Xiaohua Zhao

Bio: Xiaohua Zhao is an academic researcher from Zhejiang Normal University. The author has contributed to research in topics: Wave packet & Wave vector. The author has an hindex of 6, co-authored 11 publications receiving 105 citations. Previous affiliations of Xiaohua Zhao include Yunnan University.

ON A CLASS OF SINGULAR NONLINEAR TRAVELING WAVE EQUATIONS (II): AN EXAMPLE OF GCKdV EQUATIONS

Abstract: By using the method of dynamical systems, we continuously study the dynamical behavior for the first class of singular nonlinear traveling wave systems. As an example, the traveling wave solutions for a generalized coupled KdV equations are discussed. Exact explicit parametric representations of solitary wave solutions, periodic wave solutions and kink wave solutions are given.

Breaking wave solutions to the second class of singular nonlinear traveling wave equations

Abstract: The existence of breaking wave solutions of the second class of singular nonlinear wave equations is proved by methods from the dynamical systems theory. For the second class of singular nonlinear traveling wave equations, dynamical behaviors of the traveling wave solutions are completely classified and thoroughly discussed. Corresponding to some bounded orbits of the traveling systems, exact parametric representations of traveling wave solutions are derived within different parameter regions of the parameter space.

Classification and dynamics of stably dissipative Lotka–Volterra systems

Abstract: This paper deals with classification and dynamical behaviors of stably dissipative Lotka–Volterra (LV) systems. The sufficient and necessary conditions for a matrix being stably dissipative are discussed firstly. Then, a graph-theoretic classification method for stably dissipative matrices is proposed, based on which, for all five-order stably dissipative matrices, the associated graphs are classified completely as 27 topologically different graphs and for each graph the dynamics of the corresponding LV system is discussed. Finally, the effects of removing some links from the above graphs on the dynamics of the corresponding LV systems are discussed.

Dynamical properties of 2-torus parabolic maps

Abstract: In this paper, a class of linear maps on the 2-torus are discussed. Discussions are focused on the case that the maps are parabolic. It is shown that the maximal invariant set for a 2-torus parabolic map is indeed invariant, and is almost closed, and the Lebesgue measure restricted to a maximal invariant set is invariant. Under this invariant measure, all Lyapunov exponents of a parabolic map are zero. In certain simple cases, the Lebesgue measure of the maximal invariant sets are computed and estimated. For the case the maps are invertible, it is shown that the inverse of a non-horocyclic parabolic map is no longer a parabolic map. Interesting properties of the conjugation of invertible parabolic maps by automorphisms of the torus are characterized, and a conjugation invariant for such maps are obtained. And it is proven that all these maps can be reduced to a family of one parameter rigid rotations.

On a class of nonlinear dispersive-dissipative interactions

Abstract: The authors study the prototypical, genuinely nonlinear, equation; u{sub t} + a(u{sup m}){sub x} + (u{sup n}){sub xxx} = {mu}(u{sup k}){sub xx}, a, {mu} = consts., which encompasses a wide variety of dissipative-dispersive interactions. The parametric surface k = (m + n)/2 separates diffusion dominated from dissipation dominated phenomena. On this surface dissipative and dispersive effects are in detailed balance for all amplitudes. In particular, the m = n + 2 = k + 1 subclass can be transformed into a form free of convection and dissipation making it accessible to theoretical studies. Both bounded and unbounded oscillations are found and certain exact solutions are presented. When a = (2{mu}3/){sup 2} the map yields a linear equation; rational, periodic and aperiodic solutions are constructed.

Visualizing Lagrangian Coherent Structures and Comparison to Vector Field Topology

Abstract: This paper takes a look at the visualization side of vector field anal ysis based on Lagrangian coherent structures. The Lagrangian coherent structures are extracted as height ridges of finite-time Lyapunov exponent fields. The result ing visualizations are compared to those from traditional instantaneous vector field topology of steady and unsteady vector fields: they often provide more and better interpretable information. The examination is applied to 3D vector fields from a dynamical system and practical CFD simulations.

Contact topology and hydrodynamics III: knotted orbits

Abstract: We employ the relationship between contact structures and Beltrami fields derived in part I of this series to construct a steady nonsingular solution to the Euler equations on a Riemannian S3 whose flowlines trace out closed curves of all possible knot and link types. Using careful contacttopological controls, we can make such vector fields real-analytic and transverse to the tight contact structure on S3. Sufficient review of concepts is included to make this paper independent of the previous works in this series.

Peakon, pseudo-peakon, and cuspon solutions for two generalized Camassa- Holm equations

Abstract: In this paper, we study peakon, cuspon, and pseudo-peakon solutions for two generalized Camassa-Holm equations. Based on the method of dynamical systems, the two generalized Camassa-Holm equations are shown to have the parametric representations of the solitary wave solutions such as peakon, cuspon, pseudo-peakons, and periodic cusp solutions. In particular, the pseudo-peakon solution is for the first time proposed in our paper. Moreover, when a traveling system has a singular straight line and a heteroclinic loop, under some parameter conditions, there must be peaked solitary wave solutions appearing.