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**A fifth order semidiscrete mKdV equation.**
*(English)*
Zbl 1339.37064

Summary: In this paper, aiming to get more insight on the relation between the higher order semidiscrete mKdV equations and higher order mKdV equations, we construct a fifth order semidiscrete mKdV equation from the three known semidiscrete mKdV fluxes. We not only give its Lax pairs, Darboux transformation, explicit solutions and infinitely many conservation laws, but also show that their continuous limits yield the corresponding results for the fifth order mKdV equation. We thus conclude that the fifth order discrete mKdV equation is extremely an useful discrete scheme for the fifth order mKdV equation.

### MSC:

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |

37K05 | Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) |

### Keywords:

fifth-order semidiscrete mKdV equation; Darboux transformation; soliton solutions; conservation laws; continuous limits
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\textit{T. Zhou} et al., Sci. China, Math. 56, No. 1, 123--134 (2013; Zbl 1339.37064)

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### References:

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