KdV solitons in active media (Q1118765)

The authors investigate the normalized KdV equation \[ u_ t-6uu_ x+u_{xxx}=\epsilon \cdot R(u), \] where \(R(u)=-(b_ 1u+b_ 2u^ 2+b_ 3u^ 3)\), \(b_ 1,b_ 2,b_ 3>0\) is a cubic polynomial, \(\epsilon \ll 1.\) The perturbation of the well-known soliton solution of the unperturbed KdV equation \((\epsilon =0)\) is studied. Here, depending on the locations of the roots of the cubic polynomial R(u) it is discussed if amplification or attenuation of the soliton occurs. Further, some numerical experiments are given. This work is based on a paper of Karpman and Maslov (1978) in which a perturbation theory for the soliton of the KdV equation is given by use of the inverse scattering transform.