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Questions tagged [dispersive-pde]

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1answer
27 views

Question about schrodinger free equation

Left $u$ be a solution to free linear schrodinger equation $u_t=i\Delta u$. The momentum is defined as $$P(t)=Im \int_{\mathbb{R}^N} \overline{u}(x,t)\nabla u(x,t) dx $$ I want to prove the following ...
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2answers
61 views

On the Schrodinger fundamental solution

Let $e^{it\Delta}$ be the fundamental Schrodinger solution. If $u_0$ is the corresponding initial data to the problem associated to Schrodinger free equation $u_t = i\Delta u$ and $S(\mathbb{R}^N)$ ...
1
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1answer
21 views

An Sobolev-type inequality on $1D$ torus related to algebra properties for Sobolev spaces

In Kishimoto-Tsutsumi's paper published by Math Research Letter 2018, I see the following inequality in the last line of page 10: $\| f g \partial_x h \|_{H^{-s}(T)} \lesssim \| f \|_{H^s(T)} \| g \...
2
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1answer
32 views

Dual local smoothing and retarded local smoothing for Schrodinger equation

This exercise is from Tao's Nonlinear Dispersive Equations: Local and Global Analysis, Exercise 2.54. Let $u$ be a solution to the inhomogeneous Schrodinger equation $i\partial_t u+\Delta u=F$, ...
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0answers
20 views

What does it really mean for a wave equation to be critical?

I am trying to understand intuitively the concept of criticality in general for Wave equations. For example, consider the cauchy problem of semi-linear equation \begin{equation} \begin{cases} \...
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0answers
57 views

Physical interpretation of a singular pde

Suppose $\Omega$ is a bounded domain in $\mathbb{R}^N$ and consider the following Dirichlet boundary condition: $$ -\Delta u=\frac{f(x)}{u^\delta}\text{ in }\Omega, u>0\text{ in }\Omega; $$ where $\...
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0answers
31 views

An asymptotic for a simple oscillatory integral

Consider the oscillatory integral $$I(\lambda):=\int_{0}^{\infty}\int_{0}^{\infty} \psi(x,y) \, \mathrm{e}^{\dot{\imath}\phi(x,y)}dxdy$$ where 1) the phase $\phi$ and amplitude $\psi$ are smooth 2)...
3
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1answer
222 views

The Fourier transform is unbounded from $L^{p}$ to $L^{p^{\prime}}$ when $2<p\leq \infty$?

I previously asked this question here Haussdorff-Young inequality optimal Lebesgue exponents range and got a comment that referred me to the answer here Fourier transform in $L^p$ but I did not ...
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0answers
132 views

Virial identity for nonlinear Schrödinger equation

Suppose that $u: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$ is a classical solution to the nonlinear Schrödinger equation: \begin{align*} iu_t+\frac{1}{2}\Delta u = -u|u|^2. \tag{1} \end{align*} ...
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0answers
43 views

dispersive relation for linearized Euler-Poisson equation

Euler-poisson equation written : $$n_t+(nu)_x=0$$ $$u_t+uu_x=-\phi_x$$ $$\phi_{xx}=e^\phi-n$$ And linearized following ways: The system has a constant solution $n=1, \phi=u=0$. By assuming that the ...
1
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1answer
60 views

Euler-Lagrange equations of 1D NLS with periodic external potential

I am trying to find the Euler Lagrange (EL) equations in order to find ODEs for the parameters $a(t), \xi(t), c(t), d(t)$. Consider the following form: $$iu_t+\frac{1}{2}u_{xx}+|u|^2u=V(x)u \tag{1}$$ ...
3
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1answer
98 views

Laplace transform in the Fokker-Planck equation

Given the Fokker-Planck equation $$D\frac{\partial^2}{\partial x^2}\rho(x;t)=\frac{\partial}{\partial t}\rho(x;t)$$ the paper I'm reading said to have taken the Laplace transform, resulting $$D\frac{\...
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0answers
59 views

Understanding the dual of a weighted Fourier transform space

I'm studying Tao's Dispersive PDE book. In section 2.6, he discusses $X^{s,b}_{\tau=h(\xi)}$ spaces, which are defined by the norm $$ \| u\|_{X^{s,b}_{\tau=h(\xi)}} = \| \langle \xi \rangle^s \langle \...
3
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1answer
26 views

$u$ satisfies Schrodinger equation implies $\mathcal{F}^{-1} \left(\chi_{2}(\xi) \hat{u} \right) $ also?

Consider Schr\"odinger equation (SE): $i \frac{\partial }{\partial t}u (x,t )+ \Delta u(x,t) =0, (x, t)\in \mathbb R^{N}\times \mathbb R.$ $u(0,x)=\phi(x).$ Then, formally, the solution of (SE) ...
3
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1answer
382 views

Infinite propagation speed for the Schrödinger equation

I've seen many articles making reference to the property of the infinite propagation speed for the solution of the linear Schrödinger equation; but i can't find a book giving a 'good' definition or a ...
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0answers
36 views

On Automorphism on Space of Smooth Functions

Given a Operator $T$ (an Automorphism) on the subspace $X$ of Smooth functions on $\mathbb R ^n$, $\mathcal C^\infty(\mathbb R^n)$ $$ X=\{u \in \mathcal C^\infty (\mathbb R^n) \,|\, supp \,(u)\...
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1answer
25 views

How to find spacial periodicity

I am struggling with the following question. Consider $$U(x,t)=2\cos\left(kx-\sqrt{\frac{a}{b}}k^2+(\Delta k)^2)t\right)\cos\left((\Delta k)(x-2\sqrt{\frac{a}{b}}kt\right)$$ and suppose that $\Delta ...
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0answers
119 views

linearized KDV equation

Consider the PDE $$u_t +4u_x +6u_{xxx} =0$$ for the function $u = u(x, t).$ (a) Determine for what wave speeds $c$ there exist bounded travelling wave solutions of $u$, and find the form of the ...
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76 views

$L^1$-blow up in the free Schroedinger evolution

I would like to understand, possibly with an explicit example, how the free Schroedinger evolution does not leave $L^1$ invariant. More precisely: given $f\in L^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d)$,...
2
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1answer
64 views

Conservation law for Benjamin Ono equation

Consider the Bejamin Ono equation \begin{equation} \partial_t u + H\partial_{xx}u = u\,\partial_x u, \end{equation} where $u=u(x,t): \mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is a real scalar field, ...
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0answers
96 views

Strichartz Estimate with Fourier Transform

Let $f$ be a Schwartz function. Prove that, whenever $2\le r < \infty,$ $$\| e^{it \Delta} f\|_{L^{3r}(\mathbb{R}^2_{xt})} \le c \| \widehat{f}\|_{r'},$$ Where $1/r + 1/r' = 1.$ My Attempt My ...
3
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0answers
172 views

Wave equation on a finite domain with no internal resonance

I am considering the one-dimensional wave equation on a finite interval $[0,1]$. Loosely speaking, the formulation is: Find $u(x,t)$ solution of $$ u_{,tt}-c^2u_{,xx}=0,\quad x=(0,1),\quad t>0 $$ ...
2
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1answer
59 views

Non-compactness of support of linear KdV equation solution

The last question in Linares and Ponce's 'Introduction to Nonlinear Dispersive Equations's first chapter asks the reader to prove that, if the following IVP is given: $$\begin{cases} \partial_t u + \...
0
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0answers
196 views

Show that Airy function is bounded

I want to establish the bounds for the Airy function \begin{equation} K(x) = \frac{1}{2\pi} \int_\mathbb{R} e^{i(x\xi+\xi^3)} \,d\xi. \end{equation} I want to show that $K(x)=O_N(<x>^{-N})$ for ...
2
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1answer
97 views

Computing the total energy of Nonlinear Schrödinger (NLS) equation

NLS: $$ i\, u_t + \frac 12 u_{xx} \pm \lVert u\rVert^2u=0 $$ Show that the following energy of the nonlinear Schrödinger (NLS) equation is constant $$ E=\int\limits_{-\infty}^\infty \left( \...
1
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1answer
100 views

Free wave behaviour of $\frac{\partial^2 \phi}{\partial x^2} - \frac{\partial^2 \phi}{\partial t^2} = \lambda |\phi|^2 \phi$.

I am playing with the partial differential equation $\frac{\partial^2 \phi}{\partial x^2} - \frac{\partial^2 \phi}{\partial t^2} = \lambda |\phi|^2 \phi$. $\phi(x,t)$ is complex and the domain is not ...
0
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1answer
412 views

2D wave equation: decay estimate

Let $u$ be a solution to $$\begin{cases}\Box u =0,\; \; (t,x)\in \mathbb{R}_+ \times \mathbb{R}^2\\(u,u_t)\restriction_{t=0} = (f,g),\end{cases}$$ where $f,g$ are smooth functions with compact ...