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DCDS-B

In this paper, we investigate the traveling wave solutions of
$K(m, n)$ equation $ u_t+a(u^m)_{x}+(u^n)_{x x x}=0$ by using the
bifurcation method and numerical simulation approach of dynamical
systems. We obtain some new results as follows:

**(i)**For $K(2, 2)$ equation, we extend the expressions of the smooth periodic wave solutions and obtain a new solution, the periodic-cusp wave solution. Further, we demonstrate that the periodic-cusp wave solution may become the peakon wave solution.**(ii)**For $K(3, 2)$ equation, we extend the expression of the elliptic smooth periodic wave solution and obtain a new solution, the elliptic periodic-blow-up solution. From the limit forms of the two solutions, we get other three types of new solutions, the smooth solitary wave solutions, the hyperbolic 1-blow-up solutions and the trigonometric periodic-blow-up solutions.**(iii)**For $K(4, 2)$ equation, we construct two new solutions, the 1-blow-up and 2-blow-up solutions.
CPAA

In this paper, we study the nonlinear wave solutions of
the generalized Camassa-Holm-Degasperis-Procesi equation $
u_t-u_{x x t}+(1+b)u^2u_x=b u_x u_{x x}+u u_{x x x}$. Through phase
analysis, several new types of the explicit nonlinear wave
solutions are constructed. Our concrete results are: (i) For given
$b> -1$, if the wave speed equals $\frac{1}{1+b}$, then the
explicit expressions of the smooth solitary wave solution and the
singular wave solution are given. (ii) For given $b> -1$, if the
wave speed equals $1+b$, then the explicit expressions of the
peakon wave solution and the singular wave solution are got. (iii)
For given $b> -2$ and $b\ne -1$, if the wave speed equals
$\frac{2+b}{2}$, then the explicit smooth solitary wave solution,
the peakon wave solution and the singular wave solution are
obtained. We also verify the correctness of these solutions by
using the software Mathematica. Our work extends some previous
results.

CPAA

In this article, we study the existence and multiplicity of positive
solutions for the Kirchhoff type problem with singular and critical nonlinearities
\begin{eqnarray}
\begin{cases}
-\left(a+b\displaystyle\int_\Omega|\nabla u|^2dx\right)\Delta u=\mu u^{3}+\frac{\lambda}{|x|^{\beta}u^{\gamma}}, &\rm \mathrm{in}\ \ \Omega, \\
u>0, &\rm \mathrm{in}\ \ \Omega, \\
u=0, &\rm \mathrm{on}\ \ \partial\Omega,
\end{cases}
\end{eqnarray}
where $\Omega\subset \mathbb{R}^{4}$ is a bounded smooth domain and $a, b>0$, $\lambda>0$. For all $\mu>0$ and $\gamma\in(0,1)$, $0\leq\beta < 3$, we obtain one positive solution. Particularly, we prove that this problem
has at least two positive solutions for all $\mu> bS^{2}$ and $\gamma\in(0,\frac{1}{2})$, $2+2\gamma < \beta < 3$.

keywords:
positive solution
,
critical exponent
,
perturbation method.
,
singularity
,
Kirchhoff type problem

CPAA

In this paper, we study the nonlinear wave solutions of the
generalized $b$-equation involving two parameters $b$ and $k$. Let
$c$ be constant wave speed, $c_5=$
$\frac{1}{2}(1+b-\sqrt{(1+b)(1+b-8k)})$,
$c_6=\frac{1}{2}(1+b+\sqrt{(1+b)(1+b-8k)})$. We obtain
the following results:

Not only is the existence of these solutions shown, but their concrete expressions are presented. We also reveal the relationships among these solutions. Besides, the correctness of these solutions is tested by using the software Mathematica.

**1**. If $-\infty < k < \frac{1+b}{8}$ and $c\in (c_5, c_6)$, then there are three types of explicit nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic peakon wave solution and hyperbolic blow-up solution.**2**. If $-\infty < k < \frac{1+b}{8}$ and $c=c_5$ or $c_6$, then there are two types of explicit nonlinear wave solutions, fractional peakon wave solution and fractional blow-up solution.**3**. If $k=\frac{1+b}{8}$ and $c=\frac{b+1}{2}$, then there are two types of explicit nonlinear wave solutions, fractional peakon wave solution and fractional blow-up solution.Not only is the existence of these solutions shown, but their concrete expressions are presented. We also reveal the relationships among these solutions. Besides, the correctness of these solutions is tested by using the software Mathematica.

MCRF

A continuous-time and infinite-horizon optimal investment and consumption model with proportional transaction costs and regime-switching was considered in Liu [

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