# Tagged Questions

Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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### The Lie algebra of Harmonic functions

Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ with corresponding volume form $\omega= \sqrt{det(g_{ij})} dx \wedge dy$ and the corresponding Laplace operator $\Delta$ such that the space ...
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### Boundary conditions for Klein-Gordon equation

Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$. ...
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### Existence and uniqueness of solutions for a nonlinear elliptic PDE

The following nonlinear elliptic PDE arose in my research: $$\Delta f - e^f \partial_s f = E(s,t)\,,$$ where $f : \mathbb R(s) \times \mathbb R/\mathbb Z(t) \to \mathbb R$, $f = f(s,t)$, is the ...
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Main Question Consider a $C^2,H^2$ map $F:\mathbb{R}^m \to \mathbb{C}^n$ which satisfies the following equation: $$-\Delta F(x) + \sum_i a_i(x)\nabla_iF(x) + B(x)F(x) + |F(x)|^2F(x) = 0$$ Here $a_i:... 0answers 66 views ### I have an embedding$\iota$between two Hilbert spaces and want to know if$\iota\iota^\ast$is something simple like an orthogonal projection I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical$Q$-Wiener process$W$. It turns out that$W$is just a$...
It is known that the Korteweg-de Vries equation $$u_{t}+uu_{x}+u_{xxx} = 0,$$ with $u=u(x,t)$ smooth and with period equal to $L$, has important conservation laws, namely, E(u)=\frac{1}{2}\int_{0}^{...