The negative order KdV (nKdV) and the negative order modified KdV (nmKdV) equations have two different formulations based on different hierarchy operators. The two forms of the nKdV equation (nKdV-1 and nKdV-2) have Lagrangian formulations and there is a simple transformation that relates the two forms. Both these forms of the nKdV equation when applied to the traveling wave solutions lead to an equation that can be directly related to the equation for the static solutions of the 1+1 dimensional Formula: see text field theory. For the modified nKdV equations, only the first formulation (nmKdV-1) can be directly related, for the case of traveling waves, to the equation for the static solutions of the 1+1 dimensional Formula: see text field theory. Specifically, for the traveling wave solutions of the nKdV-1 equation, we find that the traveling waves satisfy: Formula: see text, where Formula: see text and Formula: see text is an integration constant. The nmKdV-1 equation has both focusing and defocusing branches described by Formula: see text, where the minus (plus) sign is for the focusing (defocusing) case, and Formula: see text is again an integration constant. Using this connection between the equations for the traveling waves and the static Formula: see text equation, we obtain a large number of solutions of the nKdV-1 and the nmKdV-1 equations, many of which are new. The traveling wave solutions of the nmKdV-2 equation obey a higher-order differential equation, namely, letting Formula: see text: Formula: see text where the plus (minus) sign corresponds to the focusing (defocusing) cases. Remarkably, all the solutions of the nmKdV-1 that we have found also satisfy the nmKdV-2 equation. For the nmKdV equation, we have not been able to find a Lagrangian formulation nor a simple mapping of one form of the equation to the other. Furthermore, for both forms of the nmKdV equation as well as for both focusing and defocusing cases, we show that with a suitable ansatz one can decouple the x and t dependence of the nmKdV field Formula: see text and obtain solutions of the form Formula: see text in all the cases. We also obtain rational solutions of both the nKdV and the nmKdV equations and discuss briefly the orbital stability of the periodic traveling wave solutions of the nKdV equation.
Khare et al. (Thu,) studied this question.