  We study the following family of evolutionary 1+1 PDEs that describe the
balance between convection and stretching for small viscosity in the dynamics
of 1D nonlinear waves in fluids: \[ m_t + \underbrace{um_x \}
_{(-2mm)\hbox{convection}(-2mm)} + \underbrace{b u_xm \}
_{(-2mm)\hbox{stretching}(-2mm)} = \underbrace{\nu m_{xx}\
}_{(-2mm)\hbox{viscosity}}, \quad\hbox{with}\quad u=g*m . \] Here $u=g*m$
denotes $ u(x)=\int_{-\infty}^\infty g(x-y)m(y) dy . $ We study exchanges of
stability in the dynamics of solitons, peakons, ramps/cliffs, leftons,
stationary solutions and other solitary wave solutions associated with this
equation under changes in the nonlinear balance parameter $b$.
