Long time asymptotic behavior for the nonlocal nonlinear Schr\"odinger equation with weighted Sobolev initial data

In this paper, we extend $\overline\partial$ steepest descent method to study the Cauchy problem for the nonlocal nonlinear Schr\"odinger (NNLS) equation with weighted Sobolev initial data %and finite density initial data \begin{align*} &iq_{t}+q_{xx}+2\sigma q^2(x,t)\overline{q}(-x,t)=0, & q(x,0)=q_0(x), \end{align*} where $q_0(x)\in L^{1,1}(\mathbb{R})\cap L^{2,1/2}(\mathbb{R})$. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem is expressed in terms of solutions of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. Finally, we obtain the asymptotic expansion of the Cauchy problem for the NNLS equation in solitonic region. The leading order term is soliton solutions, the second term is the error term is the interaction between solitons and dispersion, the error term comes from the corresponding $\bar{\partial}$ equation. Compared to the asymptotic results on the classical NLS equation, the major difference is the second and third terms in asymptotic expansion for the NNLS equation were affected by a function ${\rm Im}\nu(\xi)$ for the stationary phase point $\xi$.Comment: 34 page

On the global existence for the modified Camassa-Holm equation via the inverse scattering method

In this paper, we address the existence of global solutions to the Cauchy problem of the modified Camassa-Holm (mCH) equation, which is known as a model for the unidirectional propagation of shallow water waves. Based on the spectral analysis of the Lax pair, we apply the inverse scattering transform to rigorously analyze the mCH equation with zero background. By connecting the Cauchy problem to the Riemann-Hilbert (RH) problem, we establish a bijective map between potential and reflection coefficients within the $L^2$-Sobolev space framework. Utilizing a reconstruction formula and estimates on the time-dependent RH problem, we obtain a unique global solution to the Cauchy problem for the mCH equation.Comment: 29 page

The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: Long-time and Painlev\'e asymptotics

Based on the $\overline\partial$-generalization of the Deift-Zhou steepest descent method, we extend the long-time and Painlev\'e asymptotics for the Camassa-Holm (CH) equation to the solutions with initial data in a weighted Sobolev space $H^{4,2}(\mathbb{R})$. With a new scale $(y,t)$ and a RH problem associated with the initial value problem,we derive different long time asymptotic expansions for the solutions of the CH equation in different space-time solitonic regions. The half-plane $\{ (y,t): -\infty 0\}$ is divided into four asymptotic regions: 1. Fast decay region, $y/t \in(-\infty,-1/4)$ with an error $\mathcal{O}(t^{-1/2})$; 2. Modulation-solitons region, $y/t \in(2,+\infty)$, the result can be characterized with an modulation-solitons with residual error $\mathcal{O}(t^{-1/2 })$; 3. Zakhrov-Manakov region,$y/t \in(0,2)$ and $y/t \in(-1/4,0)$. The asymptotic approximations is characterized by the dispersion term with residual error $\mathcal{O}(t^{-3/4})$; 4. Two transition regions, $|y/t|\approx 2$ and $|y/t| \approx -1/4$, the results are describe by the solution of Painlev\'e II equation with error order $\mathcal{O}(t^{-1/2})$.Comment: 61 page

Optimizing Epicardial Restraint and Reinforcement Following Myocardial Infarction: Moving Towards Localized, Biomimetic, and Multitherapeutic Options

The mechanical reinforcement of the ventricular wall after a myocardial infarction has been shown to modulate and attenuate negative remodeling that can lead to heart failure. Strategies include wraps, meshes, cardiac patches, or fluid-filled bladders. Here, we review the literature describing these strategies in the two broad categories of global restraint and local reinforcement. We further subdivide the global restraint category into biventricular and univentricular support. We discuss efforts to optimize devices in each of these categories, particularly in the last five years. These include adding functionality, biomimicry, and adjustability. We also discuss computational models of these strategies, and how they can be used to predict the reduction of stresses in the heart muscle wall. We discuss the range of timing of intervention that has been reported. Finally, we give a perspective on how novel fabrication technologies, imaging techniques, and computational models could potentially enhance these therapeutic strategies. Keywords: ventricular restraint; infarct reinforcement; biomimetic