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We consider the nonlinear dynamics in a double-chain model of DNA which consists of two long elastic homogeneous strands connected with each other by an elastic membrane. By using the method of dynamical systems, the bounded traveling wave solutions such as bell-shaped solitary waves and periodic waves for the coupled nonlinear dynamical equations of DNA model are obtained and simulated numerically. For the same wave speed, bell-shaped solitary waves of different heights are found to coexist.

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In this paper, the active disturbance rejection control (ADRC) approach is applied to a class of multi-input multioutput (MIMO) uncertain stochastic nonlinear systems. An extended state observer (ESO) is first designed for estimation of both unmeasured states and stochastic total disturbance of each subsystem which represents the total effects of internal unmodeled stochastic dynamics and external stochastic disturbance with unknown statistical property. The ADRC controller based on the states of ESO is further designed to achieve the closed-loop system's output regulation performance including practical mean square reference signals tracking, disturbance attenuation, and practical mean square stability when the reference signals are zero avoiding solving any partial differential equations in the conventional output regulation theory. Some numerical simulations are presented to demonstrate the effectiveness of the proposed ADRC approach.

In this paper we study the Boussinesq equation with power law nonlinearity and dual dispersion which arises in fluid dynamics. A particular kind of product of distributions is introduced and applied to solve non-smooth solutions of this equation. It is proved that, under certain conditions, a distribution solution as a singular Dirac delta function exists for this model. For the first time, this kind of product of distributions is used to deal with a fourth order nonlinear partial differential equation.

We use bifurcation method of dynamical systems to study exact traveling wave solutions of a nonlinear evolution equation. We obtain exact explicit expressions of bell-shaped solitary wave solutions involving more free parameters, and some existing results are corrected and improved. Also, we get some new exact periodic wave solutions in parameter forms of the Jacobian elliptic function. Further, we find that the bell-shaped waves are limits of the periodic waves in some sense. The results imply that we can deduce bell-shaped waves from periodic waves for some nonlinear evolution equations.