American Institute of Mathematical Sciences

In this note we consider the 1-D cubic Schrödinger equation with data given as small perturbations of a Dirac-δ function and some other related equations. We first recall that although the problem for this type of data is ill-posed one can use the geometric framework of the Schrödinger map to define the solution beyond the singularity time. Then, we find some natural and well defined geometric quantities that are not regular at time zero. Finally, we make a link between these results and some known phenomena in fluid mechanics that inspired this note.

We construct bi-frequency solitary waves of the nonlinear Dirac equation with the scalar self-interaction, known as the Soler model (with an arbitrary nonlinearity and in arbitrary dimension) and the Dirac-Klein-Gordon with Yukawa self-interaction. These solitary waves provide a natural implementation of qubit and qudit states in the theory of quantum computing.

We show the relation of \begin{document}$± 2ω\mathrm{i}$\end{document} eigenvalues of the linearization at a solitary wave, Bogoliubov \begin{document}$\mathbf{SU}(1,1)$\end{document} symmetry, and the existence of bi-frequency solitary waves. We show that the spectral stability of these waves reduces to spectral stability of usual (one-frequency) solitary waves.

We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system, proving the existence of solutions in which the Dirac wave function is of the form \begin{document}$φ(x,ω)e^{-iω t}$\end{document}, with \begin{document}$ω∈(-m,ω_ *)$\end{document} for some \begin{document}$ω_ *>-m$\end{document}. The solutions satisfy \begin{document}$φ(\,·\,,ω)∈ H^ 1(\mathbb{R}^3,\mathbb{C}^4)$\end{document}, and are small amplitude in the sense that \begin{document}${\left\| {φ(\,·\,,ω)} \right\|}^2_{L^ 2} = O(\sqrt{m+ω})$\end{document} and \begin{document}${\left\| {φ(\,·\,,ω)} \right\|}_{L^∞} = O(m+ω)$\end{document}. The method of proof is an implicit function theorem argument based on the identification of the nonrelativistic limit as the ground state of the Choquard equation. This identification is in some ways unexpected on account of the repulsive nature of the electrostatic interaction between electrons, and arises as a manifestation of certain peculiarities (Klein paradox) which result from attempts to interpret the Dirac equation as a single particle quantum mechanical wave equation.

We consider standing wave solutions of various dispersive models with non-standard form of the dispersion terms. Using index count calculations, together with the information from a variational construction, we develop sharp conditions for spectral stability of these waves.

We study the dynamics for the focusing nonlinear Klein Gordon equation with a positive, singular, radial potential and initial data in energy space. More precisely, we deal with

\begin{document}$u_{tt}-Δ u+m^2 u=|x|^{-a}|u|^{p-1}u$ \end{document}

with \begin{document}$0 < a < 2$\end{document}. In dimension \begin{document}$d≥3$\end{document}, we establish the existence and uniqueness of the ground state solution that enables us to define a threshold size for the initial data that separates global existence and blow-up. We find a critical exponent depending on \begin{document}$a$\end{document}. We establish a global existence result for subcritical exponents and subcritical energy data. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary sets.

We consider the modified Witham equation

\begin{document}${{\partial }_{t}}v+{{\partial }_{x}}\sqrt{{{a}^{2}}-\partial _{x}^{2}v}={{\partial }_{x}}\left( {{v}^{3}} \right),\ \ \left( t,x \right)\in \mathbb{R}\times \mathbb{R},$ \end{document}

where \begin{document}$\sqrt{a^{2}-\partial _{x}^{2}}$\end{document} means the dispersion relation which correspond to nonlinear Kelvin and continental-shelf waves. We develop the factorization technique to study the large time asymptotics of solutions.

This article focuses on a quasilinear wave equation of \begin{document}$ p $\end{document}-Laplacian type:

\begin{document}$u_{tt} - Δ_p u -Δ u_t = 0$ \end{document}

in a bounded domain \begin{document}$ \Omega \subset \mathbb{R}^3 $\end{document} with a sufficiently smooth boundary \begin{document}$ \Gamma = \partial \Omega $\end{document} subject to a generalized Robin boundary condition featuring boundary damping and a nonlinear source term. The operator \begin{document}$ Δ_p $\end{document}, \begin{document}$ 2<p<3 $\end{document}, denotes the classical \begin{document}$ p$\end{document}-Laplacian. The nonlinear boundary term \begin{document}$ f(u) $\end{document} is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from \begin{document}$ {W^{1,p}}\left( \Omega \right) $\end{document} into \begin{document}$ L^2(\Gamma) $\end{document}. Under suitable assumptions on the parameters we provide a rigorous proof of existence of a local weak solution which can be extended globally in time provided the source term satisfies an appropriate growth condition.

We consider the Cauchy problem for coupled systems of nonlinear wave and Klein-Gordon equations in three space dimensions. The author previously proved the small data global existence for rapidly decreasing data under a certain condition on nonlinearity. In this paper, we show that we can weaken the condition, provided that the initial data are compactly supported.

This paper focuses on the initial boundary value problem of semilinear wave equation in exterior domain in two space dimensions with critical power. Based on the contradiction argument, we prove that the solution will blow up in a finite time. This complements the existence result of supercritical case by Smith, Sogge and Wang [20] and blow up result of subcritical case by Li and Wang [14] in two space dimensions.

We study several quantitative properties of solutions to the incompressible Navier-Stokes equation in three and higher dimensions:

\begin{document}$ \begin{align} u_t -Δ u+u· \nabla u +\nabla p = 0, \ \ {\rm div} u = 0, \ \ u(0, x) = u_0(x). \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)\label{NSa} \end{align}$ \end{document}

More precisely, for the blow up mild solutions with initial data in \begin{document}$L^{∞}(\mathbb{R}^d)$\end{document} and \begin{document}$H^{d/2 -1}(\mathbb{R}^d)$\end{document}, we obtain a concentration phenomenon and blowup profile decomposition respectively. On the other hand, if the Fourier support has the form \begin{document}${\rm supp} \ \widehat{u_0} \subset \{ξ∈ \mathbb{R}^n: ξ_1≥ L \}$\end{document} and \begin{document}$ \|u_0\|_{∞} \ll L$\end{document} for some \begin{document}$L >0$\end{document}, then (1) has a unique global solution \begin{document}$u∈ C(\mathbb{R}_+, L^∞)$\end{document}. In 3D, we show the compactness of the set consisting of minimal-\begin{document}$L^p$\end{document} singularity-generating initial data with \begin{document}$3<p< ∞$\end{document}, furthermore, if the mild solution with data in \begin{document}$L^p({{\mathbb{R}}^{3}})$\end{document} blows up in a Type-Ⅰ manner, we prove the existence of a blowup solution which is uniformly bounded in critical Besov spaces \begin{document}$\dot B^{-1+6/p}_{p/2, ∞}({{\mathbb{R}}^{3}})$\end{document}.

We study special regularity properties of solutions to the initial value problem associated to the Zakharov-Kuznetsov equation in three dimensions. We show that the initial regularity of the data in a family of half-spaces propagates with infinite speed. By dealing with the finite envelope of a class of these half-spaces we extend the result to the complement of a family of cones in $\mathbb{R}^3$.

We address various issues concerning the Cauchy problem for the Zakharov-Rubenchik system(known as the Benney-Roskes system in water waves theory), which models the interaction of short and long waves in many physical situations. Motivated by the transverse stability/instability of the one-dimensional solitary wave (line solitary), we study the Cauchy problem in the background of a line solitary wave.

In this paper, we consider the final state problem for the nonlinear Klein-Gordon equation (NLKG) with a critical nonlinearity in three space dimensions: \begin{document}$(\Box+1)u = λ|u|^{2/3}u$\end{document}, \begin{document}$t∈\mathbb{R}$\end{document}, \begin{document}$x∈\mathbb{R}^{3}$\end{document}, where \begin{document}$\Box = \partial_{t}^{2}-Δ$\end{document} is d'Alembertian. We prove that for a given asymptotic profile \begin{document}$u_{\mathrm{ap}}$\end{document}, there exists a solution \begin{document}$u$\end{document} to (NLKG) which converges to \begin{document}$u_{\mathrm{ap}}$\end{document} as \begin{document}$t\to∞$\end{document}. Here the asymptotic profile \begin{document}$u_{\mathrm{ap}}$\end{document} is given by the leading term of the solution to the linear Klein-Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on the combination of Fourier series expansion for the nonlinearity used in our previous paper [23] and smooth modification of phase correction by Ginibre and Ozawa [6].

The \begin{document}$L^∞$\end{document}-energy method is developed so as to handle nonlinear parabolic systems with convection and hysteresis effect. The system under consideration originates from a biological model where the hysteresis and convective effects are taken into account in the evolution of species. Some results for the existence of local and global solutions as well as the uniqueness of solution are presented.

We study a nonlinear Schrödinger equation with damping, detuning, and spatially homogeneous input terms, which is called the Lugiato-Lefever equation, on the unit disk with the Neumann boundary conditions. We aim at understanding bifurcations of a so-called cavity soliton which is a radially symmetric stationary spot solution. It is known by numerical simulations that a cavity soliton bifurcates from a spatially homogeneous steady state. We prove the existence of the parameter-dependent center manifold and a branch of radially symmetric steady state in a neighborhood of the bifurcation point. In order to capture further bifurcations of the radially symmetric steady state, we study a degenerate bifurcation for which two radially symmetric modes become unstable simultaneously, which is called the two-mode interaction. We derive a vector field on the center manifold in a neighborhood of such a degenerate bifurcation and present numerical simulations to demonstrate the Hopf and homoclinic bifurcations of bifurcating solutions.

The uncertainty principle of Heisenberg type can be generalized via the Boltzmann entropy functional. After reviewing the \begin{document} $L^p$ \end{document} generalization of the logarithmic Sobolev inequality by Del Pino-Dolbeault [6], we introduce a generalized version of Shannon's inequality for the Boltzmann entropy functional which may regarded as a counter part of the logarithmic Sobolev inequality. Obtaining best possible constants of both inequalities, we connect both the inequalities to show a generalization of uncertainty principle of the Heisenberg type.

We study the instability of standing wave solutions for nonlinear Schrödinger equations with a one-dimensional harmonic potential in dimension \begin{document} $N≥2$ \end{document}. We prove that if the nonlinearity is \begin{document} $L^2$ \end{document}-critical or supercritical in dimension \begin{document} $N-1$ \end{document}, then any ground states are strongly unstable by blowup.

This paper deals with the local well-posedness of free boundary problems for the Navier-Stokes equations in the case where the fluid initially occupies an exterior domain \begin{document} $Ω$ \end{document} in \begin{document} $N$ \end{document}-dimensional Euclidian space \begin{document} $\mathbb{R}^N$ \end{document}.