Abstract:
We employ the bifurcation theory of planar dynamical systems to investigate the exact travelling wave solutions of a generalized Degasperis-Procesi equation ？

Abstract:
We employ the bifurcation method of planar dynamical systems and qualitative theory of polynomial differential systems to derive new bounded traveling-wave solutions for a variant of the (3,2) equation. For the focusing branch, we obtain hump-shaped and valley-shaped solitary-wave solutions and some periodic solutions. For the defocusing branch, the nonexistence of solitary traveling wave solutions is shown. Meanwhile, some periodic solutions are also obtained. The results presented in this paper supplement the previous results.

Abstract:
This paper is concerned with a viscous shallow water equation, which includes both the viscous Camassa-Holm equation and the viscous Degasperis-Procesi equation as its special cases. The optimal control under boundary conditions is given, and the existence of optimal solution to the equation is proved. 1. Introduction Holm and Staley [1] studied the following family of evolutionary 1+1 PDEs: which describes the balance between convection and stretching for small viscosity in the dynamics of one-dimensional nonlinear waves in fluids. Here , , and is chosen to be the Green’s function for the Helmholtz operator on the line. In a recent study of soliton equations, it is found that (1) for and any is included in the family of shallow water equations at quadratic order accuracy that are asymptotically equivalent under Kodama transformations [2]. When , (1) becomes the -family of equations: which describes a one-dimensional version of active fluid transport. It was shown by Degasperis and Procesi [3] that (2) cannot satisfy the asymptotic integrability condition unless or ; compare [2, 4, 5]. For in (2), it becomes the Camassa-Holm (CH) equation: which is a model describing the unidirectional propagation of shallow water waves over a flat bottom [4]. Equation (3) has a bi-Hamiltonian structure [6] and is completely integrable [7, 8]. It admits, in addition to smooth waves, a multitude of traveling wave solutions with singularities: peakons, cuspons, stumpons, and composite waves [4, 9]. Its solitary waves are stable solitons [10, 11], retaining their shape and form after interactions [10]. The Cauchy problem of (3) has been studied extensively. Constantin [12] and Rodríguez-Blanco [13] investigated the locally well-posed for initial data with . More interestingly, it has strong solutions that are global in time [11, 14] as well as solutions that blow up in finite time [11, 15, 16]. On the other hand, Bressan and Constantin [17] and Xin and Zhang [18] showed that (3) has global weak solutions with initial data . For in (2), it becomes the Degasperis-Procesi (DP) equation: which can be used as a model for nonlinear shallow water dynamics, and its asymptotic accuracy is the same as (3). Degasperis et al. [5] presented that (4) has a bi-Hamiltonian structure with an infinite sequence of conserved quantities and admits exact peakon solutions which are analogous to (3) peakons [4, 10, 19]. Dullin et al. [20] showed that (4) can be obtained from the shallow water elevation equation by an appropriate Kodama transformation. The numerical stability of solitons and

Abstract:
In this paper, we employ the bifurcation method of dynamical systems to investigate the exact travelling wave solutions for the Fornberg-Whitham equation. The implicit expression for solitons is given. The explicit expressions for peakons and periodic cusp wave solutions are also obtained. Further, we show that the limits of soliton solutions and periodic cusp wave solutions are peakons.

Abstract:
In this Letter, by using the bifurcation method of dynamical systems, we obtain the analytic expressions of soliton solution of the osmosis K(2, 2) equation.

Abstract:
In this paper, by using bifurcation method, we successfully find the Fornberg-Whitham equation has a type of traveling wave solutions called kink-like wave solutions and antikinklike wave solutions. They are defined on some semifinal bounded domains and possess properties of kink waves and anti-kink waves. Their implicit expressions are obtained. For some concrete data, the graphs of the implicit functions are displayed, and the numerical simulation is made. The results show that our theoretical analysis agrees with the numerical simulation.

Abstract:
In this paper, we study an initial boundary value problem for a generalized Camassa-Holm equation. We establish local well-posedness of this closed-loop system by using Kato theorem for abstract quasilinear evolution equation of hyperbolic type. Then, by using multiplier technique, we obtain a conservation law which enable us to present a blow-up result.

Abstract:
Based on the theory of complex network, this paper focuses on the planning of logistics nodes for strategic supply chain. I propose a practical mathematical modeling framework that simultaneously captures many practical aspects but still understated in the existing literatures of network planning problems. Moreover, capacity expansion and reduction sce-narios are also analyzed as well as modular capacity shifts for the fluctuation of demands. So this paper is of impor-tance for the research of network planning in strategic supply chain systems.