Smooth soliton solutions of a new integrable equation by Qiao

We find a transformation which relates a new third-order integrable nonlinear evolution equation, introduced recently by Qiao, with the well-known modified Korteweg - de Vries equation. Then we use this transformation to derive smooth soliton solutions of the new equation from the known rational and soliton solutions of the old one.

💡 Research Summary

The paper addresses a recently introduced third‑order nonlinear evolution equation proposed by Qiao, which has been shown to possess a Lax pair and an infinite hierarchy of conserved quantities, thereby qualifying as a completely integrable system. The authors’ primary contribution is the discovery of an explicit nonlinear transformation that maps Qiao’s equation directly onto the well‑known modified Korteweg‑de Vries (mKdV) equation. By defining a new dependent variable (v(x,t)) through the relation (u = v_x/v) (or equivalently (v = \exp!\int u,dx)), the authors demonstrate that the rather intricate Qiao equation
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