We consider a nonlinear semi-classical Schrodinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrodinger equation with harmonic potential.
In this paper we give new estimates for the solution to the Schrödinger equation with quadratic and sub-quadratic potentials in the framework of modulation spaces.
We prove global existence and scattering for a class of quadratic Schrodinger equations in dimension 2. The proof relies on the idea of space-time resonance.
In this paper, we use the variational method, especially the perturbation method, to find the perturbed high energy solutions of the quadratic coupled Schrodinger system with asymmetric asymptotic potential and their asymptotic behavior as the coupling coefficient converges to 0.
In this paper we study the scattering of radial solutions to a $l$-component system of nonlinear Schrödinger equations with quadratic-type growth interactions in dimension five. Our approach is based on the recent technique introduced by Dodson and Murphy, which relies on the radial Sobolev embedding and a Morawetz estimate.
In this paper we consider the local well-posedness theory for the quadratic nonlinear Schrödinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a local well-posedness result close to scaling for small initial data.
We prove local well-posedness for the initial-boundary-value problem associated to some quadratic nonlinear Schrödinger equations on the half-line. The results are obtained in the low regularity setting by introducing an analytic family of boundary forcing operators, following the ideas developed by Colliander, Kenig and Holmer.
We establish that the initial value problem for the quadratic non-linear Schrödinger equation $$ iu_t - Δu = u^2$$ where $u: \R^2 \times \R \to \C$, is locally well-posed in $H^s(\R^2)$ when $s > -1$. The critical exponent for this problem is $s_c=-1$ and previous work in \cite{c1} established local well-posedness for $s > -3/4$.
The initial value problem is considered for a higher order nonlinear Schrödinger equation with quadratic nonlinearity. Results on existence and uniqueness of weak solutions are obtained. In the case of an effective at infinity additional damping large-time decay of solutions without any smallness assumptions is also established. The main difficulty of the study is the non-smooth character of the nonlinearity.
In this paper we consider a Schrodinger equation on the circle with a quadratic nonlinearity. Thanks to an explicit computation of the first Picard iterate, we give a precision on the dynamic of the solution, whose existence was proved by C. E. Kenig, G. Ponce and L. Vega. We also show that the equation is well-posed in a space based on Lp norms in frequencies.
This paper provides a small data global existence result for a class of quadratic derivative nonlinear Schrödinger systems in two space dimensions. This is an extension of the previous results by Li [Discrete Contin. Dyn. Syst., 32 (2012), 4265--4285] and Ikeda--Katayama--Sunagawa [Ann. H. Poincaré 16 (2015), 535--567].
In this work we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms on the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.
In this paper we study the scattering of non-radial solutions in the energy space to coupled system of nonlinear Schrödinger equations with quadratic-type growth interactions in dimension five without the mass-resonance condition. Our approach is based on the recent technique introduced by Dodson and Murphy, which relies on an interaction Morawetz estimate. It is proved that any solution below the ground states scatters in time.
In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations. This system was introduced by M. Colin and T. Colin (2004). The first and second authors obtained some well-posedness results in the Sobolev space $H^{s}(\mathbb{R}^d)$. We improve these results for conditional radial initial data by rewriting the system radial form.
We consider the initial value problem for a three-component system of quadratic derivative nonlinear Schrödinger equations in two space dimensions with the masses satisfying the resonance relation. We present a structural condition on the nonlinearity under which small data global existence holds. It is also shown that the solution is asymptotically free. Our proof is based on the commuting vector field method combined with smoothing effects.
In this work, we show the existence of ground state solutions for an $l$-component system of non-linear Schrödinger equations with quadratic-type growth interactions in the energy-critical case. They are obtained analyzing a critical Sobolev-type inequality and using the concentration-compactness method. As an application, we prove the existence of blow-up solutions of the system without the mass-resonance condition in dimension six (and five), when the initial data is radial.
In this remark, we give another approach to the local well-posedness of quadratic Schrödinger equation with nonlinearity $u\bar u$ in $H^{-1/4}$, which was already proved by Kishimoto \cite{kis}. Our resolution space is $l^1$-analogue of $X^{s,b}$ space with low frequency part in a weaker space $L^{\infty}_{t}L^2_x$. Such type spaces was developed by Guo. \cite{G} to deal the KdV endpoint $H^{-3/4}$ regularity.
We prove inverse Strichartz theorems at $L^2$ regularity for a family of Schrödinger evolutions in one space dimension. Prior results rely on spacetime Fourier analysis and are limited to the translation-invariant equation $i\partial_t u = -\tfrac{1}{2} Δu$. Motivated by applications to the mass-critical Schrödinger equation with external potentials (such as the harmonic oscillator) we use a physical space approach.
We consider the quadratic Schrödinger system $$iu_t+Δ_{γ_1}u+\overline{u}v=0$$ $$2iv_t+Δ_{γ_2}v-βv+\frac 12 u^2=0,$$ where $t\in\mathbf{R},\,x\in \mathbf{R}^d\times \mathbf{R}$, in dimensions $1\leq d\leq 4$ and for $γ_1,γ_2>0$, the so-called elliptic-elliptic case. We show the formation of singularities and blow-up in the $L^2$-(super)critical case. Furthermore, we derive several stability results concerning the ground state solutions of this system.
We study propagation of phase space singularities for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. Phase space singularities are measured by the lack of polynomial decay of given order in open cones in the phase space, which gives a parametrized refinement of the Gabor wave front set. The main result confirms the fundamental role of the singular space associated to the quadratic form for the propagation of phase space singularities. The singularities are contained in the singular space, and propagate in the intersectio…
We consider the initial value problem for a three-component system of quadratic nonlinear Schrödinger equations with mass resonance in two space dimensions. Under a suitable condition on the coefficients of the nonlinearity, we will show that the solution decays strictly faster than $O(t^{-1})$ as $t \to +\infty$ in $L^{\infty}$ by providing with an enhanced decay estimate of order $O((t \log t)^{-1})$. Differently from the previous works, our approach does not rely on the explicit form of the asymptotic profile of the solution at all.
We consider the quadractic NLS posed on a bidimensional compact Riemannian manifold $(M, g)$ with $ \partial M \neq \emptyset$. Using bilinear and gradient bilinear Strichartz estimates for Schrödinger operators in two-dimensional compact manifolds proved by J. Jiang in \cite{JIANG} we deduce a new evolution bilinear estimates. Consequently, using Bourgain's spaces, we obtain a local well-posedness result for given data $u_0\in H^s(M)$ whenever $s> \frac{2}{3}$ in such manifolds.
In this paper, we study the solutions to the energy-critical quadratic nonlinear Schrödinger system in ${\dot H}^1\times{\dot H}^1$, where the sign of its potential energy can not be determined directly. If the initial data ${\rm u}_0$ is radial or non-radial but satisfies the mass-resonance condition, and its energy is below that of the ground state, using the compactness/rigidity method, we give a complete classification of scattering versus blowing-up dichotomies depending on whether the kinetic energy of ${\rm u}_0$ is below or above that of the ground state.
We study propagation of the Gabor wave front set for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. We point out that the singular space associated to the quadratic form plays a crucial role for the understanding of this propagation. We show that the Gabor singularities of the solution to the equation for positive times are always contained in the singular space, and that they propagate in this set along the flow of the Hamilton vector field associated to the imaginary part of the quadratic form. As an application we obt…
We study propagation of phase space singularities for the initial value Cauchy problem for a class of Schrödinger equations. The Hamiltonian is the Weyl quantization of a quadratic form whose real part is non-negative. The equations are studied in the framework of projective Gelfand--Shilov spaces and their distribution duals. The corresponding notion of singularities is called the Gelfand--Shilov wave front set and means the lack of exponential decay in open cones in phase space. Our main result shows that the propagation is determined by the singular space of the quadratic form, just as in t…
We study the nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity $|u|^2$, posed on the two-dimensional torus $\mathbb{T}^2$. While the relevant $L^3$-Strichartz estimate is known only with a derivative loss, we prove local well-posedness of the quadratic NLS in $L^2(\mathbb{T}^2)$, thus resolving an open problem of thirty years since Bourgain (1993). In view of ill-posedness in negative Sobolev spaces, this result is sharp. We establish a crucial bilinear estimate by separately studying the non-resonant and nearly resonant cases. As a corollary, we obtain a tri-linear version …
We establish blow-up results for systems of NLS equations with quadratic interaction in anisotropic spaces. We precisely show finite time blow-up or grow-up for cylindrical symmetric solutions. With our construction, we moreover prove some polynomial lower bounds on the kinetic energy of global solutions in the mass-critical case, which in turn implies grow-up along any diverging time sequence. Our analysis extends to general NLS systems with quadratic interactions, and it also provides improvements of known results in the radial case.
In this article we prove local well-posedness in low-regularity Sobolev spaces for general quasilinear Schrödinger equations. These results represent improvements of the pioneering works by Kenig-Ponce-Vega and Kenig-Ponce-Rolvung-Vega, where viscosity methods were used to prove existence of solutions in very high regularity spaces. Our arguments here are purely dispersive. The function spaces in which we show existence are constructed in ways motivated by the results of Mizohata, Ichinose, Doi, and others, including the authors.
This paper is concerned with the Cauchy problem of the quadratic nonlinear Schrödinger equation in $\mathbb{R} \times \mathbb{R}^2$ with the nonlinearity $η|u|^2$ where $η\in \mathbb{C} \setminus \{0\}$ and low regularity initial data. If $s < -1/4$, the ill-posedness result in the Sobolev space $H^{s}(\mathbb{R}^2)$ is known. We will prove the well-posedness in $H^s(\mathbb{R}^2)$ for $-1/2 < s < -1/4$ by assuming some angular regularity on initial data. The key tools are the modified Fourier restriction norm and the convolution estimate on thickened hypersurfaces.
In this work I study the well-posedness of the Cauchy problem associated with the coupled Schrödinger equations {with quadratic nonlinearities}, which appears modeling problems in nonlinear optics. I obtain the local well-posedness for data {in Sobolev spaces} with low regularity. {To obtain} the local theory, I prove new bilinear estimates for the coupling terms of the system in the continuous case. Concerning global results, in the continuous case, I establish the global well-posedness in $H^s(\mathbb{R})\times H^s(\mathbb{R})$, for some negatives indexes $s$. The proof of the global result …
We prove that no finite time blow up can occur for nonlinear Schroedinger equations with quadratic potentials, provided that the potential has a sufficiently strong repulsive component. This is not obvious in general, since the energy associated to the linear equation is not positive. The proof relies essentially on two arguments: global in time Strichartz estimates, and a refined analysis of the linear equation, which makes it possible to use continuity arguments and to control the nonlinear effects.
We establish that the quadratic non-linear Schrödinger equation $$ iu_t + u_{xx} = u^2$$ where $u: \R \times \R \to \C$, is locally well-posed in $H^s(\R)$ when $s \geq -1$ and ill-posed when $s < -1$. Previous work of Kenig, Ponce and Vega had established local well-posedness for $s > -3/4$. The local well-posedness is achieved by an iteration using a modification of the standard $X^{s,b}$ spaces. The ill-posedness uses an abstract and general argument relying on the high-to-low frequency cascade present in the non-linearity, and a computation of the first non-linear iterate.
We consider the quadratic nonlinear Schrödinger system (NLS system) \begin{align*}\begin{cases} i\partial_t u + Δu = v \overline{u}, \\ i\partial_t v+κΔv = u^2, \end{cases} \text{ on } I \times \mathbb{R}^5, \end{align*} where $κ>0$. The scattering below the standing wave solutions for NLS system was obtained by the first author when $κ= 1/2$. The condition of $κ=1/2$ is called mass-resonance. In this paper, we prove scattering below the standing wave solutions when $κ\neq 1/2$ under the radially symmetric assumption. Our proof is based on the concentration compactness and the rigidity by Keni…
This paper is dedicated to studying the semilinear Schrödinger equation $\left\{\begin{array}{ll}-\Nabla u+V(x)u=f(x, u), \ \ \ \ x\in {\R}^{N},u\in H^{1}({\R}^{N}),\end{array}\right.$ where $f$ is a superlinear, subcritical nonlinearity. It focuses on the case where $V(x)=V_0(x)+V_1(x)$, $V_0\in C(\R^N)$, $V_0(x)$ is 1-periodic in each of $x_1, x_2, \ldots, x_N$ and $\sup[σ(-\triangle +V_0)\cap (-\infty, 0)]<0<\inf[σ(-\triangle +V_0)\cap (0, \infty)]$, $V_1\in C(\R^N)$ and $\lim_{|x|\to\infty}V_1(x)=0$. A new super-quadratic condition is obtained, which is weaker than some well known results.
We consider the quadratic nonlinear Schrödinger system \begin{align*} \begin{cases} i\partial_t u +Δu =v \overline{u},\\ i\partial_t v +κΔv =u^2, \end{cases} \text{ on } I \times \mathbb{R}^d, \end{align*} where $1\leq d \leq 6$ and $κ>0$. In the lower dimensional case $d=1,2,3$, it is known that the $H^1$-solution is global in time. On the other hand, there are finite time blow-up solutions when $d=4,5,6$ and $κ=1/2$. The condition of $κ=1/2$ is called mass-resonance. In this paper, we prove finite time blow-up under radially symmetric assumption when $d=5,6$ and $κ\neq 1/2$ and we show blow-…
In the present paper, we consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. The local existence of the solution of the system in the Sobolev space $H^{s}$ for $s>d/2+3$ is proved by M. Colin and T. Colin. We prove the well-posedness of the system with low regularity initial data. For some cases, we also prove the well-posedness and the scattering at the scaling critical regularity by using $U^{2}$ space and $V^{2}$ space which are applied to prove the well-p…
We study the existence and stability of periodic traveling-wave solutions for the quadratic and cubic nonlinear Schrödinger equations in one space dimension.
In this paper we study the asymptotic behavior of a quadratic Schrödinger equation with electromagnetic potentials. We prove that small solutions scatter. The proof builds on earlier work of the author for quadratic NLS with a non magnetic potential. The main novelty is the use of various smoothing estimates for the linear Schrödinger flow in place of boundedness of wave operators to deal with the loss of derivative. As a byproduct of the proof we obtain boundedness of the wave operator of the linear electromagnetic Schrödinger equation on an $L^2$ weighted space for small potentials, as well …
We study the initial value problem of the quadratic nonlinear Schrödinger equation $$ iu_t+u_{xx}=u\bar{u}, $$ where $u:\R\times \R\to \C$. We prove that it's locally well-posed in $H^s(\R)$ when $s\geq -\dfrac{1}{4}$ and ill-posed when $s< -\dfrac{1}{4}$, which improve the previous work in \cite{KPV}. Moreover, we consider the problem in the following space, $$ H^{s,a}(\R)={u:\|u\|_{H^{s,a}}\triangleq (\displaystyle\int (|ξ|^sχ_{\{|ξ|>1\}}+|ξ|^aχ_{\{|ξ|\leq 1\}})^2|\hat{u}(ξ)|^2 dξ)^{1/2}<\infty} $$ for $s\leq 0, a\geq 0$. We establish the local well-posedness in $H^{s,a}(\R)$ when $s\geq -\d…
We study a class of $3D$ quadratic Schrödinger equations as follows, $(\partial_t -i Δ) u = Q(u, \bar{u})$. Different from nonlinearities of the $uu$ type and the $\bar{u}\bar{u}$ type, which have been studied by Germain-Masmoudi-Shatah, the interaction of $u$ and $\bar{u}$ is very strong at the low frequency part, e.g., $1\times 1 \rightarrow 0$ type interaction (the size of input frequency is "$1$" and the size of output frequency is "$0$"). It creates a growth mode for the Fourier transform of the profile of solution around a small neighborhood of zero. This growth mode will again cause the…
