  We study exchange of stability in the dynamics of solitary wave solutions
under changes in the nonlinear balance in a 1+1 evolutionary partial
differential equation related both to shallow water waves and to turbulence. We
find that solutions of the equation $ m_t + um_x +b u_xm = \nu m_{xx} $ with $m
= u - \alpha^2 u_{xx}$ for fluid velocity $u(x,t)$ change their behavior at the
special values $b=0,\pm1,\pm2,\pm3$.
