We rely on the helicoidal Peyrard-Dauxois model of DNA [1]. A continuum approximation for solving the crucial dynamical equation of motion brings about the following solutions [2]:
$\psi_1(\xi)=\frac{1}{4}\left(1+\tanh(w)+\tanh^2(w)\right),$
$\psi_2(\xi)=\frac{1}{4}\left(3+2\tanh(w)-\tanh^2(w)\right),\quad w=\frac{5\xi}{12\rho},$
if viscosity $\rho$ is taken into consideration, and

$\psi_{10}(\xi)=\frac{1}{2}\left[-1+3\tanh^2\left(\sqrt{\frac{3}{2a_2^{(1)}}}\xi\right)\right], \quad a_2^{(1)}>0,$
$\psi_{20}(\xi)=\frac{3}{2}\left[1-\tanh^2\left(\sqrt{-\frac{3}{2a_2^{(2)}}}\xi\right)\right], \quad a_2^{(2)}<0,$

if viscosity $\rho$ is neglected. The functions $\psi_1(\xi)$ and $\psi_2(\xi)$ represent the supersonic and subsonic kinks, respectively. We show that only subsonic soliton is stable, while the solutions $\psi_{10}(\xi)$ and $ \psi_{20}(\xi)$ are unstable, which means that the viscosity enables the existence of the solitary waves in DNA.
[1] S. Zdravković, Nonlinear Dynamics of DNA Chain, In Nonlinear Dynamics of Nanobiophysics, Edited by S. Zdravković and D. Chevizovich, Springer, 2022, 29-66.
[2] S. Zdravković, D. Chevizovich, A.N. Bugay, and A. Maluckov, Chaos 29 (2019) 053118.