Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs

Consider a general linear Hamiltonian system ∂ t u = J L u \partial _{t}u=JLu in a Hilbert space X X . We assume that L : X → X ∗ \ L:X\rightarrow X^{\ast } induces a bounded and symmetric bi-linear form ⟨ L ⋅ , ⋅ ⟩ \left \langle L\cdot ,\cdot \right \rangle on X X , which has only finitely many negative dimensions n − ( L ) n^{-}(L) . There is no restriction on the anti-self-dual operator J : X ∗ ⊃ D ( J ) → X J:X^{\ast }\supset D(J)\rightarrow X . We first obtain a structural decomposition of X X into the direct sum of several closed subspaces so that L L is blockwise diagonalized and J L JL is of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy of e t J L e^{tJL} . In particular, e t J L e^{tJL} has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relate n − ( L ) n^{-}\left ( L\right ) and the dimensions of generalized eigenspaces of eigenvalues of J L \ JL , some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostly J J was assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schrödinger equations with nonzero conditions at infinity, where our general theory applies to yield stability or instability of some coherent states.

Strong Instability of Solitary Waves for Inhomogeneous Nonlinear Schrödinger Equations

This paper studies the inhomogeneous nonlinear Schrödinger equations, which may model the propagation of laser beams in nonlinear optics. Using the cross-constrained variational method, a sharp condition for global existence is derived. Then, by solving a variational problem, the strong instability of solitary waves of this equation is proved.

The number of Dirac-weighted eigenvalues of Sturm-Liouville equations with integrable potentials and an application to inverse problems

In this paper, we further Meirong Zhang, et al.’s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly $n$. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied.

Sharp condition of global well-posedness for inhomogeneous nonlinear Schrödinger equation

<p style='text-indent:20px;'>This paper studies the Cauchy problem of Schrödinger equation with inhomogeneous nonlinear term <inline-formula><tex-math id="M1">\begin{document}$ V(x)|\varphi|^{p-1}\varphi $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. For the case <inline-formula><tex-math id="M3">\begin{document}$ p&gt;1+\frac{4(1+\varepsilon_0)}{n} (0&lt;\varepsilon_0&lt;\frac{2}{n-2}) $\end{document}</tex-math></inline-formula>, by introducing a potential well, we obtain some invariant sets of solution and give a sharp condition of global existence and finite time blowup of solution; for the case <inline-formula><tex-math id="M4">\begin{document}$ p&lt;1+\frac{4}{n} $\end{document}</tex-math></inline-formula>, we obtain the global existence of solution for any initial data in <inline-formula><tex-math id="M5">\begin{document}$ H^1 (\mathbb{R}^n) $\end{document}</tex-math></inline-formula>.</p>

Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem

<p style='text-indent:20px;'>In this paper, we study the initial boundary value problem of the visco-elastic dynamical system with the nonlinear source term in control system. By variational arguments and an improved convexity method, we prove the global nonexistence of solution, and we also give a sharp condition for global existence and nonexistence.</p>

Cross-constrained variational method and nonlinear Schrödinger equation with partial confinement

<p style='text-indent:20px;'>In this paper, we study the nonlinear Schrödinger equation with a partial confinement. By applying the cross-constrained variational arguments and invariant manifolds of the evolution flow, the sharp condition for global existence and blowup of the solution is derived.</p>

A perturbation analysis based on group sparse representation with orthogonal matching pursuit

In this paper, by exploiting orthogonal projection matrix and block Schur complement, we extend the study to a complete perturbation model. Based on the block-restricted isometry property (BRIP), we establish some sufficient conditions for recovering the support of the block 𝐾-sparse signals via block orthogonal matching pursuit (BOMP) algorithm. Under some constraints on the minimum magnitude of the nonzero elements of the block 𝐾-sparse signals, we prove that the support of the block 𝐾-sparse signals can be exactly recovered by the BOMP algorithm in the case of \ell_{2} and \ell_{2}/\ell_{\infty} bounded total noise if 𝑨 satisfies the BRIP of order K+1 with\delta_{K+1}<\frac{1}{\sqrt{K+1}(1+\epsilon_{\boldsymbol{A}}^{(K+1)})^{2}}+\frac{1}{(1+\epsilon_{\boldsymbol{A}}^{(K+1)})^{2}}-1.In addition, we also show that this is a sharp condition for exactly recovering any block 𝐾-sparse signal with the BOMP algorithm. Moreover, we also give the reconstruction upper bound of the error between the recovered block-sparse signal and the original block-sparse signal. In the noiseless and perturbed case, we also prove that the BOMP algorithm can exactly recover the block 𝐾-sparse signal under some constraints on the block 𝐾-sparse signal and\delta_{K+1}<\frac{2+\sqrt{2}}{2(1+\epsilon_{\boldsymbol{A}}^{(K+1)})^{2}}-1.Finally, we compare the actual performance of perturbed OMP and perturbed BOMP algorithm in the numerical study. We also present some numerical experiments to verify the main theorem by using the completely perturbed BOMP algorithm.

Minimum Degree Conditions for Small Percolating Sets in Bootstrap Percolation

The $r$-neighbour bootstrap process is an update rule for the states of vertices in which `uninfected' vertices with at least $r$ `infected' neighbours become infected and a set of initially infected vertices is said to percolate if eventually all vertices are infected. For every $r \geq 3$, a sharp condition is given for the minimum degree of a sufficiently large graph that guarantees the existence of a percolating set of size $r$. In the case $r=3$, for $n$ large enough, any graph on $n$ vertices with minimum degree $\lfloor n/2 \rfloor +1$ has a percolating set of size $3$ and for $r \geq 4$ and $n$ large enough (in terms of $r$), every graph on $n$ vertices with minimum degree $\lfloor n/2 \rfloor + (r-3)$ has a percolating set of size $r$. A class of examples are given to show the sharpness of these results.

Equilibria of an aggregation model with linear diffusion in domains with boundaries

We investigate the effect of linear diffusion and interactions with the domain boundary on swarm equilibria by analyzing critical points of the associated energy functional. Through this process we uncover two properties of energy minimization that depend explicitly on the spatial domain: (i) unboundedness from below of the energy due to an imbalance between diffusive and aggregative forces depends explicitly on a certain volume filling property of the domain, and (ii) metastable mass translation occurs in domains without sufficient symmetry. From the first property, we present a sharp condition for existence (respectively non-existence) of global minimizers in a large class of domains, analogous to results in free space, and from the second property, we identify that external forces are necessary to confine the swarm and grant existence of global minimizers in general domains. We also introduce a numerical method for computing critical points of the energy and give examples to motivate further research.