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Nonlocal symmetry, optimal systems, and explicit solutions of the mKdV equation

Chinese Physics B 2014年第1期23卷 49-54页

作者：辛祥鹏苗倩陈勇 Shanghai Key Laboratory of Trustworthy Computing East China Normal University

The nonlocal symmetry of the mKdV equation is obtained from the known Lax pair; it is successfully localized to Lie point symmetries in the enlarged space by introducing suitable auxiliary dependent variables. For the closed prolongation of the nonlocal symmetry, the details of the construction for a one-dimensional optimal system are presented. Furthermore, using the associated vector fields of the obtained symmetry, we give the reductions by the one-dimensional sub-algebras and the explicit analytic interaction solutions between cnoidal waves and kink solitary waves, which provide a way to study the interactions among these types of ocean waves. For some of the interesting solutions, the figures are given to show their properties.

关键词： nonlocal symmetry optimal system prolonged system explicit solutions

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Nonlocal Symmetries and Explicit Solutions of the Boussinesq Equation

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Chinese Annals of Mathematics,Series B 2014年第6期35卷 841-856页

作者： Xiangpeng XIN Junchao CHEN Yong CHEN Shanghai Key Laboratory of Trustworthy Computing East China Normal University

The nonlocal symmetry of the Boussinesq equation is obtained from the known Lax pair. The explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetry. Some other types of solutions, such as rational solutions and error function solutions, are given by using the fourth Painlev~ equation with special values of the parameters. For some interesting solutions, the figures are given out to show their properties.

关键词： Nonlocal symmetry Lax pair prolonged system Explicit solution

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