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.
65
questions
3
votes
0
answers
80
views
Estimates of kernel function of Schrodinger Operator concerning Littlewood Paley projection operator
Maybe this problem is difficult to understand, because we lump integration and distribution together(usually we can distinguish them).
I have some trouble about the following proof concerning the ...
1
vote
1
answer
30
views
An inequality from Kato local smoothing effect
When I reading a book concerning the proof of Kato local smoothing effect, I'm confused by the following inequality:
$$\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\hat{a}(\eta-\xi){\color{red}{\delta(|\...
0
votes
0
answers
27
views
Commutator estimate (PDE)
While studying uniqueness of a weak solution to KdV equation (on torus), I encountered a problem to bound
$$
\int_{\mathbb{T}}f(x)^{2} \partial_{xxxxx}(\mathbb{P}_{N}f)(x) dx
$$
using $\|f\|_{H^{2}(\...
- 638
0
votes
0
answers
75
views
A reference for solution of non homogeneous heat equation on bounded domain
In PDE's book from Evans, is said that
$$
u(x,t) = \int_0^t \int_{\mathbb{R}^N} \frac{1}{(4\pi (t-s))^{N/2}} e^{-\frac{|x-y|^2}{4(t-s)}} f(y,s) dy ds
$$
is a solution of
$$
\begin{cases}
u_t - \Delta ...
- 891
0
votes
1
answer
63
views
Physical meaning of a heat equation with a term $\alpha u$ [closed]
I would like to know if there is any physical meaning for the equation
$$
\begin{cases}
u_t - u_{xx} + \alpha u = f(x), (a,b)\times(0, +\infty) \\
u(x,0)=u_0(x), x \in (a,b)\\
u(a,t)=u(b,t)=0, t \in (...
- 891
0
votes
1
answer
87
views
Existence and Uniqueness of Solution to PDE using Banach Fixed Point Theorem
I am studying some results about existence and uniqueness of solutions to some PDEs and many times Banach fixed point Theorem is used. I saw that the ideia is to consider the integral formulation and ...
- 89
0
votes
0
answers
99
views
Find the energy of a PDE, and show it is conserved
In the domain $(0,1) \times [0,T], T>0$, we consider the boundary value problem
$$V_{tt} + \eta V = (\xi V_x - \beta V_{xxx})_x,\,\,\,\,\,V(0,t)=0, V(1,t)=0, V_x(0,t)=0, V_x(1,t)=0,$$
where $\eta, \...
- 1,384
4
votes
0
answers
123
views
IBVP for the linear homogeneous 1-D Schrödinger equation
Consider the following initial boundary value problem for the linear homogeneous 1-D Schrödinger equation for a function $u(t,x)$ in the domain $\Omega=[0,T]\times[0,L]$:
$$
\begin{cases}
iu_{t}(t,x)+...
- 275
1
vote
1
answer
67
views
Describing terms of a PDE
I have seen words like 'dispersive', 'advective' and 'diffusive' when describing certain terms in PDEs - however I am unsure on the exact intuition behind this terminology and how one is to know how a ...
- 175
3
votes
0
answers
202
views
Relating a dispersion equation to an eigenvalue equation in a Fourier transformed system of PDEs
I am reading through the paper "Dynamics of Membranes Driven by Actin Polymerization" by Nir S. Gov and Ajay Gopinathan. In it a set of coupled differential equations for a mathematical ...
- 421
1
vote
0
answers
151
views
Fourier transform of $e^{-|x|^4}$
Let $f(x) = e^{-\alpha\pi|x|^4}, x \in \mathbb{R}^n$, where $\alpha > 0$ is a constant. I know that, taking $g(x) = e^{-\pi \alpha|x|^2}$, we get
$$\hat{g}(\xi) = \alpha^{-n/2}e^{-\pi|\xi|^2/\alpha}...
- 891
1
vote
1
answer
50
views
A step in obtaining Strichartz estimates for the homogeneous Schrodinger equation from the Fourier restriction
I am trying to understand the Dennis Kriventsov's note on Tomas-Stein restriction theorem.
These are available here:
https://math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Kriventsov.pdf
In page 10, ...
- 3,023
0
votes
0
answers
25
views
Elementary computation about NLS
I was trying to show
$$\lim_{t \rightarrow \infty} \int_{|x|<kt} |u(t,x)|^2 dx = \int_{|\xi|<k/2} |\hat{u_{0}}(\xi)|^2 d \xi$$
where
$$ u(t,x)=e^{it\Delta} u_{0}(x), \quad \textrm{in} \quad \...
- 122
1
vote
0
answers
51
views
How to find the relation between $\omega$ and $\kappa$?
Suppose we have a system of equations:
\begin{gather*} \left(\begin{array}{cccc} c\partial_t +\partial_x & \alpha & 0 &0 \\
N_0 & \partial_t & -\sigma_2 \frac{Q_0}{N_0} & \...
- 11
0
votes
1
answer
73
views
About superharmonic functions
Let $\Omega \subset \mathbb{R}^{N}$ a bounded smooth domain. I would like to know if is it true that we can find a nonnegative function $\varphi \in C^{2}(\Omega)\cap C(\overline{\Omega})$ such that
$$...
- 891
0
votes
0
answers
30
views
dispersion relation of $i\epsilon_{t}+\epsilon_{xx}+2(d_{1}^2+b_{2}^2)(\epsilon+\epsilon^\ast)+4d_1b_2\epsilon+4d_1b_2^2=0$?
For $\epsilon=\epsilon(x,\;t)$, I have
$i\epsilon_{t}+\epsilon_{xx}+2(d_{1}^2+b_{2}^2)(\epsilon+\epsilon^\ast)+4d_1b_2\epsilon+4d_1b_2^2=0, \;\;\; (1)$
where $d_1,\;b_2$ are constants, and $\epsilon_{...
- 11
3
votes
0
answers
53
views
Nonexistence of jump discontinuity for fractionally differentable functions
This is exercise 3.11 of the book $Introduction\ to\ Nonlinear\ Dispersive\ Equations$ written by Felipe Linares and Gustavo Ponce.
Let's define the fractional sobolev space with $0<s<1,\ p\in (...
- 313
1
vote
1
answer
66
views
Is $C(I; H2(\Omega))$ compactly embbed into $C(I; H1(\Omega))$?
I'm dealing with KdV equation with the fourth order regularizer $\epsilon \partial_{xxxx}u^\epsilon$. If we know the parametrized solution $u^\epsilon$ is bounded in $C(I; H2(\Omega))$, can we extract ...
- 81
0
votes
2
answers
85
views
How to prove that two PDE's are related?
Say that I have PDE a)
$U_x+U_y=\alpha U$
then I have PDE b)
$U_{xx}+U_{yy}=\beta U$
It is obvious that the first and the second are related by that they are composed of two operators which differ by ...
- 2,574
0
votes
1
answer
63
views
partial derivative of Bessel's operator.
Let $J^s = (I- \Delta)^{\frac{s}{2}}$ where $\Delta$ is the Laplacian, and $w(x,y) \in L^2(\mathbb{T}^2)$. During my study in some paper the author stated that
$$\int_{\mathbb{T}^2} J^s (\partial^3_x ...
- 1,384
3
votes
1
answer
163
views
Why is the solution of the periodic KdV equation unique?
Bourgain proved that the periodic KdV equation
$$\begin{align} \partial_t u+\partial_x^3 u+u\partial_x u&=0\\u(0,x)&=u_0(x)\end{align}$$
is locally well-posed in $H^s(\mathbb T)$ in [1]. Here ...
- 649
1
vote
1
answer
129
views
Non-Schwartz functions in Bourgain space X^{s,b}?
From Terence Tao's Nonlinear dispersive equations: local and global analysis, Definition 2.7:
The Bourgain space $X^{s,b}(\mathbb R\times\mathbb R^n)$ is defined to be the closure of the set of ...
- 649
27
votes
0
answers
440
views
Wave equation: predicting geometric dispersion with group theory
Context
The wave equation
$$
\partial_{tt}\psi=v^2\nabla^2 \psi
$$
describes waves that travel with frequency-independent speed $v$, ie. the waves are dispersionless. The character of solutions is ...
- 4,506
2
votes
1
answer
152
views
Correct expansion for 2D anisotropic diffusion
I am trying to model a diffusion phenomenon with anisotropic and heterogeneous diffusion.
$$\nabla \cdot (D\nabla c)$$
where
$$D = \begin{bmatrix} D_{xx}(x,y)&D_{xy}(x,y)\\ D_{yx}(x,y)&D_{yy}(...
- 21
1
vote
0
answers
152
views
Recasting a dispersion relation (determinant =0) as a matrix eigenvalue problem for numerical solution
I have a physical problem that involves a dispertion relation between two parameters, the frequency $\omega$ and the propagation coefficient $\beta_{n}$, where $n$ is an integer index. The dispersion ...
- 1,299
2
votes
0
answers
54
views
Interpretation of the conformal symmetry of Schrodinger equation
Consider the linear Schrodinger equation
$$
\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.
$$
If $v$ is a solution to the ...
- 7,158
1
vote
0
answers
70
views
Assume $u \in L^\infty ([0,T];H^{s-1})$, $u_t \in L^\infty ([0,T];H^{s-1})$ then can we show that $u \in C([0,T];H^{s-1})$
In Sogge's "Lectures on nonlinear wave equations" it states that :Assume $u \in L^\infty ([0,T];H^{s-1})$, $u_t \in L^\infty ([0,T];H^{s-1})$ then $u \in C([0,T];H^{s-1})$ .
Here $u_t$ is ...
- 135
2
votes
0
answers
81
views
Stability of PDE and the relation with Discrete spectrum/continuous spectrum
I'm reading about the stability around the solitons(or kinks) in several dispersive PDE's. At some point in the argument, the authors take time to explicit the continuous spectrum and counting the ...
- 41
2
votes
3
answers
119
views
$e^{itH}$ notation
recently I saw the notation $e^{itH}$, and just wondering how should I interpret it?
In my understanding, $u(t,x) = e^{itH} u_0$ is, for example, a solution to
Schrodinger-type equation $i\partial_tu =...
- 1,856
1
vote
0
answers
75
views
Dispersion relation for system of nonlinear equations
I have a system of four fields $\mathbf{u}(t,x)=(\alpha,\gamma,\beta_{+},\beta_{-})$ over one spatial dimension and one time dimension. The system is expressed as an inital value problem and each ...
- 11
2
votes
1
answer
190
views
How dispersive term in KdV PDE causes smoothing
In KdV equation,
$$u_t+u_{xxx}-6uu_x=0,$$
$uu_x$ is the nonlinear term which cause blow up, and $u_{xxx}$ is the dispersive term, I am wondering how the dispersive term smooth the solution, otherwise ...
- 7,554
0
votes
1
answer
59
views
How can I find $\frac{\partial y}{\partial t}$ if I know that $\frac{\partial y}{\partial t}-D\frac{\partial^2y}{\partial x}=0$ (D is a constant)
How can I find $\frac{\partial y}{\partial t}$ if I know that $\frac{\partial y}{\partial t}-D\frac{\partial^2y}{\partial x}=0$ (D is a constant) and $y(x,0)=500+A\sin(2\pi x/L)$ and $y(0, t)=y(2\pi,t)...
- 1,643
0
votes
0
answers
24
views
Given the 1- D diffusion equation $\frac{\partial y}{\partial t}-D\frac{\partial^2y}{\partial x}=0$
Given the 1- D diffusion equation $\frac{\partial y}{\partial t}-D\frac{\partial^2y}{\partial x}=0$
, predict the evolution of an initia coastline described by
$y=500+A\sin(2\pi x/L)$, where $A=200 $ ...
- 1,643
4
votes
1
answer
192
views
Justification for Uniqueness of Solutions to Dispersive PDE
For the sake of concreteness, we consider the linear Schrodinger equation
$$
\partial_t u = i\Delta u, \ \ \ \ u(0, x) = u_0(x).
$$
The solution is typically (at least, how I've seen it) obtained by ...
- 5,046
2
votes
1
answer
296
views
Re-writing a PDE in Hamiltonian form
I am reading a paper about Camassa-Holm equation, which is given by $$
u_t-u_{txx}=-2\kappa u_x-3uu_x+2u_xu_{xx}+uu_{xxx}, \qquad t,x\in\mathbb{R},
$$
where $u(t,x)$ is a real-valued function and $\...
- 1,337
2
votes
1
answer
173
views
Deriving conservation laws from stress-energy tensor
I'm going through Tao's book on nonlinear dispersive equations. In dealing with the Schrodinger equation
$$
iu_t + \frac{1}{2}\Delta u = 0,
$$
he defines the stress-energy tensor in standard ...
- 5,046
2
votes
1
answer
110
views
Doubt on momentum conservation in nonlinear schrodinger
I want to prove the momentum conservation of the nonlinear Schrodinger eq. $u_t=i\Delta u + i|u|^{p-1}u$. The momentum is gives by $$ Pu=2Im \int_{R^n} \overline{u}\nabla u\ dx$$
I have read that in ...
- 1,213
2
votes
1
answer
416
views
Interpretation of dispersion relation
I'm going through Terry Tao's book on nonlinear dispersive PDE, and am a little confused by some of the things going on. The setup is that we are working with a PDE given by
$$
\partial_t u = Lu.
$$
...
- 5,046
3
votes
0
answers
85
views
Regularity of Solution for the Kdv equation
Let
$u_{t}+u_{xxx}=f,\,\, u(x,0)=0,\,x\in(0,1), \, t\in[0,T]$
$u(0,t)=0,u(1,t)=0, u_{x}(1,t)=0$.
Prove that
\begin{equation}
\boxed{\lVert u \rVert_{L^{2}(0,T;H^{2}(0,1))}\leq C\lVert f\rVert_{ L^{...
0
votes
1
answer
88
views
Question about schrodinger free equation
Left $u$ be a solution to free linear schrodinger equation $u_t=i\Delta u$. The momentum is defined as
$$P(t)=Im \int_{\mathbb{R}^N} \overline{u}(x,t)\nabla u(x,t) dx $$
I want to prove the following ...
- 1,213
2
votes
2
answers
802
views
On the Schrodinger fundamental solution
Let $e^{it\Delta}$ be the fundamental Schrodinger solution. If $u_0$ is the corresponding initial data to the problem associated to Schrodinger free equation $u_t = i\Delta u$ and $S(\mathbb{R}^N)$ ...
- 1,213
2
votes
1
answer
192
views
An Sobolev-type inequality on $1D$ torus related to algebra properties for Sobolev spaces
In Kishimoto-Tsutsumi's paper published by Math Research Letter 2018, I see the following inequality in the last line of page 10:
$\| f g \partial_x h \|_{H^{-s}(T)} \lesssim \| f \|_{H^s(T)} \| g \...
2
votes
1
answer
162
views
Dual local smoothing and retarded local smoothing for Schrodinger equation
This exercise is from Tao's Nonlinear Dispersive Equations: Local and Global Analysis, Exercise 2.54.
Let $u$ be a solution to the inhomogeneous Schrodinger equation $i\partial_t u+\Delta u=F$, ...
- 2,899
3
votes
1
answer
236
views
What does it really mean for a wave equation to be critical?
I am trying to understand intuitively the concept of criticality in general for Wave equations.
For example, consider the cauchy problem of semi-linear equation
\begin{equation}
\begin{cases}
\phi_{tt}...
- 347
2
votes
0
answers
186
views
Physical interpretation of a singular pde
Suppose $\Omega$ is a bounded domain in $\mathbb{R}^N$ and consider the following Dirichlet boundary condition:
$$
-\Delta u=\frac{f(x)}{u^\delta}\text{ in }\Omega, u>0\text{ in }\Omega;
$$
where $\...
- 713
1
vote
0
answers
69
views
An asymptotic for a simple oscillatory integral
Consider the oscillatory integral
$$I(\lambda):=\int_{0}^{\infty}\int_{0}^{\infty} \psi(x,y) \, \mathrm{e}^{\dot{\imath}\phi(x,y)}dxdy$$
where
1) the phase $\phi$ and amplitude $\psi$ are smooth
2)...
- 3,023
3
votes
1
answer
1k
views
The Fourier transform is unbounded from $L^{p}$ to $L^{p^{\prime}}$ when $2<p\leq \infty$?
I previously asked this question here
Haussdorff-Young inequality optimal Lebesgue exponents range
and got a comment that referred me to the answer here
Fourier transform in $L^p$
but I did not ...
- 3,023
1
vote
0
answers
457
views
Virial identity for nonlinear Schrödinger equation
Suppose that $u: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$ is a classical solution to the nonlinear Schrödinger equation:
\begin{align*}
iu_t+\frac{1}{2}\Delta u = -u|u|^2. \tag{1}
\end{align*}
...
1
vote
0
answers
69
views
dispersive relation for linearized Euler-Poisson equation
Euler-poisson equation written :
$$n_t+(nu)_x=0$$
$$u_t+uu_x=-\phi_x$$
$$\phi_{xx}=e^\phi-n$$
And linearized following ways:
The system has a constant solution $n=1, \phi=u=0$. By assuming that the ...
- 13
1
vote
1
answer
259
views
Euler-Lagrange equations of 1D NLS with periodic external potential
I am trying to find the Euler Lagrange (EL) equations in order to find ODEs for the parameters $a(t), \xi(t), c(t), d(t)$.
Consider the following form:
$$iu_t+\frac{1}{2}u_{xx}+|u|^2u=V(x)u \tag{1}$$ ...
- 37