Correction to: On solutions to the nonlocal ${\overline{\partial }}$-problem and (2+1) dimensional completely integrable systems

1 Correction to: Letters in Mathematical Physics (2021) 111:16 https://doi.org/10.1007/s11005-021-01353-w

The publication of this note unfortunately contained three errors. The author apologizes for any confusions this may have caused.

The first error is that $\eta (x,y,t)$ appears in (50). The solution to the complex KP equation should be u(x, y, t) and not $\eta (x,y,t)$.

The second error is that the definition of the function

$$\begin{aligned} \phi (\lambda ,x,y,t)=\lambda x+\lambda ^2y + \lambda ^3 t \end{aligned}$$

on the lines following equations (21) and (49) leads to solutions to the scaling

$$\begin{aligned} (4u_t+6uu_x-u_{xxx})_x - 3u_{yy} =0 \end{aligned}$$

of the complex KP equation. To produce solutions to the scaling

$$\begin{aligned} (4u_t-6uu_x+u_{xxx})_x + u_{yy} = 0 \end{aligned}$$

used in this note a valid definition of $\phi$ is

$$\begin{aligned} \phi (\lambda ,x,y,t) = \lambda x + \sqrt{3}\lambda ^2 y - \lambda ^3 t. \end{aligned}$$

When this scaling is used $t_2=\sqrt{3}y$ and $t_3=-t$ in (52).

The third error is that the terms

$$\begin{aligned} -\kappa _nx+ \kappa _n^3 t, \; -sx+s^3t, \text { and } sx-s^3 t \end{aligned}$$

appearing in (38), (42), and (43) respectively should be

$$\begin{aligned} -2\kappa _n x+2 \kappa _n^3 t, \; -2sx+2s^3t, \text { and } 2sx-2s^3 t \end{aligned}$$

respectively to produce solutions to the scaling

$$\begin{aligned} 4u_t-6uu_x+u_{xxx} = 0 \end{aligned}$$

of the KdV equation used in this note.