Questions tagged [dispersive-pde]

This tag is for questions relating to dispersive partial differential equation or dispersive PDE. Informally, “dispersion” will refer to the fact that different frequencies in this equation will tend to propagate at different velocities, thus dispersing the solution over time.

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13 views

Recasting a dispersion relation (determinant =0) as a matrix eigenvalue problem for numerical solution

I have a physical problem that involves a dispertion relation between two parameters, the frequency $\omega$ and the propagation coefficient $\beta_{n}$, where $n$ is an integer index. The dispersion ...
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32 views

Dispersion relation in wave PDE

I need to understand any intermediate steps in inferring dispersion relation in a PDE. For example, in this PDE: $u_t+\rho{u_x}+\nu{u_{xxx}}=0$ we plug it into $u(t,x)=e^{i(\omega{t}-\xi{x})}$ and we ...
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29 views

Interpretation of the conformal symmetry of Schrodinger equation

Consider the linear Schrodinger equation $$ \begin{cases} i\partial_t u + \Delta u =0,\\ u|_{t=0}=u_0, \end{cases}, t\in\mathbb R,x\in \mathbb R^n, u\in \mathbb C. $$ If $v$ is a solution to the ...
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32 views

Assume $u \in L^\infty ([0,T];H^{s-1})$, $u_t \in L^\infty ([0,T];H^{s-1})$ then can we show that $u \in C([0,T];H^{s-1})$

In Sogge's "Lectures on nonlinear wave equations" it states that :Assume $u \in L^\infty ([0,T];H^{s-1})$, $u_t \in L^\infty ([0,T];H^{s-1})$ then $u \in C([0,T];H^{s-1})$ . Here $u_t$ is ...
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38 views

Stability of PDE and the relation with Discrete spectrum/continuous spectrum

I'm reading about the stability around the solitons(or kinks) in several dispersive PDE's. At some point in the argument, the authors take time to explicit the continuous spectrum and counting the ...
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3answers
86 views

$e^{itH}$ notation

recently I saw the notation $e^{itH}$, and just wondering how should I interpret it? In my understanding, $u(t,x) = e^{itH} u_0$ is, for example, a solution to Schrodinger-type equation $i\partial_tu =...
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25 views

Dispersion relations in spherical coordinates and beyond

While discovering how to compute a laplacian in any manifold $\mathcal M$ given any frame, I tried computing it in Schwarschild coordinates. If I'm not wrong, the laplacian of $f \in \mathcal C^2(\...
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17 views

An estimate on supercritical NLW

Would there exist $p>5,r>p+1$ and a fixed constant $C$ s.t. $\forall u\in C^\infty_{cpt}(\mathbb{R}\times\mathbb{R}^3)$ s.t. $$u_{tt}-\Delta u=-|u|^{p-1}u,u_t(0,x)\equiv 0$$ we have $\|u\|_{L^...
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25 views

Dispersion relation for system of nonlinear equations

I have a system of four fields $\mathbf{u}(t,x)=(\alpha,\gamma,\beta_{+},\beta_{-})$ over one spatial dimension and one time dimension. The system is expressed as an inital value problem and each ...
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27 views

Morawetz inequality with force term

Consider $u:I\times\mathbb{R}^3\rightarrow\mathbb{R}$, $u_{tt}-\Delta u-|u|^{p-1}u=f,p>1$. Assume $u$ is a sufficiently nice solution. If $f=0$, we can expect the following Morawetz bound: $$\...
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1answer
67 views

How dispersive term in KdV PDE causes smoothing

In KdV equation, $$u_t+u_{xxx}-6uu_x=0,$$ $uu_x$ is the nonlinear term which cause blow up, and $u_{xxx}$ is the dispersive term, I am wondering how the dispersive term smooth the solution, otherwise ...
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1answer
56 views

How can I find $\frac{\partial y}{\partial t}$ if I know that $\frac{\partial y}{\partial t}-D\frac{\partial^2y}{\partial x}=0$ (D is a constant)

How can I find $\frac{\partial y}{\partial t}$ if I know that $\frac{\partial y}{\partial t}-D\frac{\partial^2y}{\partial x}=0$ (D is a constant) and $y(x,0)=500+A\sin(2\pi x/L)$ and $y(0, t)=y(2\pi,t)...
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24 views

Given the 1- D diffusion equation $\frac{\partial y}{\partial t}-D\frac{\partial^2y}{\partial x}=0$

Given the 1- D diffusion equation $\frac{\partial y}{\partial t}-D\frac{\partial^2y}{\partial x}=0$ , predict the evolution of an initia coastline described by $y=500+A\sin(2\pi x/L)$, where $A=200 $ ...
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1answer
121 views

Justification for Uniqueness of Solutions to Dispersive PDE

For the sake of concreteness, we consider the linear Schrodinger equation $$ \partial_t u = i\Delta u, \ \ \ \ u(0, x) = u_0(x). $$ The solution is typically (at least, how I've seen it) obtained by ...
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47 views

A small result of Barry Simon's “Schrödinger Operators”

I would like some help understanding this part of a theorem in a paper of Barry Simon[*], and while I speak english quite fluently, there is a particular phrase, combined with the fact that I'm not ...
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1answer
91 views

Re-writing a PDE in Hamiltonian form

I am reading a paper about Camassa-Holm equation, which is given by $$ u_t-u_{txx}=-2\kappa u_x-3uu_x+2u_xu_{xx}+uu_{xxx}, \qquad t,x\in\mathbb{R}, $$ where $u(t,x)$ is a real-valued function and $\...
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1answer
75 views

Deriving conservation laws from stress-energy tensor

I'm going through Tao's book on nonlinear dispersive equations. In dealing with the Schrodinger equation $$ iu_t + \frac{1}{2}\Delta u = 0, $$ he defines the stress-energy tensor in standard ...
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1answer
56 views

Doubt on momentum conservation in nonlinear schrodinger

I want to prove the momentum conservation of the nonlinear Schrodinger eq. $u_t=i\Delta u + i|u|^{p-1}u$. The momentum is gives by $$ Pu=2Im \int_{R^n} \overline{u}\nabla u\ dx$$ I have read that in ...
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1answer
201 views

Interpretation of dispersion relation

I'm going through Terry Tao's book on nonlinear dispersive PDE, and am a little confused by some of the things going on. The setup is that we are working with a PDE given by $$ \partial_t u = Lu. $$ ...
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61 views

Regularity of Solution for the Kdv equation

Let $u_{t}+u_{xxx}=f,\,\, u(x,0)=0,\,x\in(0,1), \, t\in[0,T]$ $u(0,t)=0,u(1,t)=0, u_{x}(1,t)=0$. Prove that \begin{equation} \boxed{\lVert u \rVert_{L^{2}(0,T;H^{2}(0,1))}\leq C\lVert f\rVert_{ L^{...
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1answer
46 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
227 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 $S(\mathbb{R}^N)$ ...
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1answer
42 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 \...
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1answer
104 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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1answer
74 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
94 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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54 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)...
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1answer
551 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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255 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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58 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 ...
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1answer
135 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
198 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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1answer
111 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
36 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) ...
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1answer
655 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
45 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
26 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
179 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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78 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
107 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
124 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 ...
2
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1answer
90 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 + \...
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0answers
296 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
140 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
131 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 ...
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1answer
660 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 ...