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I am trying to understand if the deterministic 2+1D Kuramoto-Sivashinsky equation $$ \partial_t h = -\nu \nabla^2 h - K \nabla^4 h + \frac{\lambda}{2} (\nabla h)^2, $$ where $\nu$, $K$, $\lambda$ are constants and $h=h(x,y,t)$ is a time-dependent real field in two spatial dimensions, can be seen in the light of a statistical field theory. It seems to me that this equation corresponds to a non-conservative system (similar to the Korteweg–de Vries equation, see wiki). Thus, my question is the following:

Is there a Hamiltonian or a free energy that has been written down and analysed for the deterministic Kuramoto-Sivashinsky? If not, has another similar equation been analyzed in terms of statistical field theory (phases, free energy, phase transitions, critical behavior etc.)?

Qmechanic
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jarm
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1 Answers1

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Comments to the question (v2):

  1. On one hand, the Kuramoto-Sivashinsky (KS) equation is a dissipative differential equation (DE). Each term has an even number of spatial derivatives. It's a non-linear version of the heat equation. Dissipative systems rarely have variational action formulations nor Hamiltonian formulations.

  2. On the other hand, in the Korteweg de Vries (KdV) equation, each term has an odd number of derivatives. The KdV equation is not a dissipative DE. It has both a Lagrangian and a Hamiltonian formulation. The energy is a conserved quantity.

Qmechanic
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