Mathematical Research Letters 26(1) 253-279 Jun 2019 [Refereed]

We consider the final-data problem for systems of nonlinear Schrödinger
equations with subcritical nonlinearity. An asymptotically free solution
is uniquely obtained for almost every randomized asymptotic profile in
, ex...

J. Differential Equations 265(11) 5656-5675 Dec 2018 [Refereed]

We consider the estimates for the solutions to the wave and
Schrödinger equations in high dimensions. For the homogeneous estimates, we
show estimates fail at the critical regularity in high
dimensions by using sta...

Communications in Mathematical Physics 359(1) 265-295 Apr 2018 [Refereed]

We study the Cauchy problem for the 3D Gross-Pitaevskii equation. The global
well-posedness in the natural energy space was proved by Gérard
\cite{Gerard}. In this paper we prove scattering for small data in the same
space with some additional a...

This article resolves some errors in the paper "Scattering threshold for the
focusing nonlinear Klein-Gordon equation", Analysis & PDE 4 (2011) no. 3,
405-460. The errors are in the energy-critical cases in two and higher
dimensions.

Ioan Bejenaru, Zihua Guo, Sebastian Herr, Kenji Nakanishi

Analysis & PDE 8-8 (2015), 2029-2055 Apr 2015

The Cauchy problem for the Zakharov system in four dimensions is considered.
Some new well-posedness results are obtained. For small initial data, global
well-posedness and scattering results are proved, including the case of initial
data in the e...

International Mathematics Research Notices 2015(22) 11549-11565 Feb 2015 [Refereed]

The aim of this article is to prove that for the 2+1-dimensional equivariant
Faddeev model, which is a quasilinear generalization of the corresponding
nonlinear sigma model, small initial data in critical Besov spaces evolve into
global solutions ...

Zihua Guo, Sanghyuk Lee, Kenji Nakanishi, Chengbo Wang

Communications in Mathematical Physics 331(1) 239-259 Oct 2014 [Refereed]

We obtain scattering for the 3D Zakharov system with non-radial small data in
the energy space with angular regularity of degree one. The main ingredient is
a generalized Strichartz estimate for the Schrödinger equation in the space
of ang...

Communications in Partial Differential Equations 39(6) 1158-1184 May 2014 [Refereed]

We consider the global dynamics below the ground state energy for the
Klein-Gordon-Zakharov system in the 3D radial case; and obtain the dichotomy
between scattering and finite time blow up.

Mathematische Annalen 361(1-2) 1-50 Feb 2015 [Refereed]

We construct a center-stable manifold of the ground state solitons in the
energy space for the critical wave equation without imposing any symmetry, as
the dynamical threshold between scattering and blow-up, and also as a
collection of solutions w...

Chongsheng Cao, Slim Ibrahim, Kenji Nakanishi, Edriss S. Titi

Communications in Mathematical Physics 337(2) 473-482 Jul 2015 [Refereed]

In an earlier work we have shown the global (for all initial data and all
time) well-posedness of strong solutions to the three-dimensional viscous
primitive equations of large scale oceanic and atmospheric dynamics. In this
paper we show that for...

Communications in Mathematical Physics 327(1) 309-332 Apr 2014 [Refereed]

For the critical focusing wave equation on in the
radial case, we establish the role of the "center stable" manifold
constructed in \cite{KS} near the ground state as a threshold between
type I blowup and...

Mathematical Research Letters 21(4) 733-755 2014 [Refereed]

We prove small energy scattering for the 3D Klein-Gordon-Zakharov system with
radial symmetry. The idea of proof is the same as the Zakharov system studied
in \cite{GN}, namely to combine the normal form reduction and the
radial-improved Strichart...

Advances in Mathematics 238(1) 412-441 May 2013 [Refereed]

We consider the global dynamics below the ground state energy for the
Zakharov system in the 3D radial case. We obtain dichotomy between the
scattering and the growup.

International Mathematics Research Notices 2014(9) 2327-2342 Jan 2013 [Refereed]

We prove small energy scattering for the 3D Zakharov system with radial
symmetry. The main ingredients are normal form reduction and the
radial-improved Strichartz estimates.

Discrete and Continuous Dynamical Systems 33(6) 2423-2450 2013 [Refereed]

In this paper we establish the existence of certain classes of solutions to
the energy critical nonlinear wave equation in dimensions 3 and 5 assuming that
the energy exceeds the ground state energy only by a small amount. No radial
assumption is ...

Trans. Amer. Math. Soc. 366 5653-5669 May 2014 [Refereed]

We study global dynamics for the focusing nonlinear Klein-Gordon equation
with the energy-critical nonlinearity in two or higher dimensions when the
energy equals the threshold given by the ground state of a mass-shifted
equation, and prove that t...

Journal of the European Mathematical Society 17(4) 819-835 2015 [Refereed]

Trudinger-Moser inequality is a substitute to the (forbidden) critical
Sobolev embedding, namely the case where the scaling corresponds to .
It is well known that the original form of the inequality with the sharp
exponent (proved by Mos...

Dan-Andrei Geba, Kenji Nakanishi, Sarada G. Rajeev

Communications on Pure & Applied Analysis 11(5) 1923-1933 Sep 2012 [Refereed]

We study equivariant maps corresponding to the classical Skyrme model and the
Adkins-Nappi model, for which we prove global existence and scattering in
critical Sobolev-Besov spaces.

SIAM Journal on Mathematical Analysis 44(2) 1175-1210 Apr 2012 [Refereed]

We construct center-stable and center-unstable manifolds, as well as stable
and unstable manifolds, for the nonlinear Klein-Gordon equation with a focusing
energy sub-critical nonlinearity, associated with a family of solitary waves
which is gener...

Mathematische Zeitschrift 272(1-2) 297-316 Oct 2012 [Refereed]

In this paper we obtain a global characterization of the dynamics of even
solutions to the one-dimensional nonlinear Klein-Gordon (NLKG) equation on the
line with focusing nonlinearity |u|^{p-1}u, p>5, provided their energy exceeds
that of the gro...

American Journal of Mathematics 135(4) 935-965 Aug 2013 [Refereed]

We study global behavior of radial solutions for the nonlinear wave equation
with the focusing energy critical nonlinearity in three and five space
dimensions. Assuming that the solution has energy at most slightly more than
the ground states and ...

Journal of Functional Analysis 263(3) 703-781 Aug 2012 [Refereed]

Consider a nonlinear Schrödinger equation in whose linear part has
three or more eigenvalues satisfying some resonance conditions. Solutions which
are initially small in and inside a neighborhood of the
first excited st...

Calculus of Variations and Partial Differential Equations 44(1-2) 1-45 May 2012 [Refereed]

We extend our previous result on the nonlinear Klein-Gordon equation to the
nonlinear Schrodinger equation with the focusing cubic nonlinearity in three
dimensions, for radial data of energy at most slightly above that of the ground
state. We prov...

Journal of Differential Equations 250(5) 2299-2333 Mar 2011 [Refereed]

We study the focusing, cubic, nonlinear Klein-Gordon equation in 3D with
large radial data in the energy space. This equation admits a unique positive
stationary solution, called the ground state. In 1975, Payne and Sattinger
showed that solutions...

We show scattering versus blow-up dichotomy below the ground state energy for
the focusing nonlinear Klein-Gordon equation, in the spirit of Kenig-Merle for
the critical wave and Schrödinger equations. Our result includes the
critica...

Communications in Partial Differential Equations 36(5) 797-818 Jan 2011 [Refereed]

We derive a uniform exponential decay of the total energy for the nonlinear
Klein-Gordon equation with a damping around spatial infinity in the whole space
or in the exterior of a star shaped obstacle.

We investigate existence and asymptotic completeness of the wave operators
for nonlinear Klein-Gordon and Schrödinger equations with a defocusing
exponential nonlinearity in two space dimensions. A certain threshold is
defined based on the value...

Shu-Ming Chang, Stephen Gustafson, Kenji Nakanishi, Tai-Peng Tsai

SIAM Journal on Mathematical Analysis 39(4) 1070-1111 Nov 2006 [Refereed]

Nonlinear Schrödinger (NLS) equations with focusing power nonlinearities
have solitary wave solutions. The spectra of the linearized operators around
these solitary waves are intimately connected to stability properties of the
solitary waves, an...

International Mathematics Research Notices (66) 3559-3584 Jan 2004 [Refereed]

In this paper we study a class of nonlinear Schrödinger equations which
admit families of small solitary wave solutions. We consider solutions which
are small in the energy space , and decompose them into solitary wave and
dispersive wave c...