Exact stationary-wave solutions in the standard model of the Kerr-Nonlinear optical fiber with the Bragg grating

By using dynamical system method to the standard model of the Kerr-Nonlinear optical fiber with the Bragg grating, under fixed parameter conditions, all possible exact parametric representations of the bounded stationary wave solutions are obtained from the double sine-Gordon equation.     

Existence of travelling wavefronts of the KdV–Burgers equation

In this letter, we study the existence of travelling wavefronts of the KdV–Burgers equation from a monotone dynamical systems point of view. For any constant wave velocity, we obtain a sufficient condition for the existence. The conditions under which some exact travelling wavefronts have been found are special cases of ours.     

Persistence of travelling wavefronts in a generalized Burgers-Huxley equation with long-range diffusion

In this paper, we study the persistence of travelling wavefronts in a generalized Burgers-Huxley equation with long-range diffusion. When the influence of long-range diffusion effect is sufficiently small, we prove the persistence of these waves by using geometric singular perturbation theory. When the influence becomes large, the behavior of these waves can only be investigate numerically. In this case, we find that the solutions lose monotonicity by using Matlab program bvp4c. Some previous results are extended.     

Bifurcations of invariant torus and knotted periodic orbits for the generalized Hopf–Langford system

Nonlinear Dynamics - In this paper, we study the bifurcations of invariant torus and knotted periodic orbits for the generalized Hopf-Langford system. By using bifurcation theory of dynamical...     

Non-uniform dependence on initial data for the periodic modified Camassa–Holm equation

In this paper, we study the periodic Cauchy problem for the modified Camassa–Holm equation $$m_t+um_x+2u_xm=0,\quad m=(1-\partial_x^2)^2u$$ , and show that the solution map is not uniformly continuous in Sobolev spaces ${H^s(\mathbb T)}$ for s > 7/2. Our proof is based on the method of approximate solutions and well-posedness estimates for the actual solutions.     

错用罗素悖论-康托在集合论中的两个逻辑性错误

分析了罗素悖论与康托的实数集合不可数证明及康托定理S〈P（S）证明之间的本质性联系，发现康托的这两个非构造性证明与罗素悖论有完全相同的思路，但是康托犯了两个逻辑性错误而使他误用了这个悖论思路。得到明确的结论：康托在集合论中如上两个证明里的核心部分实际上是罗素悖论的翻版，这两个证明中的思路与做法是错误的，这样的证明结果没有科学性。     

Persistence of travelling fronts of KdV-Burgers-Kuramoto equation

We study the travelling fronts of KdV-Burgers-Kuramoto equation from geometric singular perturbation point of view. Motivated by the analogue between travelling fronts and heteroclinic orbits of the corresponding ordinary differential equations, we prove the persistence of these waves for sufficiently small dissipation. The result of numerical investigation also establishes our analysis.     

Orbital stability of peakons for the Degasperis-Procesi equation with strong dispersion

In this paper, we study the orbital stability of the peakons for the Degasperis-Procesi equation with a strong dispersive term on the line. Using the method in [Z. Lin, Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math. 62 (2009) 125-146], we prove that the shapes of these peakons are stable under small perturbations. Some previous results are extended.     

Non-uniform dependence on initial data for the periodic Degasperis-Procesi equation

In this paper, we show that the solution map of the periodic Degasperis-Procesi equation is not uniformly continuous in Sobolev spaces Hs(T) for s>3/2. This extends previous result for s?2 to the whole range of s for which the local well-posedness is known. Our proof is based on the method of approximate solutions and well-posedness estimates for the actual solutions.