Nonlinear Systems and Their Remarkable Mathematical Structures, Vol 3, Contributions from China (Eds. Norbert Euler and Dajun Zhang), CRC Press (2021) 201-226.

Darboux transformation for integrable systems with symmetries
Zi-Xiang Zhou

Abstract The Darboux transformation method is useful in getting explicit solutions of integrable systems. When the system has no symmetries (i.e., all the entries of the coefficients of the Lax pair are independent), the construction of Darboux transformation is well known, which was given by Neugebauer and Meinel for 2x2 systems, then by Gu, Sattinger and Zurkowski for general systems. However, in practice, there should be some symmetries. To keep all these symmetries, the construction is much more difficult. In this chapter, we shall first review the general construction of Darboux transformation for systems without symmetries. Then the Darboux transformation for systems with unitary symmetry is discussed. The equivalent construction given by Zakharov and Mikhailov is also reviewed. By analyzing the spectrum, soliton solutions and breather solutions of MKdV equation are obtained. Finally, the Darboux transformation for a two-dimensional affine Toda equation, which has more complicated symmetries, is constructed.

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