New reductions of integrable matrix PDEs: $Sp(m)$-invariant systems

We propose a new type of reduction for integrable systems of coupled matrix PDEs; this reduction equates one matrix variable with the transposition of another multiplied by an antisymmetric constant matrix. Via this reduction, we obtain a new integrable system of coupled derivative mKdV equations and a new integrable variant of the massive Thirring model, in addition to the already known systems. We also discuss integrable semi-discretizations of the obtained systems and present new soliton solutions to both continuous and semi-discrete systems. As a by-product, a new integrable semi-discretization of the Manakov model (self-focusing vector NLS equation) is obtained.

💡 Research Summary

The paper introduces a novel reduction scheme for integrable systems of coupled matrix partial differential equations (PDEs). Traditional reductions often impose symmetric relations such as (V = U^{\mathrm T}) between two matrix fields, which restrict the underlying symmetry to orthogonal or unitary groups. In contrast, the authors propose a reduction that ties one matrix variable to the transpose of the other multiplied by a constant antisymmetric matrix (A) (i.e., (V = A,U^{\mathrm T}) with (A^{\mathrm T} = -A)). This relation preserves the symplectic group (Sp(m)) invariance and introduces a new structural degree of freedom that is absent in earlier works.

Applying this reduction to the Lax pair of a generic matrix integrable hierarchy yields two previously unknown integrable systems. The first is a multi‑component derivative modified Korteweg–de Vries (dmKdV) equation. While the scalar dmKdV reads (\partial_t u + 6u^2\partial_x u + \partial_x^3 u = 0), the matrix version obtained here contains additional cross‑terms involving (A) and the transpose of (U): \