Focusing mKdV Breather Solutions with Nonvanishing Boundary Conditions by the Inverse Scattering Method

Using the Inverse Scattering Method with a nonvanishing boundary condition, we obtain the square k^2 of a focusing modified Korteweg-de Vries (mKdV) breather solution with non zero vacuum parameter b^2 . We are able to factorize and simplify it in order to get explicitly the associated mKdV breather solution k with non zero vacuum parameter b. Moreover, taking the limiting case of zero frequency, we obtain a generalization of the Double Pole solution introduced by M.Wadati et al.

💡 Research Summary

The paper tackles the problem of constructing explicit breather solutions of the focusing modified Korteweg‑de Vries (mKdV) equation under non‑vanishing boundary conditions (NVBC), i.e., when the field approaches a constant non‑zero value (b) as (|x|\to\infty). Traditional inverse scattering methods (ISM) for mKdV assume a zero background, which simplifies the definition of Jost functions, scattering data, and the associated Lax pair. However, many physical contexts—such as nonlinear optics with a constant bias field, plasma waves in a uniform background, or magnetic media with a static magnetization—require a non‑zero asymptotic state. The authors therefore modify the Lax pair to incorporate the constant vacuum parameter (b) and develop a scattering theory that remains consistent with the mKdV equation.

Model and Lax pair modification
Starting from the focusing mKdV equation
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