Multiple-soliton and Periodic Solutions to Space-time Fractional Whitham-Broer-Kaup Equations

In this paper, the space-time fractional Whitham-Broer-Kaup equations are investigated. By means of new fractional scaling transformations, the fractional nonlinear system of different time and space orders is transformed to the integer one. The multiple solitary solutions and periodic solutions are obtained respectively. All those solutions are given exactly by introducing new scaling transformations, which makes our study is unique and different from most existing studies. It is expected that exact solutions for nonlinear wave system of fractional order can be handled in the similar way.

Global solutions to a nonlinear wave system arising from cholesteric liquid crystals

We investigate a one-dimensional nonlinear wave system which is the Euler–Lagrange equations of a variational principle modeling a type of cholesteric liquid crystals. Due to the chiral effect, the Lagrangian density of the variational principle includes linear terms as well as quadratic terms in derivatives of the field, which is different from the nematic liquid crystals. Moreover, the linear terms and zero term in derivatives of the field might cause the energy to be negative in some cases. By using the method of energy-dependent coordinates and the Young measure theory, we establish the global existence of weak solutions to its Cauchy problem under some conditions on the coefficients.