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I'm a physics student and this question troubled me while reading solitons in $1+1$ dimensions.

Consider a real-valued, continuous, and bounded function $\phi(x,t)$, of two real variavles $x,t$ such that the quantity $$E[\phi]=\int\limits_{-\infty}^{+\infty} dx\, \left[\frac{1}{2}\left(\frac{\partial\phi}{\partial t}\right)^2+\frac{1}{2}\left(\frac{\partial \phi}{\partial x} \right)^2+a\phi^2+b\phi^4 \right]$$ is finite. Here, $a,b$ are non-zero real numbers but can be positive or negative.

  1. What is the necessary condition that $\phi$, and its derivatives $\frac{\partial\phi}{\partial t}$ and $\frac{\partial \phi}{\partial x}$ must satisfy as $x\rightarrow\pm\infty$ so that $E[\phi]$ is finite?

  2. Can we also say something about the sufficiency condition?

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    Do we have some information about $\phi $? For example, is it a polynomial function? – 2017-01-24
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    @Anatoly Not necessarily. I know of a hyperbolic tangent solution of $\phi$. – 2017-01-24
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    Are you asking specifically about the set of soliton solutions, or are you asking a general question about this integral (which looks like an integral Hamiltonian) independent of the type of solution? You mention the [kink solution](https://en.wikipedia.org/wiki/Scalar_field_theory#Kink_solutions), for which this integral of course does not converge. If all you care about is the condition at $\infty$, I don't see why the answer wouldn't be the usual restriction that each of these terms falls off faster than $1/x$, which implies (I think) that $\phi$ must fall off faster than $1/\sqrt{x}$. – 2017-01-24

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