Publications (60)
Asymptotic stability of solitary waves for the NLS with an attractive delta potential
Satoshi Masaki, Jason Murphy, Jun-ichi Segata
We consider the one-dimensional nonlinear Schrödinger equation with an attractive delta potential and mass-supercritical nonlinearity. This equation admits a one-parameter family…
Threshold dynamics for the 4 mass-energy double critical NLS
Alex H. Ardila, Zuyu Ma, Jason Murphy +1
We consider the 4 mass-energy double critical NLS \[ (i\partial_t+Î)u = -|u|^2 u + |u| u. \] In Luo (2024) and Cheng--Miao--Zhao (2016), the authors established a scattering/bl…
Scattering for the non-radial energy-critical inhomogeneous NLS
Carlos M. Guzmán, Jason Murphy
We prove scattering below the ground state threshold for an energy-critical inhomogeneous nonlinear Schrödinger equation in three space dimensions. In particular, we extend result…
Almost sure scattering for the energy-critical NLS with radial data below
Rowan Killip, Jason Murphy, Monica Visan
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…
Scattering for the non-radial inhomogeneous NLS
Changxing Miao, Jason Murphy, Jiqiang Zheng
We extend the result of Farah and Guzmán on scattering for the cubic inhomogeneous NLS to the non-radial setting. The key new ingredient is a construction of scattering solut…
Well-posedness and blowup for the dispersion-managed nonlinear Schrödinger equation
Jason Murphy, Tim Van Hoose
We consider the nonlinear Schrödinger equation with periodic dispersion management. We first establish global-in-time Strichartz estimates for the underlying linear equation with…
Scattering below the ground state for the intercritical non-radial inhomogeneous NLS
Mykael Cardoso, Luiz Gustavo Farah, Carlos M. Guzmán +1
We consider the focusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Îu + |x|^{-b}|u|^αu = 0\quad\text{on}\quad\mathbb{R}\times\mathbb{R}^N, \] with $N\geq 2…
The energy-critical nonlinear wave equation with an inverse-square potential
Changxing Miao, Jason Murphy, Jiqiang Zheng
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 in…
Cubic-quintic NLS: scattering beyond the virial threshold
Rowan Killip, Jason Murphy, Monica Visan
We consider the nonlinear Schrödinger equation in three space dimensions with combined focusing cubic and defocusing quintic nonlinearity. This problem was considered previously b…
Scattering below the ground state for the 2 radial nonlinear Schrödinger equation
Anudeep Kumar Arora, Benjamin Dodson, Jason Murphy
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 simp…
Averaging for the dispersion-managed NLS
Luccas Campos, Jason Murphy, Tim Van Hoose
We establish global-in-time averaging for the -critical dispersion-managed nonlinear Schrödinger equation in the fast dispersion management regime. In particular, in the case…
Determination of radial nonlocal nonlinearities from the scattering map
Luccas Campos, Rowan Killip, Jason Murphy +1
We show that the small-data scattering map uniquely determines the nonlinearity for a class of nonlinear Schrödinger equations with radial, Hartree-type nonlinearities. Our assump…
Dynamics of subcritical threshold solutions for the 4d energy-critical NLS
Zuyu Ma, Changxing Miao, Jason Murphy +1
We study dynamics of the 4 energy-critical nonlinear Schrödinger equation at the ground state energy. Previously, Duyckaerts and Merle [Geom. Funct. Anal. (2009)] proved that a…
Threshold solutions for the intercritical inhomogeneous NLS
Luccas Campos, Jason Murphy
We consider the focusing inhomogeneous nonlinear Schrödinger equation in , \begin{equation} i\partial_t u + Îu + |x|^{-b}|u|^{2}u=0,{equation} where $0 < b <\t…
Threshold solutions for the cubic INLS: the energy-subcritical case
Luccas Campos, Luiz Gustavo Farah, Jason Murphy
We revisit the work [L. Campos and J. Murphy, SIAM J. Math. Anal., 55 (2023), pp. 3807--3843], which classified the dynamics of solutions at the ground state threshold for cu…
The focusing cubic NLS with inverse-square potential in three space dimensions
Rowan Killip, Jason Murphy, Monica Visan +1
We consider the focusing cubic nonlinear Schrödinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup,…
Determination of Schrödinger nonlinearities from the scattering map
Rowan Killip, Jason Murphy, Monica Visan
We prove that the small-data scattering map uniquely determines the nonlinearity for a wide class of gauge-invariant, intercritical nonlinear Schrödinger equations. We use the Bor…
Recovery of a cubic nonlinearity for the nonlinear Schrödinger equation
Christopher C. Hogan, Jason Murphy, David Grow
We consider the problem of recovering a spatially-localized cubic nonlinearity in a nonlinear Schrödinger equation in dimensions two and three. We prove that solutions with data g…
Modified scattering for the cubic NLS with a repulsive delta potential
Satoshi Masaki, Jason Murphy, Jun-ichi Segata
We consider the initial-value problem for the cubic nonlinear Schrödinger equation with a repulsive delta potential. We prove that small initial data in a weighted Sobolev sp…
The defocusing energy-supercritical NLS in four space dimensions
Changxing Miao, Jason Murphy, Jiqiang Zheng
We consider a class of defocusing energy-supercritical nonlinear Schrödinger equations in four space dimensions. Following a concentration-compactness approach, we show that for $…
Scattering for the NLS with inhomogeneous nonlinearities
Luke Baker, Jason Murphy
We prove large-data scattering in for inhomogeneous nonlinear Schrödinger equations in one space dimension for powers . We assume the inhomogeneity is nonnegative and r…
A simple proof of scattering for the intercritical inhomogeneous NLS
Jason Murphy
We adapt the argument of Dodson-Murphy to give a simple proof of scattering below the ground state for the intercritical inhomogeneous nonlinear Schrödinger equation. The decaying…
Global dynamics below excited solitons for the non-radial NLS with potential
Satoshi Masaki, Jason Murphy, Jun-ichi Segata
We consider the global dynamics of solutions to the cubic nonlinear Schrödinger equation in the presence of an external potential, in the setting in which the equation admits…
Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations
Jason Murphy, Kenji Nakanishi
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, so…
Almost global existence for cubic nonlinear Schrödinger equations in one space dimension
Jason Murphy, Fabio Pusateri
We consider non-gauge-invariant cubic nonlinear Schrödinger equations in one space dimension. We show that initial data of size in a weighted Sobolev space lead to s…
Large scale limit for a dispersion-managed NLS
Jason Murphy
We derive the standard power-type NLS as a scaling limit of the Gabitov--Turitsyn dispersion-managed NLS, using the defocusing, cubic equation as a model case. In particular,…
Threshold dynamics for the radial cubic NLS with repulsive inverse-square potential
Luke Baker, Luccas Campos, Jason Murphy +1
We classify the dynamics of radial solutions at the (radial) ground state threshold for the cubic NLS in the presence of a repulsive inverse-square potential.
Scattering for defocusing energy subcritical nonlinear wave equations
Benjamin Dodson, Andrew Lawrie, Dana Mendelson +1
We consider the Cauchy problem for the defocusing power type nonlinear wave equation in -dimensions for energy subcritical powers in the range . We prove that…
Stability of small solitary waves for the 1 NLS with an attractive delta potential
Satoshi Masaki, Jason Murphy, Jun-ichi Segata
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…
Inter-critical NLS: critical -bounds imply scattering
Jason Murphy
We consider a class of power-type nonlinear Schrödinger equations for which the power of the nonlinearity lies between the mass- and energy-critical exponents. Following the conce…
Stability of nonlinear recovery from scattering and modified scattering maps
Gong Chen, Jason Murphy
We prove stability estimates for the recovery of the nonlinearity from the scattering or modified scattering map for one-dimensional nonlinear Schrödinger equations. We consider n…
Recovery of the nonlinearity from the modified scattering map
Gong Chen, Jason Murphy
We consider a class of one-dimensional nonlinear Schrödinger equations of the form \[ (i\partial_t+Î)u = [1+a]|u|^2 u. \] For suitable localized functions , such equations adm…
Approximate traveling waves for the focusing NLS with an external potential
Luke Baker, Xuemei Li, Jason Murphy
We construct approximate traveling wave solutions for the focusing cubic NLS perturbed by a rapidly decaying external potential. As an application, we demonstrate that there e…
The initial-value problem for the cubic-quintic NLS with non-vanishing boundary conditions
Rowan Killip, Jason Murphy, Monica Visan
We consider the initial-value problem for the cubic-quintic NLS \[ (i\partial_t+Î)Ï=α_1 Ï-α_{3}\vert Ï\vert^2 Ï+α_5\vert Ï\vert^4 Ï\] in three spatial dimensions in the c…
Numerical simulations for the energy-supercritical nonlinear wave equation
Jason Murphy, Yanzhi Zhang
We carry out numerical simulations of the defocusing energy-supercritical nonlinear wave equation for a range of spherically-symmetric initial conditions. We demonstrate numericall…
Threshold dynamics for the 3 radial NLS with combined nonlinearity
Alex H. Ardila, Jason Murphy, Jiqiang Zheng
We consider the nonlinear Schrödinger equation with focusing quintic and defocusing cubic nonlinearity in three space dimensions: \[ (i\partial_t+Î)u = |u|^2 u - |u|^4 u. \] In […
The cubic-quintic nonlinear Schrödinger equation with inverse-square potential
Alex H. Ardila, Jason Murphy
We consider the nonlinear Schrödinger equation in three space dimensions with a focusing cubic nonlinearity and defocusing quintic nonlinearity and in the presence of an external…
Stability estimates for the recovery of the nonlinearity from scattering data
Gong Chen, Jason Murphy
We prove stability estimates for the problem of recovering the nonlinearity from scattering data. We focus our attention on nonlinear Schrödinger equations of the form \[ (i\parti…
Random data final-state problem for the mass-subcritical NLS in
Jason Murphy
We study the final-state problem for the mass-subcritical NLS above the Strauss exponent. For , we perform a physical-space randomization, yielding random final states…
Large data mass-subcritical NLS: critical weighted bounds imply scattering
Rowan Killip, Satoshi Masaki, Jason Murphy +1
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\…
A new proof of scattering below the ground state for the 3d radial focusing cubic NLS
Benjamin Dodson, Jason Murphy
We revisit the scattering result of Holmer and Roudenko on the radial focusing cubic NLS in three space dimensions. Using the radial Sobolev embedding and a virial/Morawetz estimat…
Recovery of a spatially-dependent coefficient from the NLS scattering map
Jason Murphy
We follow up on work of Strauss, Weder, and Watanabe concerning scattering and inverse scattering for nonlinear Schrödinger equations with nonlinearities of the form $α(x)|u|^p u…
The radial mass-subcritical NLS in negative order Sobolev spaces
Rowan Killip, Satoshi Masaki, Jason Murphy +1
We consider the mass-subcritical NLS in dimensions with radial initial data. In the defocusing case, we prove that any solution that remains bounded in the critical Sobol…
Threshold solutions for the 3d cubic-quintic NLS
Alex H. Ardila, Jason Murphy
We study the cubic-quintic NLS in three space dimensions. It is known that scattering holds for solutions with mass-energy in a region corresponding to positive virial, the boundar…
Transmission of fast solitons for the NLS with an external potential
Christopher C. Hogan, Jason Murphy
We consider the dynamics of a boosted soliton evolving under the cubic NLS with an external potential. We show that for sufficiently large velocities, the soliton is effectively tr…
Threshold solutions for the cubic INLS: the energy-critical case
Luccas Campos, Luiz Gustavo Farah, Jason Murphy
We study the energy-critical cubic inhomogeneous NLS equation . In this work, we prove the existence of special solutions with…
The scattering map determines the nonlinearity
Rowan Killip, Jason Murphy, Monica Visan
Using the two-dimensional nonlinear Schrödinger equation (NLS) as a model example, we present a general method for recovering the nonlinearity of a nonlinear dispersive equation f…
Small and large data scattering for the dispersion-managed NLS
Jumpei Kawakami, Jason Murphy
We prove several scattering results for dispersion-managed nonlinear Schrödinger equations. In particular, we establish small-data scattering for both `intercritical' and `mass-su…
Invariance of white noise for KdV on the line
Rowan Killip, Jason Murphy, Monica Visan
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 th…
Threshold scattering for the focusing NLS with a repulsive potential
Changxing Miao, Jason Murphy, Jiqiang Zheng
We adapt the arguments in the recent work of Duyckaerts, Landoulsi, and Roudenko to establish a scattering result at the sharp threshold for the focusing cubic NLS with a repu…
The defocusing quintic NLS in four space dimensions
Benjamin Dodson, Changxing Miao, Jason Murphy +1
We consider the defocusing quintic nonlinear Schrödinger equation in four space dimensions. We prove that any solution that remains bounded in the critical Sobolev space must be g…
Scattering in for the intercritical NLS with an inverse-square potential
Jing Lu, Changxing Miao, Jason Murphy
We study the nonlinear Schrödinger equation with an inverse-square potential in dimensions . We consider both focusing and defocusing nonlinearities in the mass-su…
The radial defocusing nonlinear Schrödinger equation in three space dimensions
Jason Murphy
We study the defocusing nonlinear Schrödinger equation in three space dimensions. We prove that any radial solution that remains bounded in the critical Sobolev space must be glob…
Modified scattering for a dispersion-managed nonlinear Schrödinger equation
Jason Murphy, Tim Van Hoose
We prove sharp decay and modified scattering for a one-dimensional dispersion-managed cubic nonlinear Schrödinger equation with small initial data chosen from a weighte…
The final-state problem for the cubic-quintic NLS with non-vanishing boundary conditions
Rowan Killip, Jason Murphy, Monica Visan
We construct solutions with prescribed scattering state to the cubic-quintic NLS in three spatial di…
Threshold scattering for the 2d radial cubic-quintic NLS
Jason Murphy
We consider the cubic-quintic nonlinear Schrödinger equation in two space dimensions. For this model, X. Cheng established scattering for data with mass strictly below that…
The defocusing -critical NLS in high dimensions
Jason Murphy
We consider the defocusing -critical nonlinear Schrödinger equation in dimensions . In the spirit of Kenig and Merle [Trans. Amer. Math. Soc. 362 (2010), 1…
A new proof of scattering below the ground state for the non-radial focusing NLS
Benjamin Dodson, Jason Murphy
We revisit the scattering result of Duyckaerts, Holmer, and Roudenko for the non-radial -critical focusing NLS. By proving an interaction Morawetz inequality, we give…
A note on averaging for the dispersion-managed NLS
Jason Murphy
We discuss averaging for dispersion-managed nonlinear Schrödinger equations in the fast dispersion management regime, with an application to the problem of constructing soliton-li…
Modified scattering for the cubic dispersion-managed NLS
Jason Murphy, Jiqiang Zheng
We establish a small-data modified scattering result for the cubic dispersion-managed NLS (with time-dependent dispersion map) for initial data in a weighted space.