Rogon-like solutions excited in the two-dimensional nonlocal nonlinear Schrödinger equation

We report the analytical one- and two-rogon-like solutions for the two-dimensional nonlocal nonlinear Schrödinger equation by means of the similarity transformation. These obtained solutions can be used to describe the possible physical mechanisms for rogue-like wave phenomenon. Moreover, the free function of space y involved in the obtained solutions excites the abundant structures of rogue-like wave propagations. The Hermite-Gaussian function of space y (normalized function) is, in particular, chosen to depict the dynamical behaviors for rogue-like wave phenomenon.     

Zhiber-Shabat方程的孤立波解与周期波解

结合齐次平衡法原理并利用F展开法,再次研究了Zhiber-Shabat方程的各种椭圆函数周期解.当椭圆函数的模m分别趋于1或0时,利用这些椭圆函数周期解,得到了Zhiber-Shabat方程的各种孤子解和三角函数周期解,从而丰富了相关文献中关于Zhiber-Shabat波方程的解的类型.     

Solitary Wave Solutions and Kink Wave Solutions for a Generalized PC Equation

We present in this paper a generalised PC (GPC) equation which includes several known models. The corresponding traveling wave system is derived and we show that the homoclinic orbits of the traveling wave system correspond to the solitary waves of GPC equation, and the heteroclnic orbits correspond to the kink waves. Under some parameter conditions, the existence of above two types of orbits is demonstrated and the explicit expressions of the two solutions are worked out.     

Solitary wave solutions and kink wave solutions for a generalized KDV-mKDV equation

Bifurcation method of dynamical systems is employed to investigate solitary wave solutions and kink wave solutions of the generalized KDV-mKDV equation. Under some parameter conditions, their explicit expressions are obtained.     

Global solutions for a nonlinear wave equation

In this work the existence of a global solution for the mixed problem associated to the nonlinear equation

is proved in a Hilbert space framework by using Galerkin method.     

Solitary wave solutions and kink wave solutions for a generalized Zakharov-Kuznetsov equation

Bifurcation method of dynamical systems is employed to investigate solitary wave solutions and kink wave solutions in the generalized Zakharov-Kuznetsov equation. Under some parameter conditions, their explicit expressions are obtained.     

Bifurcation of travelling wave solutions for （2＋1）-dimension nonlinear dispersive long wave equation

In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for （2＋1）-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.     

Bifurcations of travelling wave solutions for modified nonlinear dispersive phi-four equation

By using the bifurcation theory of dynamical systems to modified nonlinear dispersive phi-four equation, we analysis all bifurcations and phase portraits in the parametric space, the existence of solitary wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some explicit exact solution formulas are acquired for some special cases.     

非线性时滞差分议程的全局渐近稳定性

In this paper,a sufficient condition for the global asymptotic stability of the solutions of the following nonlinear delay difference equation is obtained,xn 1=xn xn-1xn-2 a/xmxm-1 xn-2 a,n=0,1…,where a∈(0,∞) and the initial values x-2,x-1,x0∈(0,∞).As a special case,a conjecture by Ladas is confirmed.     

Bifurcation studies on travelling wave solutions for an integrable nonlinear wave equation

By using the method of planar dynamical systems to an integrable nonlinear wave equation, the existence of periodic travelling wave, solitary wave and kink wave solutions is proved in the different parametric conditions. The phase portraits of the travelling wave system are given. It can be shown that the existence of singular curves in the travelling wave system is the reason why the travelling wave solutions lose their smoothness. Moreover, the so-called W/M-shaped solitary wave solutions are obtained.     

Qualitative analysis and solutions of bounded traveling wave for B-BBM equation

In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter conditions. Then, the relations between the behaviors of bounded traveling wave solutions and the dissipation coeffiicient μ are investigated. We find that a bounded traveling wave solution appears as a kink profile solitary wave solution when μ is more than the critical value obtained in this paper, while a bounded traveling wave solution appears as a damped oscillatory solution when μ is less than it. Furthermore, we explain the solitary wave solutions obtained in previous literature, and point out their positions in global phase portraits. In the meantime, approximate damped oscillatory solutions are given by means of undetermined coefficients method. Finally, based on integral equations that reflect the relations between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate solutions are presented.     

A note on a singular integral equation arising in water wave scattering

A quick method of solution of a singular integral equationof the first kind involving both logarithmic singularity aswell as Cauchy-type singularity is explained.     

Evolution of solitary waves for a perturbed nonlinear Schrödinger equation

Soliton perturbation theory is used to determine the evolution of a solitary wave described by a perturbed nonlinear Schrödinger equation. Perturbation terms, which model wide classes of physically relevant perturbations, are considered. An analytical solution is found for the first-order correction of the evolving solitary wave. This solution for the solitary wave tail is in integral form and an explicit expression is found, for large time. Singularity theory, usually used for combustion problems, is applied to the large time expression for the solitary wave tail. Analytical results are obtained, such as the parameter regions in which qualitatively different types of solitary wave tails occur, the location of zeros and the location and amplitude of peaks, in the solitary wave tail. Two examples, the near-continuum limit of a discrete NLS equation and an explicit numerical scheme for the NLS equation, are considered in detail. For the discrete NLS equation it is found that three qualitatively different types of solitary wave tail can occur, while for the explicit finite-difference scheme, only one type of solitary wave tail occurs. An excellent comparison between the perturbation solution and numerical simulations, for the solitary wave tail, is found for both examples.     

Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction

In this paper, we derive a lattice model for a single speciesin a one-dimensional patchy environment with infinite numberof patches connected locally by diffusion. Under the assumptionthat the death and diffusion rates of the mature populationare age independent, we show that the dynamics of the maturepopulation is governed by a lattice delay differential equationwith global interactions. We study the well-posedness of theinitial-value problem and obtain the existence of monotone travellingwaves for wave speeds c > c_*. We show that the minimal wavespeed c_* is also the asymptotic speed of propagation, whichdepends on the maturation period and the diffusion rate of maturepopulation monotonically.     

Symbolic computation on the Darboux transformation for a generalized variable-coefficient higher-order nonlinear Schrödinger equation from fiber optics

Considering the propagation of ultrashort pulse in the realistic fiber optics, a generalized variable-coefficient higher-order nonlinear Schrödinger equation is investigated in this paper. Under certain constraints, a new 3×3 Lax pair for this equation is obtained through the Ablowitz-Kaup-Newell-Segur procedure. Furthermore, with symbolic computation, the Darboux transformation and nth-iterated potential transformation formula for such a model are explicitly derived. The corresponding features of ultrashort pulse in inhomogeneous optical fibers are graphically discussed by the one- and two-soliton-like solutions.     

Bäcklund transformation in bilinear form for a higher-order nonlinear Schrödinger equation

Bäcklund transformation in bilinear form is presented for a higher-order nonlinear Schrödinger equation, which describes the propagation of ultrashort light pulses in optical fibers. With symbolic computation and starting from the Bäcklund transformation, the analytical soliton solution is obtained from a trivial solution and the inverse scattering transform scheme is also derived. Furthermore, the N$N$-soliton solution in double Wronskian form is given, and the value of the arbitrary constant appearing in the Bäcklund transformation is determined for a transformation between the (N−1)$(N-1)$ and N$N$-soliton solutions. The results obtained from the Bäcklund transformation might be valuable in optical communications.     

Global existence for rough solutions of a fourth-order nonlinear wave equation

In this paper, we prove that the cubic fourth-order wave equation is globally well-posed in Hs(Rn) for by following the Bourgain's Fourier truncation idea in Bourgain (1998) [2]. To avoid some troubles, we technically make use of the Strichartz estimate for low frequency part and high frequency part, respectively. As far as we know, this is the first result on the low regularity behavior of the fourth-order wave equation.