A Class of Soliton Solutions for the N=2 Super mKdV/Sinh-Gordon Hierarchy

Employing the Hirota’s method, a class of soliton solutions for the N=2 super mKdV equations is proposed in terms of a single Grassmann parameter. Such solutions are shown to satisfy two copies of N=1 supersymmetric mKdV equations connected by nontrivial algebraic identities. Using the super Miura transformation, we obtain solutions of the N=2 super KdV equations. These are shown to generalize solutions derived previously. By using the mKdV/sinh-Gordon hierarchy properties we generate the solutions of the N=2 super sinh-Gordon as well.

💡 Research Summary

The paper presents a systematic construction of soliton solutions for the N=2 supersymmetric modified Korteweg‑de Vries (mKdV) and sinh‑Gordon hierarchies using Hirota’s direct method. The authors begin by recalling the N=2 supersymmetric mKdV equation, which involves two bosonic fields, two fermionic (Grassmann) fields, and auxiliary components. Traditional approaches to this system required several independent Grassmann parameters, leading to cumbersome expressions.

To simplify the problem, the authors introduce a single Grassmann parameter η (with η²=0) and rewrite the N=2 mKdV equation as a pair of coupled N=1 supersymmetric mKdV equations. Each N=1 equation is cast into Hirota bilinear form by defining two τ‑functions, τ₁ and τ₂. The τ‑functions are taken as a sum of a constant term, an exponential bosonic term, and a linear Grassmann contribution proportional to η. The bilinear equations impose algebraic constraints on the amplitudes and wave numbers, which the authors solve explicitly. A crucial observation is that the two N=1 subsystems are linked by a non‑trivial algebraic identity (essentially a product of two linear combinations that vanishes), ensuring that the single η simultaneously satisfies both supersymmetries.

Having obtained the τ‑function representation, the authors apply the supersymmetric Miura transformation, which maps the mKdV fields to those of the N=2 supersymmetric KdV equation. In the τ‑function language this transformation is remarkably simple: the new KdV τ‑functions are constructed from the same exponential factors, with modified coefficients that incorporate the Grassmann contribution. The resulting expressions are shown to satisfy the bilinear form of the N=2 KdV equation, thereby providing explicit one‑soliton and two‑soliton solutions. Importantly, these solutions reduce to previously known N=2 KdV solitons when the Grassmann parameter is set to zero, and they extend them by introducing a fermionic dressing that is controlled by a single η.

The final part of the work exploits the well‑known relationship between the mKdV and sinh‑Gordon hierarchies. By analytically continuing the time variable and adjusting the dispersion relation to include a mass parameter μ, the authors generate τ‑functions that solve the N=2 supersymmetric sinh‑Gordon equation. The same single‑Grassmann‑parameter ansatz works without modification, demonstrating that the method is robust across the entire hierarchy. The resulting sinh‑Gordon solitons exhibit the expected hyperbolic dependence and retain the fermionic dressing, offering a unified description of bosonic and fermionic excitations in the supersymmetric setting.

Overall, the paper achieves three significant advances: (1) it reduces the complexity of N=2 supersymmetric soliton construction to a single Grassmann parameter; (2) it clarifies the algebraic structure linking two N=1 subsystems within the N=2 framework; and (3) it provides a coherent pipeline from mKdV to KdV and finally to sinh‑Gordon, all within Hirota’s bilinear formalism. These results open the door to systematic studies of higher‑order supersymmetric hierarchies, multi‑soliton interactions, and potential quantum extensions where the fermionic degrees of freedom play a central role.