of the diffusive behavior. We also compare our results with two alternative scenarios: superdiffusive behavior and enhanced diffusion with a logarithmic correction. We finally discuss the evolution of the entanglement entropy.

Front progression in the East model

The East model is a one-dimensional, non-attractive interacting particle system with Glauber dynamics, in which a flip is prohibited at a site x if the right neighbour x+1 is occupied. Starting from a configuration entirely occupied on the left half-line, we prove a law of large numbers for the position of the left-most zero (the front), as well as ergodicity of the process seen from the front. For want of attractiveness, the one-dimensional shape theorem is not derived by the usual coupling arguments, but instead by quantifying the local relaxation to the non-equilibrium invariant measure for the process seen from the front. This is the first proof of a shape theorem for a kinetically constrained spin model.

Meromorphic solutions to the KdV equation with non-decaying initial data supported on a left half line

We show that under the Korteweg–de Vries flow an initial profile supported on (−∞, 0) from a very broad function class (without any decay assumption) instantaneously evolves into a meromorphic function with no poles on the real line. Our treatment is based on a suitable modification of the inverse scattering transform and a detailed investigation of the Titchmarsh–Weyl m-function. As a by-product, we improve some related results of others.

The Initial-Boundary Value Problem for the Korteweg–de Vries Equation

We prove local well-posedness of the initial-boundary value problem for the Korteweg–de Vries equation on right half-line, left half-line, and line segment, in the low regularity setting. This is accomplished by introducing an analytic family of boundary forcing operators.

Semicontinuous utility functions in topological spaces

Rader [7] in 1963 gave a proof of a theorem about the existence of an upper semicontinuous function on a second countable topological spaceX with a total preorder\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec } \), provided that the left half lines\(\left\{ {x : x \prec y} \right\}\) are open for eachy belonging toX. The proof of Rader was constructive and direct, without any use of theorems like the Continuoum Characterization Theorem.

Non-equilibrium behaviour of a many particle process: Density profile and local equilibria

One considers a simple exclusion particle jump process on ℤ, where the underlying one particle motion is a degenerate random walk that moves only to the right. One starts with the configuration in which the left halfline is completely occupied and the right one free. It is shown that the number of particles at time t between site [u t] and [v t], divided by t, converges a.s. to $$\int\limits_u^\nu {f(w)dw}$$ , where f might be called the density profile. It is explicitely determined and shown to be an affine function. Secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u, converges weakly to the Bernoulli measure with density f(u), as the time tends to infinity.