Talk:Dispersive shock waves
The paper is well written and cover the state of the art i including recent (remarkable) advances n the field. It should be definitely accepted.
However, before accepting the paper in final form I suggest that the authors address the following points:
1. eq. (6): please check that the square root of the denominator is r_3 - r_2 consistently with the second of eq. (3)
2. the reasoning before eq. (6) with which the authors introduce the constant period 2 \pi is not clear to me. Could the authors be more explicit and detailed especially on the problem of the secular growth that they mention ? Does the value 2 \pi present some degree of arbitrariness ?
3. LHS of first of eqs. (16) subscript should be \pm instead of plus
4. for the NLS the authors report as wavelength the ratio L/\epsilon. please clarify if this is simply due to a different choice of variable with respect to the KdV case where \epsilon is included in the definition of variable \xi
5. as far as I understand much of the analysis of the authors is concerned with the Whitham averaging technique with step-like initial data (global regularization). Experiments, however, are better described in terms of initial value problem with smooth data. I suggest to add a discussion on this, explaining how well the aforementioned method can describe (recent) experimental data in BEC and optics, and what are the problems and limitations of the whitham method as far as a generic IVP is concerned.
6. concerning focusing NLS the authors should quote also G. Assanto,et al., PHYSICAL REVIEW A 78, 063808 (�2008�)
Thank you for your suggestions and the careful reading of the work. 1. It is correct 2. The asymptotic theory dictates that a constant (arbitrary) period be imposed. The equation above (8) shows that secular growth (e.g. n -> infinity) can occur if a non-constant period is taken. 3. Fixed 4. The wavelength for KdV in Eq. (6) is in the fast variable \xi. 5. The matched regularization section has been expanded. A section titled "General Initial Value Problem" has been added to discuss the points you raise. 6. Reference added
User 2: Comments
The article gives a nice survey on the dispersive shocks. I have the following suggestions:
1. Add a little more details on the parameters, $r_1,r_2$ and $r_3$ in Eq.(3) in terms of the elliptic integral obtained from (1) for a steady propagation. Then the modulation of the oscillation becomes clear as a slow change of those parameters. Namely it is better to explain some details on the derivation of (3):
2. On Figure 7 and 2 DSWs: I am not sure that there is 2 DSWs in some regions in Figure 7. One needs 2 DSWs (i.e. 6 Riemann invariants) to regularize the initial data. However those 2 DSWs never appear as time evolves (this is a degenerate genus 2 case). The oscillation pattern in Figure 8 shows only one oscillation. Maybe one needs to add a little more details on the global regularization process for the NLS case.
Thank you for your comments and suggestions. 1. We have added a brief derivation of (3). 2. You are correct, this is a degenerate genus 2 case. By "two DSWs", we mean two separate modulated 1-phase regions with well defined boundaries (speeds). We are not referring to the interaction of 2 DSWs because these DSW regions never overlap. In Figure 8, this is the case in the top row. Note that the upper left figure shows two DSWs separated by a non-modulated, non-constant 1-phase region. All 4 Riemann invariants are constant in this middle region.
User 3: Comments
Overall, the article reads very well. Some minor changes were made Wiki-style to improve readability.
The authors should include the reference Z. Dutton et al., Science 293, 663 (2001). This paper re-energized the field by observing quantum shock waves in Bose-Einstein condensates and occurred well before the recent BEC works cited.
Thank you for your edits. We have added the very important reference.