Solutions with compact time spectrum to nonlinear Klein--Gordon and Schroedinger equations and the Titchmarsh theorem for partial convolution

We prove that finite energy solutions to the nonlinear Schroedinger equation and nonlinear Klein--Gordon equation which have the compact time spectrum have to be one-frequency solitary waves. The argument is based on the generalization of the Titchmarsh convolution theorem to partial convolutions.Comment: 15 page

L\sp p-L\sp q regularity of Fourier integral operators with caustics

The caustics of Fourier integral operators are defined as caustics of the corresponding Schwartz kernels (Lagrangian distributions on $X\times Y$). The caustic set $\Sigma(C)$ of the canonical relation $C$ is characterized as the set of points where the rank of the projection $\pi:C\to X\times Y$ is smaller than its maximal value, $dim(X\times Y)-1$. We derive the L\sp p(Y)\to L\sp q(X) estimates on Fourier integral operators with caustics of corank 1 (such as caustics of type A\sb{m+1}, $m\in\N$). For the values of $p$ and $q$ outside of certain neighborhood of the line of duality, $q=p'$, the L\sp p\to L\sp q estimates are proved to be caustics-insensitive. We apply our results to the analysis of the blow-up of the estimates on the half-wave operator just before the geodesic flow forms caustics.Comment: 24 pages, 1 figur

Optimal regularity of Fourier integral operators with one-sided folds

We obtain optimal continuity in Sobolev spaces for the Fourier integral operators associated to singular canonical relations, when one of the two projections is a Whitney fold. The regularity depends on the type, $k$, of the other projection from the canonical relation ($k=1$ for a Whitney fold). We prove that one loses $(4+\frac{2}{k})^{-1}$ of a derivative in the regularity properties. The proof is based on the $L^2$ estimates for oscillatory integral operators

Damping estimates for oscillatory integral operators with finite type singularities

We derive damping estimates and asymptotics of $L^p$ operator norms for oscillatory integral operators with finite type singularities. The methods are based on incorporating finite type conditions into $L^2$ almost orthogonality technique of Cotlar-Stein.Comment: 17 page

On asymptotic stability of ground states of some systems of nonlinear Schr\"odinger equations

We extend to a specific class of systems of nonlinear Schr\"odinger equations (NLS) the theory of asymptotic stability of ground states already proved for the scalar NLS. Here the key point is the choice of an adequate system of modulation coordinates and the novelty, compared to the scalar NLS, is the fact that the group of symmetries of the system is non-commutative

Linear instability of nonlinear Dirac equation in 1D with higher order nonlinearity

We consider the nonlinear Dirac equation in one dimension, also known as the Soler model in (1+1) dimensions, or the massive Gross-Neveu model: $i\partial_t\psi=-i\alpha\partial_x\psi+m\beta\psi-f(\psi^\ast\beta\psi)\beta\psi$, \psi(x,t)\in\C^2, $x\in\R$, $f\in C^\infty(\R)$, $m>0$, where $\alpha$, $\beta$ are $2\times 2$ hermitian matrices which satisfy $\alpha^2=\beta^2=1$, $\alpha\beta+\beta\alpha=0$. We study the spectral stability of solitary wave solutions $\phi_\omega(x)e^{-i\omega t}$. More precisely, we study the presence of point eigenvalues in the spectra of linearizations at solitary waves of arbitrarily small amplitude, in the limit $\omega\to m$. We prove that if $f(s)=s^k+O(s^{k+1})$, $k\in\N$, with $k\ge 3$, then one positive and one negative eigenvalue are present in the spectrum of linearizations at all solitary waves with $\omega$ sufficiently close to $m$. This shows that all solitary waves of sufficiently small amplitude are linearly unstable. The approach is based on applying the Rayleigh-Schr\"odinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov-Kolokolov stability criterion. Let us mention a similar independent result [Guan-Gustafson] on linear instability for the nonlinear Dirac equation in three dimensions, with cubic nonlinearity (this result is also in formal agreement with the Vakhitov-Kolokolov stability criterion).Comment: 15 page

Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator

We consider the U(1)-invariant nonlinear Klein-Gordon equation in discrete space and discrete time, which is the discretization of the nonlinear continuous Klein-Gordon equation. To obtain this equation, we use the energy-conserving finite-difference scheme of Strauss-Vazquez. We prove that each finite energy solution converges as $t\to\pm\infty$ to the finite-dimensional set of all multifrequency solitary wave solutions with one, two, and four frequencies. The components of the solitary manifold corresponding to the solitary waves of the first two types are generically two-dimensional, while the component corresponding to the last type is generically four-dimensional. The attraction to the set of solitary waves is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent radiation. For the proof, we develop the well-posedness for the nonlinear wave equation in discrete space-time, apply the technique of quasimeasures, and also obtain the version of the Titchmarsh convolution theorem for distributions on the circle.Comment: 39 page

On the meaning of the Vakhitov-Kolokolov stability criterion for the nonlinear Dirac equation

We consider the spectral stability of solitary wave solutions \phi(x)e^{-i\omega t} to the nonlinear Dirac equation in any dimension. This equation is well-known to theoretical physicists as the Soler model (or, in one dimension, the Gross-Neveu model), and attracted much attention for many years. We show that, generically, at the values of where the Vakhitov-Kolokolov stability criterion breaks down, a pair of real eigenvalues (one positive, one negative) appears from the origin, leading to the linear instability of corresponding solitary waves. As an auxiliary result, we state the virial identities ("Pohozhaev theorem") for the nonlinear Dirac equation. We also show that \pm 2\omega i are the eigenvalues of the nonlinear Dirac equation linearized at \phi(x)e^{-i\omega t}, which are embedded into the continuous spectrum for |\omega| > m/3. This result holds for the nonlinear Dirac equation with any nonlinearity of the Soler form ("scalar-scalar interaction") and in any dimension.Comment: 13 pages, minor correction

Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity

We study the point spectrum of the linearization at a solitary wave solution $\phi_\omega(x)e^{-\mathrm{i}\omega t}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the nonlinear term given by $f(\psi^*\beta\psi)\beta\psi$ (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 $\omega\lesssim m$, in the case when $f\in C^1(\mathbb{R}\setminus\{0\})$, $f(\tau)=|\tau|^k+O(|\tau|^K)$ for $\tau\to 0$, with $0<k<K$. For $n\ge 1$, we prove the spectral stability of small amplitude solitary waves ($\omega\lesssim m$) for the charge-subcritical cases $k\lesssim 2/n$ ($1<k\le 2$ when $n=1$) and for the "charge-critical case" $k=2/n$, $K>4/n$. An important part of the stability analysis is the proof of the absence of bifurcations of nonzero-real-part eigenvalues from the embedded threshold points at $\pm 2m\mathrm{i}$. Our approach is based on constructing a new family of exact bi-frequency solitary wave solutions in the Soler model, using this family to determine the multiplicity of $\pm 2\omega\mathrm{i}$ eigenvalues of the linearized operator, and the analysis of the behaviour of "nonlinear eigenvalues" (characteristic roots of holomorphic operator-valued functions).Comment: 55 page

Small amplitude solitary waves in the Dirac-Maxwell system

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 $\phi(x,\omega)e^{-i\omega t}$, $\omega\in(-m,\omega_*)$, with some $\omega_*>-m$, such that $\phi_\omega\in H^1(\mathbb{R}^3,\mathbb{C}^4)$, $\Vert\phi_\omega\Vert^2_{L^2}=O(m-|\omega|)$, and $\Vert\phi_\omega\Vert_{L^\infty}=O(m-|\omega|)$. The method of proof is an implicit function theorem argument based on an identification of the nonrelativistic limit as the ground state of the Choquard equation.Comment: 19 pages (minor changes