Abstract.In this paper, we study the periodic Cauchy problem for the modified Camassa-Holm equationx ) 2 u, and show that the solution map is not uniformly continuous in Sobolev spaces H s (T) for s > 7/2. Our proof is based on the method of approximate solutions and well-posedness estimates for the actual solutions. Mathematics Subject Classification (2010). 35A07, 35B30, 35Q53.
In this paper, we exploit the generalized bifurcation method to study space-time fractional Drinfel'd-Sokolov-Wilson equation and derive its various new exact explicit periodic wave solutions. Especially and interestingly, we find the so-called M/W-shaped periodic wave solutions, which were not found in previous studies. Furthermore, we uncover their inside limit relations as well as their limit relations with other solutions under corresponding parameters conditions. The previous results are extended.
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