We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function which is related to the so-called flow force and has several surprising properties. In addition to solitary waves, our nonexistence result applies to "half-solitary" waves (e.g. bores) which decay in only one direction.
We consider a perturbed energy critical focusing Nonlinear Schrödinger Equation in three dimensions. We construct solitary wave solutions for focusing subcritical perturbations as well as defocusing supercritical perturbations. The construction relies on the resolvent expansion, which is singular due to the presence of a resonance. Specializing to pure power focusing subcritical perturbations we demonstrate, via variational arguments, and for a certain range of powers, the existence of a ground state solitary wave, which is then shown to be the previously constructed solution. Finally, we pres…
In this paper, we study solitary waves propagating along the surface of an infinitely deep body of water in two or three dimensions. The waves are acted upon by gravity and capillary effects are allowed --- but not required --- on the interface. We assume that the vorticity is localized in the sense that it satisfies certain moment conditions, and we permit there to be finitely many point vortices in the bulk of the fluid in two dimensions. We also consider a two-fluid model with a vortex sheet. Under mild decay assumptions, we obtain precise asymptotics for the velocity field and free surface…
In this paper, we study the existence of positive solutions to the nonlinear elliptic system, which is derived from taking the nonrelativistic limit of the nonlinear Maxwell-Klein-Gordon equations under the decomposition of waves functions into positron and electron parts. We characterize the existence and nonexistence of positive vector solutions, depending on parameters $p$ and $μ_{ij}$.
In this paper we prove the existence of positive ground state solution for a class of linearly coupled systems involving Kirchhoff-Schrödinger equations. We study the subcritical and critical case. Our approach is variational and based on minimization technique over the Nehari manifold. We also obtain a nonexistence result using a Pohozaev identity type.
We consider the KP-I and gKP-I equations in $\mathbb{R}\times (\mathbb{R}/2π\mathbb{Z})$. We prove that the KdV soliton with subcritical speed $0<c<c^*$ is orbitally stable under the global KP-I flow constructed by Ionescu and Kenig \cite{IK}. For supercritical speeds $c>c^*$, in the spirit of the work by Duyckaerts and Merle \cite{DM}, we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity. This last result also holds for the gKP-I equation.
We describe completely 2-solitary waves related to the ground state of the nonlinear damped Klein-Gordon equation \begin{equation*} \partial_{tt}u+2α\partial_{t}u-Δu+u-|u|^{p-1}u=0 \end{equation*} on $\bf R^N$, for $1\leq N\leq 5$ and energy subcritical exponents $p>2$. The description is twofold. First, we prove that 2-solitary waves with same sign do not exist. Second, we construct and classify the full family of 2-solitary waves in the case of opposite signs. Close to the sum of two remote solitary waves, it turns out that only the components of the initial data in the unstable direction of…
This paper considers two-dimensional gravity solitary waves moving through a body of density stratified water lying below vacuum. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the water and vacuum is a free boundary where the pressure is constant. We prove that, for any smooth choice of upstream velocity field and density function, there exists a continuous curve of such solutions that includes large-amplitude surface waves. Furthermore, following this solution curve, one encounters waves that come arbitrarily close to possessing points of…
In this paper we consider the blow-up for solutions to a weakly coupled system of semilinear damped wave equations of derivative type in the scattering case. After introducing suitable functionals proposed by Lai-Takamura for the corresponding single semilinear equation, we employ Kato's lemma to derive the blow-up result in the subcritical case. On the other hand, in the critical case an iteration procedure based on the slicing method is employed. Let us point out that we find as critical curve in the p-q plane for the pair of exponents (p, q) in the nonlinear terms the same one as for the we…
In this paper, we study the focusing and defocusing energy--subcritical, nonlinear wave equation in $\mathbb{R}^{1+d}$ with radial initial data for $d = 4,5$. We prove that if a solution remains bounded in the critical space on its interval of existence, then the solution exists globally and scatters at $\pm \infty$. The proof follows the concentration compactness/rigidity method initiated by Kenig and Merle, and the main obstacle is to show the nonexistence of nonzero solutions with a certain compactness property. A main novelty of this work is the use of a simple virial argument to rule out …
We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces ${\mathrm{H}^s}$, ${ s > 0 }$, to a class of nonlinear, dispersive evolution equations of the form \begin{equation*} u_t + \left(Lu+ n(u)\right)_x = 0, \end{equation*} where the dispersion ${L}$ is a negative-order Fourier multiplier whose symbol is of KdV type at low frequencies and has integrable Fourier inverse ${ K }$ and the nonlinearity ${n}$ is inhomogeneous, locally Lipschitz and of superlinear growth at the origin. This generalises earlier work by Ehrnström, Groves & Wahlén on a class of equati…
$\newcommand\normt[1]{\left\lVert#1\right\rVert_{L^2}} \newcommand\normo[1]{\left\lVert#1\right\rVert_{H^1}} \newcommand\normpro[1]{\left\lVert#1\right\rVert_{E}}$ We consider the focusing nonlinear Klein-Gordon (NLKG) equation \begin{equation*} \partial_{tt}u - Δu + u - |u|^{p-1}u = 0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^d \end{equation*} for $1\leq d\leq 5$ and $p>2$ subcritical for the $\dot H^1$ norm. In this paper we show the existence of a solution $u(t)$ of the equation such that \begin{equation*} \normo{u(t) - \sum_{k=1,2}Q_k(t)} + \normt{\partial_t u(t)} \to 0\quad \mbox{as $t\t…
We study the point spectrum of the linearization at a solitary wave solution $ϕ_ω(x)e^{-\mathrm{i}ωt}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the nonlinear term given by $f(ψ^*βψ)βψ$ (known as the Soler model). We focus on the spectral stability, that is, the absence of eigenvalues with nonzero real part, in the non-relativistic limit $ω\lesssim m$, in the case when $f\in C^1(\mathbb{R}\setminus\{0\})$, $f(τ)=|τ|^k+O(|τ|^K)$ for $τ\to 0$, with $0<k<K$. For $n\ge 1$, we prove the spectral stability of small amplitude solitary waves ($ω\lesssim m$) for the charge-subcr…
In this paper, we characterize a family of solitary waves for NLS with derivative (DNLS) by the structue analysis and the variational argument. Since (DNLS) doesn't enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters $4ω>c^2$ and the critical parameters $4ω=c^2, c>0$, we show the existence and uniqueness of the solitary waves for (DNLS), up to the phase rotation and spatial translation sym…
In this paper we consider the blow-up of solutions to a weakly coupled system of semilinear damped wave equations in the scattering case with nonlinearities of mixed type, namely, in one equation a power nonlinearity and in the other a semilinear term of derivative type. The proof of the blow-up results is based on an iteration argument. As expected, due to the assumptions on the coefficients of the damping terms, we find as critical curve in the p-q plane for the pair of exponents (p,q) in the nonlinear terms the same one found by Hidano-Yokoyama and, recently, by Ikeda-Sobajima-Wakasa for th…
This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of traveling water waves; analytical and numerical studies of periodic waves with critical layers that may overhang; existence, nonexistence, and qualitative theory of solitary waves and fronts; traveling waves with localized vorticity or density stratification; and waves in three dimensions.
Some effects of surface tension on fully-nonlinear, long, surface water waves are studied by numerical means. The differences between various solitary waves and their interactions in subcritical and supercritical surface tension regimes are presented. Analytical expressions for new peaked travelling wave solutions are presented in the case of critical surface tension. The numerical experiments were performed using a high-accurate finite element method based on smooth cubic splines and the four-stage, classical, explicit Runge-Kutta method of order four.
We study the linear stability of localized modes in self-interacting spinor fields, analyzing the spectrum of the operator corresponding to linearization at solitary waves. Following the generalization of the Vakhitov--Kolokolov approach, we show that the bifurcation of real eigenvalues from the origin is completely characterized by the Vakhitov--Kolokolov condition $dQ/dω=0$ and by the vanishing of the energy functional. We give the numerical data on the linear stability in the generalized Gross--Neveu model and the generalized massive Thirring model in the charge-subcritical, critical, and s…
We study solutions of a semilinear elliptic equation with prescribed mass and Dirichlet homogeneous boundary conditions in the unitary ball. Such problem arises in the search of solitary wave solutions for nonlinear Schrödinger equations (NLS) with Sobolev subcritical power nonlinearity on bounded domains. Necessary and sufficient conditions are provided for the existence of such solutions. Moreover, we show that standing waves associated to least energy solutions are always orbitally stable when the nonlinearity is L^2-critical and subcritical, while they are almost always stable in the L^2-s…
Nonexistence for effectively damped waves with time-dependent mass
Banking Academy of Vietnam（越南银行学院）；School of Mathematics and Computer Science, Hanoi National University of Education（河内国家教育大学数学与计算机科学学院）；Thang Long University（升龙大学）
