Is the $\mathbb{Z}_2\times \mathbb{Z}_2$-graded sine-Gordon equation integrable?

We examine the question of the integrability of the recently defined Z2 × Z2-graded sine-Gordon model, which is a natural generalisation of the supersymmetric sine-Gordon equation. We do this via appropriate auto-Bäcklund transformations, construction of conserved spinor-valued currents and a pair of infinite sets of conservation laws.

[1]  J. Grabowski,et al.  The category of Z n 2-supermanifolds , 2016 .

[2]  Z. Kuznetsova,et al.  On Supergroups with Odd Clifford Parameters and Supersymmetry with Modified Leibniz Rule , 2007, 0705.4007.

[3]  S. Ferrara,et al.  Classical Solutions in Two-Dimensional Supersymmetric Field Theories , 1977 .

[4]  L. Girardello,et al.  An Infinite Set of Conservation Laws of the Supersymmetric {Sine-Gordon} Theory , 1978 .

[5]  F. Esposito,et al.  Theory and applications of the sine-gordon equation , 1971 .

[6]  A. Bruce.  On a Z n 2-Graded Version of Supersymmetry , 2019 .

[7]  S. Bertrand.  On integrability aspects of the supersymmetric sine-Gordon equation , 2017, 1703.07925.

[8]  Pfaffian solutions for the Manin-Radul-Mathieu SUSY KdV and SUSY sine-Gordon equations , 1998, solv-int/9806005.

[9]  N. Aizawa,et al.  Z2×Z2-graded mechanics: The quantization , 2021 .

[10]  M. Chaichian,et al.  On the method of inverse scattering problem and Bäcklund transformations for supersymmetric equations , 1978 .

[11]  S. Duplij,et al.  Double-graded supersymmetric quantum mechanics , 2019, Journal of Mathematical Physics.

[12]  Uwe Franz,et al.  Generalized Lie Algebras , 1993 .

[13]  N. Aizawa,et al.  N-extension of double-graded supersymmetric and superconformal quantum mechanics , 2019, Journal of Physics A: Mathematical and Theoretical.

[14]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[15]  N. Poncin.  Towards Integration on Colored Supermanifolds , 2016 .

[16]  A. Tseytlin,et al.  Pohlmeyer reduction of AdS5×S5 superstring sigma model , 2007, 0711.0155.