| Abstract: |
| We study the finite-dimensional integrable reduction for the Jaulent--Miodek (JM) equation associated with the energy-dependent Schr\{o}dinger operator, in which a traveling periodic wave expressed by Jacobi elliptic functions is obtained for the JM equation. Solutions of the linearized JM equation are represented as squared eigenfunctions of the Lax system, so that the stability spectrum are connected with the Lax spectrum via a characteristic polynomial. The Lax spectrum are numerically computed by using the Floquet--Bloch decomposition of periodic solutions of Lax system, while the stability spectrum are traced out via the characteristic polynomial. Since the band of stability spectrum lies on the imaginary axis, the traveling periodic wave of JM equation is proved to be modulationally stable. Followed by a gauge transformation, the Darboux transformation is retrieved in a different way, from which a new algebraic soliton is obtained at the endpoint of continuous spectral band, and three new periodic waves are derived with three discrete eigenvalues. |
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