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I'm not really sure that my answer fits with your request of a "uniform" background pressure gradient, but anyway it's a related subject. Have you considered perturbations of a fluid parcel in a hydrostatically balanced atmosphere? Conservation of momentum here relates the pressure gradient to the gravitational acceleration. It can be shown that perturbed ...

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I think the Rayleigh-Taylor instability may be considered to be amplified under a pressure gradient. Due to the difference in densities, there is a pressure gradient across the interface which becomes unstable after a certain time. The instability grows exponentially in time according to an amplitude on the order of: $$a\propto\exp\left(\sqrt{A}\right)$$ ...

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Just to complement Ali Moh's answer: We can define a topological current in the same way we do for the $\phi^4$ kink, $$J_\mu=C\epsilon_{\mu\nu}\partial^\nu\phi(t,x),$$ where $C$ is a normalization constant and $\epsilon_{01}= -1$. The topological charge then is Q=\int_{-\infty}^\infty J_t dx=C\int_{-\infty}^\infty\partial_x\phi dx=C\left[\phi(\infty,t)-\...

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