Can one hear the full nonlinearity of a PDE from its small excitations?

This paper explores a classical inverse problem for PDEs of general sine-Gordon type, $u_{xx}-u_{tt} = F(u)$ using a succinct construction relating a stationary (kink) solution to the ground state of an associated linear Schrödinger equation.

By appealing to the Goldstone Theorem, translational invariance of the PDE is used to write down an integral equation of the (stationary) kink solution involving the ground state; conditions on the nonlinearity of $F$ are also made explicit via a function of the kink and ground state.

I recommend this report for publication with only minor corrections, listed alongside some general comments to be considered, below.

The authors assume the potential $V(x)$ supports at least one bound state, as supported by the examples on Table 1. It would be valuable to know if there are any examples of physical interest for which this is *not* the case; any comments that could be made on the regularity or smoothness of the potential (and consequently the bound state) would be nice to add here, though not absolutely essential.

As the Goldstone Theorem relies on the existence of continuous symmetries of the PDE, it would be good to add a remark on the conditions on F, even if only for the explicit examples considered.

In Table 1, how were A and B determined? It does seem that there are multiple choices for these constants which could do the job and it may not be obvious whether the PDE produced will be immediately recognizable. This is mentioned in the final paragraph of the conclusion, so a brief remark on how the constants were found for these examples would be helpful to the reader.

When applying Goldstone's Theorem, the translational invariance of the PDE in (3) is used. It would be interesting to apply the Theorem to those equations in Table 1 which contain both translational and rotational invariance and compare the resulting integral equations; this could lead to a different understanding of the mechanics of the inverse problem.

In a similar vein, the principles of this paper could also be applied, for example, to elliptic equations, i.e., by changing the signature $u_{tt}-u_{xx}$ to $u_{tt}+u_{xx}$. At first glance, this would be resolved by a simple Wick rotation $t \rightarrow$ $it$, but the resulting elliptic equation could have interesting ramifications for the inverse problem over $\mathbb{C}$ instead of $\mathbb{R}$.

Typos
Page 3 - Last paragraph of section 3, 4 lines from the end of the paragraph – “Golstone” should be “Goldstone”
Page 4 – “Several direction OF future research can be foreseen”
Page 4 – Line above (16) “modifies” should be “modified”
Page 5 – Paragraph just before (18) “fist” should be “first”
Page 5 – Paragraph just after (18) “eigensate” should be “eigenstate”

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This is an interesting article which flips the usual connection between a kink like state in a translation invariant system and the corresponding eigenfunction, its derivative, with eigenvalue zero (sometimes called the Goldstone mode). After performing what seems to be a sound derivation, the authors provide several examples in Klein-Gordon type examples where explicit kink-like solutions are known. I think it is a worthwhile idea to share in such a venue but I have a few comments and one question which could clarify and improve the article.

1. The inverse $\chi$ for $\bar u$ is only defined on $u$ values in the range of $\bar u$. How would this allow one to obtain the values of F(\bar u) for \bar u outside this range? While I do not doubt the authors worked out examples, I wonder how, in general, one can obtain information about the nonlinear system outside this range. Namely, could F be defined differently outside this range and not affect the inverse problem? Certainly something the authors should discuss. This also connects to the authors comment Sec. 5 that this idea indicates that low energy dynamics indicate the higher energy dynamics.

2. The authors give some nice discussion in Sec. 5, but I think more discussion on the use of this idea would be helpful to the reader. Namely, is there an example of a known kink-like coherent structure and associated linear equation where the associated nonlinear system is not known? Further, where do the authors expect this to be useful/helpful? Is there an example where one has an (approximate) understanding of some ground state eigenfunction (either numerically or experimentally) but not the underlying nonlinear equation?

3. This seems to work only when the ground state is algebraically simple. Are there relevant cases (maybe for a different linear operator) where degenerate zero eigenvalues occur? This might possibly preclude the analysis done here.

4. How much of this analysis do the authors expect to be able to be lifted to higher spatial dimensions (where there are more translation invariant norms? If the kink is localized in multiple directions, then I would expect multiple 0-eigenvalues to exist.

A few small typos:
Pg. 3, below the proof: “above tell one nothing” to “above tells one nothing”
Pg. 4, above (16): “modifies” to “modified”

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