Comment on “Vortex arrays for sinh-Poisson equation of two-dimensional fluids: Equilibria and stability” [Phys. Fluids 16, 3296 (2004)] Free

The authors of Ref. 1 discuss a category of solutions of the sinh-Poisson equation that can be expressed in terms of the inverse hyperbolic tangent of products of elliptic functions. We would like to point out that a number of known solutions from this category are described by Eq. (3) that has been derived in Ref. 2 (see Eqs. (2)–(4) of this reference). In particular, Eqs. (6) and (9) of Ref. 1 that were claimed to be new ones [Eq. (6) appeared previously in Ref. 3], reduce to Eq. (3) after straightforward transformations of parameters.

Equation (3) is written in terms of Jacobi elliptic cn functions. The ascending Landen transformation [see equation (16.14.3) of Ref. 4]

expresses cn functions in terms of dn functions with increased modulus. Here, $v=u∕(1+k1′)$⁠, $k1′=(1−k)∕(1+k)$⁠, and $k′=1−k1′2$⁠. The application of this transformation to Eq. (3) yields Eq. (6).

If the parameters of the elliptic functions in Eq. (3) fall in the range $k2<0$ and $k12>1$⁠, then the first and second cn functions in Eq. (3) become cd and dn functions, respectively, i.e., $cn(u∣−k2)=cd(u∕k′∣k′2)$ with $k′2=k2∕(1+k2)$ and $cn(u∣k2)=cd(uk′∣k′2)$ with $k′=1∕k$ [see equations (16.10.3) and (16.11.3) of Ref. 4]. Further, by shifting the $x$ coordinate (see Table 16.8 of Ref. 4) one converts cd into sn: $cd(u−K∣k2)=sn(u∣k2)$⁠. Here, $K=K(k)$ is the complete elliptic integral of the first kind, which is the real quarter period of elliptic functions. After these transformations Eq. (3) coincides with Eq. (9).

Solutions given by Eqs. (4) and (5) are also related. The sinh-Poisson equation remains invariant under the transformations

Consequently, new solutions can be constructed from existing ones by applying Eq. (2). According to Eq. (16.21.4) and Table 16.8 of Ref. 4, $dn(iu∣k2)=1∕cd(u∣k′2)=k′cd(u+K′∣k′)$⁠, where $k′=(1−k2)1∕2$ and $K′=K(k′)$⁠. As we have mentioned, the cd function reduces to sn function under the coordinate translation on $K$⁠. Using these formulas one can easily check that Eq. (5) is equivalent to Eq. (4) written in terms of $x̂$⁠, $ŷ$⁠, and $σ̂$⁠.

To conclude, the classification scheme suggested by the authors consists actually of two solutions. Different forms can be generated by renormalizing parameters and making the transition from one elliptic function to another. However, this can only lead to different representations of the same result.