Abstract
The Dbar dressing method is extended to study the focusing/defocusing nonlinear Schrödinger (NLS) equation with nonzero boundary condition. A special type of complex function is considered. The function is meromorphic outside an annulus with center 0 and satisfies a local Dbar problem inside the annulus. The theory of such function is extended to construct the Lax pair of the NLS equation with nonzero boundary condition. In this procedure, the relation between the NLS potential and the solution of the Dbar problem is established. A certain distribution for the Dbar problem is introduced to obtain the focusing/defocusing NLS equation and the conservation laws.
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This work was supported by the National Natural Science Foundation of PR China [Grant Numbers 11471295, 11971442].
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Appendix
Appendix
Proof of Lemma 1
Suppose C is a simple closed contour in D and encloses the circle \(\Gamma _\varepsilon \). For all z in the region with boundary C, and admitting \(|z-z_0|>\varepsilon \), we have
Since \(z_0\) is a mth-order pole of f, and \(0<|k-z_0|<\varepsilon \), then
In addition, for \(|z-z_0|>\varepsilon \), we have
Thus,
Substituting (A.2) into (A.1) implies that
On the other hand, for the region with boundary C and \(\Gamma _\varepsilon \), we apply the Cauchy’s formula and find
The Lemma 1 is proved by comparing equations (A.3) and (A.4). It is noted that
Proof of Lemma 2
To consider the contour integral on the circle \(\Gamma _r=\{k:|k|=r, \epsilon<r<R\}\), we let \(z=\zeta ^{-1}\) and make the transformation \(k=\mu ^{-1}, f(\mu ^{-1})=g(\mu )\). Then the circle \(\Gamma _r\) is transformed to the circle \(C=\{\mu :|\mu |=r^{-1}\}\). Noting the direction of the contours will change after the transformation, we have
Since \(0<|\zeta |<r^{-1}\) and
then
For fixed \(\zeta \), we know that
Substituting (A.7) into (A.6), we find
Then, from (A.8) and (A.5), we have
On the other hand, in the annulus \(r<|z|<R\), the Cauchy’s formula implies that
Lemma 2 is proved by comparing (A.9) and (A.10).
Lemmas 1 and 2 give the proof the Lemma 3 in the Introduction.
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Zhu, J., Jiang, X. & Wang, X. Nonlinear Schrödinger equation with nonzero boundary conditions revisited: Dbar approach. Anal.Math.Phys. 13, 51 (2023). https://doi.org/10.1007/s13324-023-00816-8
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DOI: https://doi.org/10.1007/s13324-023-00816-8