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Fixed references.
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benedict-96 committed Dec 21, 2023
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# The Linear Wave Equation

The **linear wave equation** is the prototypical example for a Hamiltonian PDE. It is given by (see (Buchfink et al, 2023) and (Peng and Mohseni, 2016)):
The **linear wave equation** is the prototypical example for a Hamiltonian PDE. It is given by (see [buchfink2023symplectic](@cite) and [peng2016symplectic](@cite)):
```math
\mathcal{H}(q, p; \mu) := \frac{1}{2}\int_\Omega\mu^2(\partial_\xi{}q(t,\xi;\mu))^2 + p(t,\xi;\mu)^2d\xi,
```
Expand Down Expand Up @@ -70,6 +70,11 @@ u_0(\mu)(\omega) = h(s(\omega, \mu)), \quad s(\omega, \mu) = 20 \mu |\omega +
```

## References
- Buchfink, Patrick, Silke Glas, and Bernard Haasdonk. "Symplectic model reduction of Hamiltonian systems on nonlinear manifolds and approximation with weakly symplectic autoencoder." SIAM Journal on Scientific Computing 45.2 (2023): A289-A311.
- Peng, Liqian, and Kamran Mohseni. "Symplectic model reduction of Hamiltonian systems." SIAM Journal on Scientific Computing 38.1 (2016): A1-A27.
- Greif C, Urban K. Decay of the Kolmogorov N-width for wave problems[J]. Applied Mathematics Letters, 2019, 96: 216-222.
```@bibliography
Pages = []
Canonical = false
buchfink2023symplectic
peng2016symplectic
greif2019decay
```

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