Searcharxiv⌕ Search

arXiv subjects

Tai-Peng Tsai

Publications and source records attributed to Tai-Peng Tsai.

At least 55 records · Page 3Linked to original sources
arXiv2016

Stability of periodic waves of 1D cubic nonlinear Schr{ö}dinger equations

We study the stability of the cnoidal, dnoidal and snoidal elliptic functions as spatially-periodic standing wave solutions of the 1D cubic nonlinear Schr{ö}dinger equations. First, we give global variational characterizations of each of these periodic waves, which in particular provide alternate proofs of their orbital stability with respect to same-period perturbations, restricted to certain subspaces. Second, we prove the spectral stability of the cnoidal waves against same-period perturbations (in a certain parameter range), and provide an alternate proof of this (known) fact for the snoidal waves, which does not rely on complete integrability. Third, we give a rigorous version of a formal asymptotic calculation of Rowlands to establish the instability of a class of real-valued periodic waves in 1D, which includes the cnoidal waves of the 1D cubic focusing nonlinear Schr{ö}dinger equation, against perturbations with period a large multiple of their fundamental period. Finally, we develop a numerical method to compute the minimizers of the energy with fixed mass and momentum constraints. Numerical experiments support and complete our analytical results.

math.AP↗
arXiv2016

Green tensor of the Stokes system and asymptotics of stationary Navier-Stokes flows in the half space

We derive refined estimates of the Green tensor of the stationary Stokes system in the half space. We then investigate the spatial asymptotics of stationary solutions of the incompressible Navier-Stokes equations in the half space. We also discuss the asymptotics of fast decaying flows in the whole space and exterior domains. In the Appendix we consider axisymmetric self-similar solutions.

math.AP↗
arXiv2016

Finite and infinite soliton and kink-soliton trains of nonlinear Schrödinger equations

We will first review known results on multi-solitons of dispersive partial differential equations, which are special solutions behaving like the sum of many weakly-interacting solitary waves. We will then describe our recent joint work with Dong Li on nonlinear Schrödinger equations: Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of a soliton train which is a multi-soliton composed of infinitely many solitons. In the 1D case, we can add to the infinite train an additional half-kink, which is a solution with a non-zero background at minus infinity.

math.AP↗
arXiv2015

Forward discretely self-similar solutions of the Navier-Stokes equations II

For any discretely self-similar, incompressible initial data $v_0$ which satisfies $\|v_0 \|_{L^3_w(\mathbb R^3)}\leq c_0$ where $c_0$ is allowed to be large, we construct a forward discretely self-similar local Leray solution in the sense of Lemarié-Rieusset to the 3D Navier-Stokes equations in the whole space. No further assumptions are imposed on the initial data; in particular, the data is not required to be continuous or locally bounded on $\mathbb R^3\setminus \{0\}$. The same method is used to construct self-similar solutions for any $-1$-homogeneous initial data in $L^3_w$.

math.AP↗
arXiv2015

Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations

In this note we construct mixed dimensional infinite soliton trains, which are solutions of nonlinear Schrödinger equations whose asymptotic profiles at time infinity consist of infinitely many solitons of multiple dimensions. For example infinite line-point soliton trains in 2D space, and infinite plane-line-point soliton trains in 3D space. This note extends the works of Le Coz, Li and Tsai [5, 6], where single dimensional trains are considered. In our approach, spatial $L^\infty$ bounds for lower dimensional trains play an essential role.

math.AP↗
arXiv2014

Remark on Luo-Hou's ansatz for a self-similar solution to the 3D Euler equations

In this note we show that Luo-Hou's ansatz for the self-similar solution to the axisymmetric solution to the 3D Euler equations leads to triviality of the solution under suitable decay condition of the blow-up profile. The equations for the blow-up profile reduces to an over-determined system of partial differential equations, whose only solution with decay is the trivial solution. We also propose a generalization of Luo-Hou's ansatz. Using the vanishing of the normal velocity at the boundary, we show that this generalized self-similar ansatz also leads to a trivial solution. These results show that the self-similar ansatz may be valid either only in a time-dependent region which shrinks to the boundary circle at the self-similar rate, or under different boundary conditions at spatial infinity of the self-similar profile.

math.AP↗
arXiv2014

Regularity criteria in weak $L^3$ for 3D incompressible Navier-Stokes equations

We study the regularity of a distributional solution $(u,p)$ of the 3D incompressible evolution Navier-Stokes equations. Let $B_r$ denote concentric balls in $\mathbb{R}^3$ with radius $r$. We will show that if $p\in L^{m} (0,1; L^1(B_2))$, $m>2$, and if $u$ is sufficiently small in $L^{\infty} (0,1; L^{3,\infty}(B_2))$, without any assumption on its gradient, then $u$ is bounded in $B_1\times (\frac{1}{10},1)$. It is an endpoint case of the usual Serrin-type regularity criteria, and extends the steady-state result of Kim-Kozono to the time dependent setting. In the appendix we also show some nonendpoint borderline regularity criteria.

math.AP↗
arXiv2013

Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations

We look for solutions to generic nonlinear Schrödinger equations build upon solitons and kinks. Solitons are localized solitary waves and kinks are their non localized counter-parts. We prove the existence of infinite soliton trains, i.e. solutions behaving at large time as the sum of infinitely many solitons. We also show that one can attach a kink at one end of the train. Our proofs proceed by fixed point arguments around the desired profile. We present two approaches leading to different results, one based on a combination of dispersive estimates and Strichartz estimates, the other based only on Strichartz estimates.

math.AP↗
arXiv2013

Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations

We study *infinite soliton trains* solutions of nonlinear Schrödinger equations (NLS), i.e. solutions behaving at large time as the sum of infinitely many solitary waves. Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of such a soliton train. We also give a new construction of multi-solitons (i.e. finite trains) and prove uniqueness in an exponentially small neighborhood, and we consider the case of solutions composed of several solitons and kinks (i.e. solutions with a non-zero background at infinity).

math.AP↗
arXiv2013

Local Dynamics Near Unstable Branches of NLS Solitons

Consider a branch of unstable solitons of NLS whose linearized operators have one pair of simple real eigenvalues in addition to the zero eigenvalue. Under radial symmetry and standard assumptions, solutions to initial data from a neighbourhood of the branch either converge to a soliton, or exit a larger neighbourhood of the branch transversally. The qualitative dynamic near a branch of unstable solitons is irrespective of whether blowup eventually occurs, which has practical implications for the description of blowup of NLS with supercritical nonlinearity.

math.AP↗
arXiv2011

Asymptotics of small exterior Navier-Stokes flows with non-decaying boundary data

We prove the unique existence of solutions of the 3D incompressible Navier-Stokes equations in an exterior domain with small non-decaying boundary data, for $t \in R$ or $t \in (0,\infty)$. In the latter case it is coupled with small initial data in weak $L^{3}$. As a corollary, the unique existence of time-periodic solutions is shown for the small periodic boundary data. We next show that the spatial asymptotics of the periodic solution is given by the same Landau solution at all times. Lastly we show that if the boundary datum is time-periodic and the initial datum is asymptotically discretely self-similar, then the solution is asymptotically the sum of a time-periodic vector field and a forward discretely self-similar vector field as time goes to infinity. It in particular shows the stability of periodic solutions in a local sense.

math.AP↗
arXiv2010

Small solutions of nonlinear Schrödinger equations near first excited states

Consider a nonlinear Schrödinger equation in $R^3$ whose linear part has three or more eigenvalues satisfying some resonance conditions. Solutions which are initially small in $H^1 \cap L^1(R^3)$ and inside a neighborhood of the first excited state family are shown to converge to either a first excited state or a ground state at time infinity. An essential part of our analysis is on the linear and nonlinear estimates near nonlinear excited states, around which the linearized operators have eigenvalues with nonzero real parts and their corresponding eigenfunctions are not uniformly localized in space.

math.AP↗
arXiv2009

Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies either $|v (x,t)| \le C_*{|t|^{-1/2}} $ or, for some $\e > 0$, $|v (x,t)| \le C_* r^{-1+ε} |t|^{-ε/2}$ for $-T_0\le t < 0$ and $0<C_*<\infty$ allowed to be large. We prove that $v$ is regular at time zero.

math.AP↗