# 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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### 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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### 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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### 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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### 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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### 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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### 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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### $L^1$-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 $f\in L^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d)$,...
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### 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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### 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 ...
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### 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 ...
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 ...