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Jason Murphy

Publications and source records attributed to Jason Murphy.

At least 55 records · Page 3Linked to original sources
arXiv2020

Scattering below the ground state for the intercritical non-radial inhomogeneous NLS

We consider the focusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b}|u|^αu = 0\quad\text{on}\quad\mathbb{R}\times\mathbb{R}^N, \] with $N\geq 2$, $0<b<\min\{\tfrac{N}{2},2\}$, and $\tfrac{4-2b}{N}<α<\tfrac{4-2b}{N-2}$. These constraints make the equation mass-supercritical and energy-subcritical. We extend the results of Farah-Guzmán and Miao-Murphy-Zheng and prove scattering below the ground state with general initial data.

math.AP↗
arXiv2020

Numerical simulations for the energy-supercritical nonlinear wave equation

We carry out numerical simulations of the defocusing energy-supercritical nonlinear wave equation for a range of spherically-symmetric initial conditions. We demonstrate numerically that the critical Sobolev norm of solutions remains bounded in time. This lends support to conditional scattering results that have been recently established for nonlinear wave equations.

math.NA↗
arXiv2019

Scattering for the non-radial inhomogeneous NLS

We extend the result of Farah and Guzmán on scattering for the $3d$ cubic inhomogeneous NLS to the non-radial setting. The key new ingredient is a construction of scattering solutions corresponding to initial data living far from the origin.

math.AP↗
arXiv2019

Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations

We consider nonlinear Schrödinger equations with either power-type or Hartree nonlinearity in the presence of an external potential. We show that for long-range nonlinearities, solutions cannot exhibit scattering to solitary waves or more general localized waves. This extends the well-known results concerning non-existence of non-trivial scattering states for long-range nonlinearities.

math.AP↗
arXiv2019

Scattering below the ground state for the 2$d$ radial nonlinear Schrödinger equation

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. The key ingredient is a localized virial/Morawetz estimate; the radial assumption aids in controlling the error terms resulting from the spatial localization.

math.AP↗
arXiv2019

Invariance of white noise for KdV on the line

We consider the Korteweg--de Vries equation with white noise initial data, posed on the whole real line, and prove the almost sure existence of solutions. Moreover, we show that the solutions obey the group property and follow a white noise law at all times, past or future. As an offshoot of our methods, we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting.

math.AP↗
arXiv2018

Scattering for defocusing energy subcritical nonlinear wave equations

We consider the Cauchy problem for the defocusing power type nonlinear wave equation in $(1+3)$-dimensions for energy subcritical powers $p$ in the range $3 < p< 5$. We prove that any solution is global-in-time and scatters to free waves in both time directions as long as its critical Sobolev norm stays bounded on the maximal interval of existence.

math.AP↗
arXiv2018

The energy-critical nonlinear wave equation with an inverse-square potential

We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocusing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold.

math.AP↗
arXiv2018

Stability of small solitary waves for the 1$d$ NLS with an attractive delta potential

We consider the initial-value problem for the one-dimensional nonlinear Schrödinger equation in the presence of an attractive delta potential. We show that for sufficiently small initial data, the corresponding global solution decomposes into a small solitary wave plus a radiation term that decays and scatters as $t\to\infty$. In particular, we establish the asymptotic stability of the family of small solitary waves.

math.AP↗
arXiv2018

The radial mass-subcritical NLS in negative order Sobolev spaces

We consider the mass-subcritical NLS in dimensions $d\geq 3$ with radial initial data. In the defocusing case, we prove that any solution that remains bounded in the critical Sobolev space throughout its lifespan must be global and scatter. In the focusing case, we prove the existence of a threshold solution that has a compact flow.

math.AP↗
arXiv2018

The initial-value problem for the cubic-quintic NLS with non-vanishing boundary conditions

We consider the initial-value problem for the cubic-quintic NLS \[ (i\partial_t+Δ)ψ=α_1 ψ-α_{3}\vert ψ\vert^2 ψ+α_5\vert ψ\vert^4 ψ\] in three spatial dimensions in the class of solutions with $|ψ(x)|\to c >0$ as $|x|\to\infty$. Here $α_1$, $α_3$, $α_5$ and $c$ are such that $ψ(x)\equiv c$ is an energetically stable equilibrium solution to this equation. Normalizing the boundary condition to $ψ(x)\to 1$ as $|x|\to\infty$, we study the associated initial-value problem for $u=ψ-1$ and prove a scattering result for small initial data in a weighted Sobolev space.

math.AP↗
arXiv2017

Almost sure scattering for the energy-critical NLS with radial data below $H^1(\mathbb{R}^4)$

We prove almost sure global existence and scattering for the energy-critical nonlinear Schrödinger equation with randomized spherically symmetric initial data in $H^s(\mathbb{R}^4)$ with $\frac56<s<1$. We were inspired to consider this problem by the recent work of Dodson--Lührmann--Mendelson, which treated the analogous problem for the energy-critical wave equation.

math.AP↗
arXiv2017

Random data final-state problem for the mass-subcritical NLS in $L^2$

We study the final-state problem for the mass-subcritical NLS above the Strauss exponent. For $u_+\in L^2$, we perform a physical-space randomization, yielding random final states $u_+^ω\in L^2$. We show that for almost every $ω$, there exists a unique, global solution to NLS that scatters to $u_+^ω$. This complements the deterministic result of Nakanishi, which proved the existence (but not necessarily uniqueness) of solutions scattering to prescribed $L^2$ final states.

math.AP↗
arXiv2017

Scattering in $H^1$ for the intercritical NLS with an inverse-square potential

We study the nonlinear Schrödinger equation with an inverse-square potential in dimensions $3\leq d \leq 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in $H^1$ for arbitrary data.

math.AP↗
arXiv2016

Large data mass-subcritical NLS: critical weighted bounds imply scattering

We consider the mass-subcritical nonlinear Schrödinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity $s_c\in(\max\{-1,-\frac{d}{2}\},0)$, we prove that any solution satisfying $\|\, |x|^{|s_c|}e^{-itΔ} u\|_{L_t^\infty L_x^2} <\infty$ on its maximal interval of existence must be global and scatter.

math.AP↗