arXiv subjects
Publications and source records attributed to Gavin Stewart.
We study the scattering problem for a long range potential, which is time dependent. We prove the existence and completeness of the scattering wave operators, and find some properties of the weakly localized, non-scattering part of the solution. The method we use follows recent methods introduced and applied to short range systems.
We study the asymptotics of the Schr\"odinger equation with time-dependent potential in dimension one. Assuming that the potential decays sufficiently rapidly as $|x| \to \infty$, we prove that the solution can be written as the sum of a free wave $e^{-it\Delta} u_+$ and a weakly bound component $u_{\text{wb}}(t)$. Moreover, we show that the weakly bound part decomposes as $u_{\text{wb}}(t) = u_{\text{loc}}(t) + o_{\dot{H}^1}(1)$, where $\partial_x u_\text{loc}(t)$ is localized near the origin uniformly in time. Since decay conditions on the potential do not preclude resonances unless $d \geq 5$, our results can be seen as a natural extension of [Terence Tao. Dynamics of Partial Differential Equations, 5(2), 2008] and [Avy Soffer, Xiaoxu Wu. arXiv:2304.04245] to the lower-dimensional case.
We consider solutions to the Benjamin-Ono equation $$\partial_t u - H \partial_x^2 u = -\partial_x(u^2)$$ that are localized in a reference frame moving to the right with constant speed. We show that any such solution that decays at least like $\langle x\rangle^{-1-\epsilon}$ for some $\epsilon > 0$ in a comoving coordinate frame must in fact decay like $\langle x\rangle^{-2}$. In view of the explicit soliton solutions, this decay rate is sharp. Our proof has two main ingredients. The first is microlocal dispersive estimates for the Benjamin-Ono equation in a moving frame, which allow us to prove spatial decay of the solution provided the nonlinearity has sufficient decay. The second is a careful normal form analysis, which allows us to obtain rapid decay of the nonlinearity for a transformed equation assuming only modest decay of the solution. Our arguments are entirely time dependent, and do not require the solution to be an exact traveling wave.
We consider the asymptotics of the one-dimensional cubic nonlinear Schr\"odinger equation with an external potential $V$ that does not admit bound states. Assuming that $\jBra{x}^{2+}V(x) \in L^1$ and that $u$ is orthogonal to any zero-energy resonances, we show that solutions exhibit modified scattering behavior. Our decay assumptions are weaker than those appearing in previous work and are likely nearly optimal for exceptional potentials, since the M\o{}ller wave operators are not known to be bounded on $L^p$ spaces unless $\jBra{x}^2V \in L^1$. Our method combines two techniques: First, we prove a new class of linear estimates, which allow us to relate vector fields associated with the equation with potential to vector fields associated with the flat equation, up to localized error terms. By using the improved local decay coming from the zero-energy assumption, we can treat these errors perturbatively, allowing us to treat the weighted estimates using the same vector field arguments as in the flat case. Second, we derive the asymptotic profile using the wave packet testing method introduced by Ifrim and Tataru. Again, by taking advantage of improved local decay near the origin, we are able to treat the potential as a lower-order perturbation, allowing us to use tools developed to study the flat problem.
We study the large-time behavior of global energy class ($H^1$) solutions of the one-dimensional nonlinear Schr\"odinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction Morawetz estimate localized to an exterior region, we prove that these solutions decompose into a free wave and a weakly localized part which is asymptotically orthogonal to any fixed free wave. We further show that the $L^2$ norm of this weakly localized part is concentrated in the region $|x| \leq t^{1/2+}$, and that the energy ($\dot{H}^1$) norm is concentrated in $|x| \leq t^{1/3+}$. Our results hold for solutions with arbitrarily large initial data.
We study the asymptotics for the Ablowitz-Ladik equation. By taking appropriate continuum limits, it can be shown that the behavior of the equation near degenerate frequencies is well approximated by a complex modified Korteweg-de Vries equation. Using this connection, we use the method of space-time resonances to derive a description of the modified scattering behavior of the Ablowitz-Ladik equation, which includes two regions where the solution behaves like a self-similar solution to the complex mKdV equation.
We consider a periodic nonlinear Schr\"odinger equation with white noise dispersion and a power nonlinearity given by \begin{equation*} idu = \Delta u \circ dW_t + |u|^{p-1}u\;dt \end{equation*} By proving stochastic Strichartz estimates, we are able to prove almost sure global wellposedness of this equation with $L^2$ initial data for nonlinearities with exponent $1 < p \leq 3$. By generalizing the Fourier restriction spaces $X^{s,b}$ to the stochastic setting, we also prove that our solutions agree with the ones constructed by Chouk and Gubinelli using rough path techniques. We also consider the quintic equation ($p=5$), and show that it is analytically illposed in $L^1_\omega C_t L^2_x$.
We study the asymptotics of the complex modified Korteweg-de Vries equation $\partial_t u + \partial_x^3 u = -|u|^2 \partial_x u$, which can be used to model vortex filament dynamics. In the real-valued case, it is known that solutions with small, localized initial data exhibit modified scattering for $|x| \geq t^{1/3}$ and behave self-similarly for $|x| \leq t^{1/3}$. We prove that the same asymptotics hold for complex mKdV. The major difficulty in the complex case is that the nonlinearity cannot be expressed as a derivative, which makes the low-frequency dynamics harder to control. To overcome this difficulty, we introduce the decomposition $u = S + w$, where $S$ is a self-similar solution with the same mean as $u$ and $w$ is a remainder that has better decay. By using the explicit expression for $S$, we are able to get better low-frequency behavior for $u$ than we could from dispersive estimates alone. An advantage of our method is its robustness: It does not depend on the precise algebraic structure of the equation, and as such can be more readily adapted to other contexts.
We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real $n$-space proposed by Allen Knutson and Jean-Claude Hausmann. This correspondence defines a natural probability measure on plane polygons. In these terms, we answer Caroll's question. We then explore the Grassmannian geometry of planar quadrilaterals, providing an answer to Sylvester's four-point problem, and describing explicitly the moduli space of unordered quadrilaterals. All of this provides a concrete introduction to a family of metrics used in shape classification and computer vision.