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In this paper, the method of bifurcation of planar dynamic systems is used to investigate bounded traveling wave solutions for the two-component Degasperis–Procesi shallow water equation. By using the phase portrait bifurcation of the traveling wave system, solitary wave solutions, periodic wave solutions and breaking kink (or anti-kink) wave solutions are considered under different parameter conditions. And the loop soliton solution is also discussed. By the use of numerical simulation, the graphs of some solutions are obtained under specific parameter conditions.

All possible optical waves in the nonlinear Schrodinger equation with a combined dispersion term are determined according to different parameters’ regions. First, we find this equation has some popular exotic solutions, such as peaked waves, looped and cusped waves. What is more interesting, this equation admits some very particular waves, such as double kinked waves and butterfly-like waves. These new two types of solutions have not been reported in the literature regarding the study of other nonlinear equations.

This paper is concerned with the attractor for a viscous two-component generalization of the Camassa–Holm equation subject to an external force, where the viscosity term is given by a second order differential operator. The global existence of solution to the viscous two-component Camassa–Holm equation with the periodic boundary condition is studied. We obtain the compact and bounded absorbing set and the existence of the global attractor in H 2 × H 2 for the viscous two-component Camassa–Holm equation by uniform prior estimate and many inequalities.

Considered herein is the orbital stability of floating periodic peakons for the Camassa–Holm (CH) equation, which describes one-dimensional surface waves at a free surface of shallow water under the influence of gravity. The floating periodic peakons shift up or down according to the change of the parameter. The result shows that the floating periodic peakons are orbitally stable and their stability is independent of the parameter in the CH equation.

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the traveling wave solutions of a 2-component of the Degasperis-Procesi equation. The expressions for smooth soliton, kink and antikink solutions are obtained., Comment: 16 pages, 18 figures

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the travelling-wave solutions to a dual equation of the Kaup–Boussinesq system. The expressions for smooth solitary-wave solutions are obtained.

Considered herein is the linear stability of compacton solutions to the nonlinear dispersive Ostrovsky equation, which is also a modification of the K ( m , n ) equation. We show that the equation does not pass the Painleve test for integrability and has compacton solutions which are different from conventional forms. Furthermore, compactons are proved to be weak solutions. A new equation similar to the nonlinear dispersive Ostrovsky equation is derived from Lagrangian, and some important conservation laws as well as the Hamiltonian structure are given. Finally, the stability of the compacton solutions to the similar equations is studied via using linear stability analysis. The result shows that the linear stability analysis it follows that, unlike compactons to the K ( m , n ) equation, the weak-compacton solutions are linear stable under certain conditions.

This paper investigates the problem of boundary control of the viscous Fornberg–Whitham equation. The existence and uniqueness of weak solution to the viscous Fornberg–Whitham equation are proved in a short interval. Moreover, we prove the Fornberg–Whitham equation global exponential stability in L 2 , H 1 , H 2 , H 3 under the given boundary condition.

This paper studies the problem for optimal control of the viscous DGH equation. The existence and uniqueness of weak solution to the equation are proved in a short interval. The optimal control of the viscous DGH equation under boundary condition is given and the existence of optimal solution to the equation is proved.

The existence of the global attractor of the viscous damped forced Ostrovsky equation in L 2 ( R ) is proved. An asymptotic smoothing effect of the equation is also shown, namely, that for the forces in L 2 ( R ) , the global attractor in L 2 ( R ) is actually a compact set in H 3 ( R ) . The energy equation method is used in conjunction with a suitable splitting of the solutions; the regularization properties of the equation in the context of Bourgain spaces are extensively exploited.

This paper analyses the dynamic behavior of an energy resources system with parametric perturbations. By adding the small sinusoidal perturbations to the three-dimensional energy resource system established for two regions of China, the autonomous system becomes the non-autonomous system which has richer dynamical behaviors. Periodic, chaotic and hyperchaos behaviors are discovered in the system by means of Lyapunov exponents and bifurcation diagrams. A new energy resources hyperchaos attractor is obtained, which has not yet been reported in present literature. Furthermore, effective non-autonomous feedback controllers are designed for stabilizing hyperchaos to unstable periodic orbits and quasi-periodic orbits. Numerical simulations are presented to show these results. This study will be instructive for the energy resource demand-supply in some regions of China.

In the paper long-time behavior of solutions derived from the dissipative MKdV equation in the unbounded domain is studied. The Sobolev interpolation inequality and prior estimate on time t are applied to show the existence of solution in unbounded domain. Moreover, operator decomposition techniques and Kuratowskii α -noncompacted measures are applied to study the smooth property of the solution. Existence of global attractor for dissipative MKdV equation in H 2 ( R 1 ) is proved. Then, the computational stability of universal double-time differential format is proved by using nature analysis approach and dissipative conservative scheme theory. Finally, accordant conclusion with theoretical proof is drawn by numerical simulation.

This paper studies the problem of optimal control of the viscous Camassa–Holm equation. The existence and uniqueness of weak solution to the viscous Camassa–Holm equation are proved in a short interval. According to variational method, optimal control theories and distributed parameter system control theories, we can deduce that the norm of solution is related to the control item and initial value in the special Hilbert space. The optimal control of the viscous Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous Camassa–Holm equation is proved.

In this paper the dynamical behaviors of a dispersive shallow water equation with viscosity, viscosity Degasperis–Procesi equation, are investigated. The existence of global solution to viscosity Degasperis–Procesi equation in L 2 under the periodical boundary condition is studied and the existence of the global attractor of semi-group to solution on viscosity Degasperis–Procesi equation in H 2 is obtained.

This paper aims at presenting a proof of the existence of the attractor for the two-dimensional weakly damped KdV equations in belt field. By using the Bourgain spaces and the orthogonal projection, we prove the existence of the attractor for the reduced equation which is obtained from the two-dimensional weakly damped KdV equation in belt field. By using the time estimate of the blow-up of the KdV equation, we get the compact and bounded aborting set and prove the existence of the attractor for the two-dimensional weakly damped KdV equation in belt field.