ABSTRACT
In this paper we review our earlier results on the modified Korteweg-deVries equation in [Gesl], [Ge-Si], [GSS] and continue these investigations into several directions. Specifically, in Section 2, we review the connections between the Korteweg-deVries (KdV) and the modified Korteweg-deVries (mKdV) equations based on Miura’s transformation [Miu], and commutation methods. Appendix A summarizes the necessary commutation formulas needed in Section 2. In Section 3 we study soliton-like solutions of the mKdV-equation (i.e., solutions that tend to (time-independent) finite asymptotic values as x —> Too sufficiently fast). In particular, due to our more general Hypothesis (H.3.1), Theorem 3.2 considerably extends our earlier findings in [GSS]. Section 4 reviews our derivation of pure soliton solutions in [GSS]. Both, Sections 3 and 4 are supported by Appendix B which summarizes spectral and scattering properties of one-dimensional Schrodinger and Dirac operators with nontrivial spatial asymptotics in the corresponding potential terms. Section 5 is devoted to spatially periodic solutions of the mKdV-equation. While Theorem 5.3 summarizes our results on periodic solutions in [GSS], the rest of this section presents new material.
