DCDS-B
Some new results on explicit traveling wave solutions of $K(m, n)$ equation
Rui Liu
Discrete & Continuous Dynamical Systems - B 2010, 13(3): 633-646 doi: 10.3934/dcdsb.2010.13.633
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.
keywords: bifurcation method $K(m blow-up solutions. n)$ equation new explicit solutions
CPAA
Several new types of solitary wave solutions for the generalized Camassa-Holm-Degasperis-Procesi equation
Rui Liu
Communications on Pure & Applied Analysis 2010, 9(1): 77-90 doi: 10.3934/cpaa.2010.9.77
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.
keywords: generalized Camassa-Holm equation phase analysis. generalized Degasperis-Procesi equation New explicit solutions
CPAA
Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four
Rui-Qi Liu Chun-Lei Tang Jia-Feng Liao Xing-Ping Wu
Communications on Pure & Applied Analysis 2016, 15(5): 1841-1856 doi: 10.3934/cpaa.2016006
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
The explicit nonlinear wave solutions of the generalized $b$-equation
Liu Rui
Communications on Pure & Applied Analysis 2013, 12(2): 1029-1047 doi: 10.3934/cpaa.2013.12.1029
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:

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.
keywords: dynamical system Explicit nonlinear wave solution solitary bifurcation method. generalized $b$-equation
MCRF
Investment and consumption in regime-switching models with proportional transaction costs and log utility
Jiapeng Liu Ruihua Liu Dan Ren
Mathematical Control & Related Fields 2017, 7(3): 465-491 doi: 10.3934/mcrf.2017017

A continuous-time and infinite-horizon optimal investment and consumption model with proportional transaction costs and regime-switching was considered in Liu [4]. A power utility function was specifically studied in [4]. This paper considers the case of log utility. Using a combination of viscosity solution to the Hamilton-Jacobi-Bellman (HJB) equation and convex analysis of the value function, we are able to derive the characterizations of the buy, sell and no-transaction regions that are regime-dependent. The results generalize Shreve and Soner [6] that deals with the same problem but without regime-switching.

keywords: Investment and consumption model proportional transaction costs regime-switching model Hamilton-Jacobi-Bellman equation viscosity solution log utility
DCDS-B
Thermodynamical potentials of classical and quantum systems
Ruikuan Liu Tian Ma Shouhong Wang Jiayan Yang
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-38 doi: 10.3934/dcdsb.2018214

The aim of the paper is to systematically introduce thermodynamic potentials for thermodynamic systems and Hamiltonian energies for quantum systems of condensates. The study is based on the rich previous work done by pioneers in the related fields. The main ingredients of the study consist of 1) SO(3) symmetry of thermodynamical potentials, 2) theory of fundamental interaction of particles, 3) the statistical theory of heat developed recently [23], 4) quantum rules for condensates that we postulate in Quantum Rule 4.1, and 5) the dynamical transition theory developed by Ma and Wang [20]. The statistical and quantum systems we study in this paper include conventional thermodynamic systems, thermodynamic systems of condensates, as well as quantum condensate systems. The potentials and Hamiltonian energies that we derive are based on first principles, and no mean-field theoretic expansions are used.

keywords: Potential-descending principle order parameters control parameters SO(3) symmetry spinor representation PVT system N-component system magnetic system dielectric system Bose-Einstein condensates (BEC) superfluid superconductor liquid helium-4 liquid helium-3

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