Soliton solutions of a coupled Ramani equation

In this work, multi-soliton solutions of the coupled Ramani equations$u_{xxxxxx}+15u_{xx}{u}_{xxx}+15u_x{u}_{xxxx}+45u_x^2{u}_{xx}-5(u_{xxxt}+3u_{xx}{u}_t+3u_x{u}_{xt})-5u_{tt}+18w_x=0\text{,}$$w_t-w_{xxx}-3w_x{u}_x-3w{u}_{xx}=0\text{,}$ are derived and expressed using Pfaffians in a compact form.     

On a nonlinear wave equation involving the term −∂∂x(μ(x,t,u,6ux62)ux): Linear approximation and asymptotic expansion of solution in many small parameters

In this paper, we consider the following nonlinear Kirchhoff wave equation (1)$\{\begin{array}{c}{u}_{tt}?\frac{?}{?x}\left(\mu (x,t,u,{6u_{x}6}^2)u_x\right)=f(x,t,u,u_x,u_t)\text{,}0<x<1,0<t<T\text{,}\hfill \\ u(0,t)=g_0\left(t\right)\text{,}u(1,t)=g_1\left(t\right)\text{,}\hfill \\ u(x,0)={\stackrel{?}{u}}_0\left(x\right)\text{,}{u}_t(x,0)={\stackrel{?}{u}}_1\left(x\right)\text{,}\hfill \end{array}$ where ${\stackrel{?}{u}}_0$, ${\stackrel{?}{u}}_1$, $\mu$, $f$, $g_0$, $g_1$ are given functions and ${6u_{x}6}^2={\int }_0^1{u}_x^2(x,t)dx$. First, combining the linearization method for nonlinear term, the Faedo–Galerkin method and the weak compact method, a unique weak solution of problem (1) is obtained. Next, by using Taylor’s expansion of the function $\mu (x,t,y,z)$ around the point $(x,t,y_0,z_0)$ up to order $N+1$, we establish an asymptotic expansion of high order in many small parameters of solution.     

ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR GENERALIZED BURGERS EQUATION

This paper is concerned with the stability of the rarefaction wave for the Burgers equationwhere 0 ≤ a < 1/4p (q is determined by (2.2)). Roughly speaking, under the assumption that u_ < u , the authors prove the existence of the global smooth solution to the Cauchy problem (I), also find the solution u(x, t) to the Cauchy problem (I) satisfying sup |u(x, t) -uR(x/t)| → 0 as t → ∞, where uR(x/t) is the rarefaction wave of the non-viscous Burgersequation ut f(u)x = 0 with Riemann initial data u(x, 0) =     

Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $b(G_1,G_2,\dots ,G_k)$ is the least positive integer $b$ so that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a copy of $G_i$ in the $i$th color for some $i$. In this paper, our main focus will be to bound the following numbers: $b(C_{2t_1},C_{2t_2})$ and $b(C_{2t_1},C_{2t_2},C_{2t_3})$ for all $t_i\ge 3,$$b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4})$ for $3\le t_i\le 9,$ and $b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4},C_{2t_5})$ for $3\le t_i\le 5.$ Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.     

Blow-up for coupled nonlinear wave equations with damping and source

We consider the system of nonlinear wave equations $\{\begin{array}{cc}{u}_{tt}+u_t+{\left|u_t\right|}^{m?1}{u}_t=\mathrm{div}\left({\rho }_1\left({\left|?u\right|}^2\right)?u\right)+f_1(u,v)\text{,}\hfill & (x,t)\in \Omega \times (0,T)\text{,}\hfill \\ v_{tt}+v_t+{\left|v_t\right|}^{r?1}{v}_t=\mathrm{div}\left({\rho }_2\left({\left|?v\right|}^2\right)?v\right)+f_2(u,v)\text{,}\hfill & (x,t)\in \Omega \times (0,T)\text{,}\hfill \end{array}$ with initial and Dirichlet boundary conditions. Under some suitable assumptions on the functions$f_1$, $f_2$, ${\rho }_1$, ${\rho }_2$, parameters $r,m$ and the initial data, the result on blow-up of solutions and upper bound of blow-up time are given.