Super Brockett equations: a graded gradient integrable system
The authors introduce a supersymmetric extension of the Brockett hierarchy consisting of equations of Lax type with double commutator \(\vartheta_k\Lambda= [\Lambda, [\Lambda, \Lambda_+^{k-1}]]\); \(k=1,2,\dots,\) see also the first author [Physica D 132, 287-297 (1999; Zbl 0933.35184)]. Main results: The super Brocket hierarchy is completely integrable (i.e., it admits an infinite family of conservation laws) and it defines a gradient system to the graded Adler functional \(F_k(S)=-\int\text{Res}_\vartheta \Lambda_-^{k-1} \circ\Lambda d(\xi,x)\). We denote \(\vartheta= \partial_\xi +\partial_x\) with \(\xi\) an odd and \(x\) an even variable, \(\Lambda= \vartheta+ \sum_{i\leq 0}b_i \vartheta^i\) a fixed odd graded Lax operator (where \(b_{-1}+ \frac 12\vartheta b_0=0)\), and \(\vartheta_{2k}= \partial_{t_{2k}}\), \(\vartheta_{2k-1}= \partial_{t_{2k-1}} +\sum_{j=1} t_{2j-1} \partial_{t_{2(k-j-1)}}\) are derivatives (involving new time variables \(t_i)\) that supercommute with \(\vartheta\).
supersymmetric extension of the Brockett hierarchy
scientific article; zbMATH DE number 1742842