Benjamin Dodson
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View article: Quantitative Scattering for the Energy-Critical Wave Equation on Asymptotically Flat Spacetimes
Quantitative Scattering for the Energy-Critical Wave Equation on Asymptotically Flat Spacetimes Open
The scattering theory for the energy-critical wave equation on asymptotically flat spacetimes has, to date, been qualitative. While the qualitative scattering of solutions is well-understood, explicit bounds on the solution's global spacet…
View article: Rigidity for the non self-dual Chern--Simons--Schr{ö}dinger equation at the level of the soliton
Rigidity for the non self-dual Chern--Simons--Schr{ö}dinger equation at the level of the soliton Open
In this paper we prove a rigidity result for a solution to the non self-dual Chern--Simons--Schr{ö}dinger equation at the level of the soliton.
View article: A determination of the blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton
A determination of the blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton Open
In this paper we prove that the only blowup solutions to the focusing, quintic nonlinear Schrödinger equation with mass equal to the mass of the soliton are rescaled solitons or the pseudoconformal transformation of those solitons.
View article: Sharp Global well-posedness and scattering for the radial nonlinear intercritical wave equation
Sharp Global well-posedness and scattering for the radial nonlinear intercritical wave equation Open
In this paper we prove global well-posedness and scattering for the defocusing, intercritical nonlinear wave equation in dimensions $d \geq 4$ with radial initial data. We prove this for sharp initial data.
View article: A Liouville theorem for the Chern–Simons–Schrödinger equation
A Liouville theorem for the Chern–Simons–Schrödinger equation Open
In this paper we prove a Liouville theorem for the Chern–Simons–Schrödinger equation. This result is consistent with the soliton resolution conjecture for initial data that does not lie in a weighted space. See [10] for the soliton resolut…
View article: Sequential convergence of a solution to the Chern--Simons--Schrodinger equation
Sequential convergence of a solution to the Chern--Simons--Schrodinger equation Open
In this paper we prove a sequential convergence result for blowup solutions to the $m$-equivariant, self-dual Chern--Simons--Schr{ö}dinger equation. We show that if $u$ has mass less than twice the mass of the soliton, a blowup solution co…
View article: Sharp Global well-posedness and scattering for the radial conformal nonlinear wave equation
Sharp Global well-posedness and scattering for the radial conformal nonlinear wave equation Open
In this paper we prove global well-posedness and scattering for the conformal, defocusing, nonlinear wave equation with radial initial data in the critical Sobolev space, for dimensions $d \geq 4$. This result extends a previous result pro…
View article: A Liouville theorem for the Chern--Simons--Schr{ö}dinger equation
A Liouville theorem for the Chern--Simons--Schr{ö}dinger equation Open
In this paper we prove a Liouville theorem for the Chern--Simons--Schr{ö}dinger equation. This result is consistent with the soliton resolution conjecture for initial data that does not lie in a weighted space. See [KKO22] for the soliton …
View article: Global well-posedness of the radial conformal nonlinear wave equation with initial data in a critical space
Global well-posedness of the radial conformal nonlinear wave equation with initial data in a critical space Open
In this note we prove global well-posedness and scattering for the conformal, defocusing, nonlinear wave equation with radial initial data in a critical Besov space. We also prove a polynomial bound on the scattering norm.
View article: Instability of the soliton for the focusing, mass-critical generalized KdV equation
Instability of the soliton for the focusing, mass-critical generalized KdV equation Open
In this paper we prove instability of the soliton for the focusing, mass-critical generalized KdV equation. We prove that the solution to the generalized KdV equation for any initial data with mass smaller than the mass of the soliton and …
View article: Scattering for the defocusing, cubic nonlinear Schr{ö}dinger equation with initial data in a critical space
Scattering for the defocusing, cubic nonlinear Schr{ö}dinger equation with initial data in a critical space Open
In this note we prove scattering for a defocusing nonlinear Schr{ö}dinger equation with initial data lying in a critical Besov space. In addition, we obtain polynomial bounds on the scattering size as a function of the critical Besov norm.
View article: Global well-posedness of the energy subcritical nonlinear wave equation with initial data in a critical space
Global well-posedness of the energy subcritical nonlinear wave equation with initial data in a critical space Open
In this paper we prove global well-posedness for the defocusing, energy-subcritical, nonlinear wave equation on $\mathbb{R}^{1 + 3}$ with initial data in a critical Besov space. No radial symmetry assumption is needed.
View article: Global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space
Global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space Open
In this note we prove global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data lying in a critical Sobolev space.
View article: A determination of the blowup solutions to the focusing NLS with mass equal to the mass of the soliton
A determination of the blowup solutions to the focusing NLS with mass equal to the mass of the soliton Open
In this paper we prove rigidity for blowup solutions to the focusing, mass-critical nonlinear Schr{ö}dinger equation in dimensions $2 \leq d \leq 15$ with mass equal to the mass of the soliton. We prove that the only such solutions are the…
View article: A determination of the blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton
A determination of the blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton Open
In this paper we prove that the only blowup solutions to the focusing, quintic nonlinear Schr{ö}dinger equation with mass equal to the mass of the soliton are rescaled solitons or the pseudoconformal transformation of those solitons.
View article: The $L^{2}$ sequential convergence of a solution to the mass-critical NLS above the ground state
The $L^{2}$ sequential convergence of a solution to the mass-critical NLS above the ground state Open
In this paper we generalize a weak sequential result of \cite{fan20182} to a non-scattering solutions in dimension $d \geq 2$. No symmetry assumptions are required for the initial data. We build on a previous result of \cite{dodson20202} f…
View article: The $L^{2}$ Sequential Convergence of a Solution to the One-Dimensional, Mass-Critical NLS above the Ground State
The $L^{2}$ Sequential Convergence of a Solution to the One-Dimensional, Mass-Critical NLS above the Ground State Open
In this paper we generalize a weak sequential result of \cite{fan20182} to any non-scattering solutions in one dimension. No symmetry assumptions are required for the initial data.
View article: Global well-posedness for the cubic nonlinear Schr{ö}dinger equation with initial lying in $L^{p}$-based Sobolev spaces
Global well-posedness for the cubic nonlinear Schr{ö}dinger equation with initial lying in $L^{p}$-based Sobolev spaces Open
In this paper we continue our study [DSS20] of the nonlinear Schrödinger equation (NLS) with bounded initial data which do not vanish at infinity. Local well-posedness on $\mathbb{R}$ was proved for real analytic data. Here we prove global…
View article: Instability of the soliton for the focusing, mass-critical generalized\n KdV equation
Instability of the soliton for the focusing, mass-critical generalized\n KdV equation Open
In this paper we prove instability of the soliton for the focusing,\nmass-critical generalized KdV equation. We prove that the solution to the\ngeneralized KdV equation for any initial data with mass smaller than the mass\nof the soliton a…
View article: Spacetime integral bounds for the energy-critical nonlinear wave equation
Spacetime integral bounds for the energy-critical nonlinear wave equation Open
In this paper we prove a global spacetime bound for the quintic, nonlinear wave equation in three dimensions. This bound depends on the $L_{t}^{\infty} L_{x}^{2}$ and $L_{t}^{\infty} \dot{H}^{2}$ norms of the solution to the quintic proble…
View article: Global well-posedness for the defocusing, cubic nonlinear\n Schr{\\"o}dinger equation with initial data in a critical space
Global well-posedness for the defocusing, cubic nonlinear\n Schr{\\"o}dinger equation with initial data in a critical space Open
In this note we prove global well-posedness for the defocusing, cubic\nnonlinear Schr{\\"o}dinger equation with initial data lying in a critical\nSobolev space.\n
View article: Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data
Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data Open
We consider the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^4$ and establish almost sure global existence and scattering for randomized radially symmetric initial data in $H^s_x(\mathbb{R}^4) \times H^{s-1}_x(\mathbb{…
View article: Scattering below the ground state for the 2$d$ radial nonlinear Schrödinger equation
Scattering below the ground state for the 2$d$ radial nonlinear Schrödinger equation Open
We revisit the problem of scattering below the ground state threshold for the mass-supercritical focusing nonlinear Schrödinger equation in two space dimensions. We present a simple new proof that treats the case of radial initial data. Th…
View article: Caustics and the Indefinite Signature Schrödinger Equation, Linear and Nonlinear
Caustics and the Indefinite Signature Schrödinger Equation, Linear and Nonlinear Open
The evolution of surface waves in deep water is given by a Schrodinger-like equation. In deep water surface water waves evolve under the nonlinear equation 2i u =1/4(uxx - 2uyy) + q |u|2u Where x, y are coordinates in R2, q is a constant. …
View article: Scattering below the ground state for the 2$d$ radial nonlinear\n Schr\\"odinger equation
Scattering below the ground state for the 2$d$ radial nonlinear\n Schr\\"odinger equation Open
We revisit the problem of scattering below the ground state threshold for the\nmass-supercritical focusing nonlinear Schr\\"odinger equation in two space\ndimensions. We present a simple new proof that treats the case of radial\ninitial da…
View article: Global Well-posedness for the Logarithmically Energy-Supercritical Nonlinear Wave Equation with Partial Symmetry
Global Well-posedness for the Logarithmically Energy-Supercritical Nonlinear Wave Equation with Partial Symmetry Open
We establish global well-posedness and scattering results for the logarithmically energy-supercritical nonlinear wave equation, under the assumption that the initial data satisfies a partial symmetry condition. These results generalize and…
View article: Global well-posedness and scattering for the radial, defocusing, cubic wave equation with initial data in a critical Besov space
Global well-posedness and scattering for the radial, defocusing, cubic wave equation with initial data in a critical Besov space Open
In this paper we prove that the cubic wave equation is globally well - posed\nand scattering for radial initial data lying in a slightly supercritical\nSobolev space, and a weighted Sobolev space.\n
View article: Global well-posedness for the radial, defocusing, nonlinear wave equation for $3 < p < 5$
Global well-posedness for the radial, defocusing, nonlinear wave equation for $3 < p < 5$ Open
In this paper we continue the study of the defocusing, energy-subcritical nonlinear wave equation with radial initial data lying in the critical Sobolev space. In this case we prove scattering in the critical norm when $3 < p < 5$.