一类广义色散方程的行波解A Study on Exact Travelling Wave Solutions of Generalized Dispersive Equations

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The aim of this paper is devoted to the study of physical structures of solutions for generalized nonlinear KdV-type equations in two-dimensional space. The variable replacement method is used to get compactons, solitons, solitary patterns and periodic solutions. Furthermore, we point out that the different exponents and coefficients lead to different results of physical structures for this kind of equations with positive or negative exponents.

1. 引言

${u}_{t}+{\left({u}^{m}\right)}_{x}+{\left({u}^{m}\right)}_{xxx}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}m>1,\text{\hspace{0.17em}}1 (1)

Wazwaz在文 [4] 中研究了两个如下形式的(2 + 1)维Boussinesq方程

${u}_{tt}-{u}_{xx}-{u}_{yy}-a{\left({u}^{2n}\right)}_{xx}-b{\left[{u}^{n}{\left({u}^{n}\right)}_{xx}\right]}_{xx}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}n>1,$ (2)

${u}_{tt}-{u}_{xx}-{u}_{yy}-a{\left({u}^{-2n}\right)}_{xx}-b{\left[{u}^{-n}{\left({u}^{-n}\right)}_{xx}\right]}_{xx}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}n>1.$ (3)

Wazwaz总结了正余弦拟设法的主要步骤，并用它获得了方程(2) (3)的紧孤子、孤立子、孤立波相似解和周期解。他进一步指出，方程中函数的指数和系数比a/b、以及正负号的变化都会导致解的物理结构产生质的变化。在文 [3] [4] [5] [6] 中，Wazwaz还运用该方法彻底地讨论了一维及更高维的KdV、mKdV、KP方程。

${u}_{t}+a{u}_{x}+b{\left({u}^{n}\right)}_{x}+b{\left({u}^{n}\right)}_{y}+{\left({u}^{n}\right)}_{xxx}+{\left({u}^{n}\right)}_{yyy}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}n>0,$ (4)

${u}_{t}+a{u}_{x}+b{\left({u}^{-n}\right)}_{x}+b{\left({u}^{-n}\right)}_{y}+{\left({u}^{-n}\right)}_{xxx}+{\left({u}^{-n}\right)}_{yyy}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}n>1,$ (5)

${\text{sinh}}_{q}\left(x\right)=\frac{\mathrm{exp}\left(x\right)-q\mathrm{exp}\left(-x\right)}{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\text{cosh}}_{q}\left(x\right)=\frac{\mathrm{exp}\left(x\right)+q\mathrm{exp}\left(-x\right)}{2},$

${\text{csch}}_{q}\left(x\right)=\frac{1}{{\text{sinh}}_{q}\left(x\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\text{sech}}_{q}\left(x\right)=\frac{1}{{\text{cosh}}_{q}\left(x\right)},$

2. 主要结论

2.1. 具有正指数的KdV型方程

$\xi =\mu x+\eta y-ct$$\mu ,\eta ,c$ 是不为0的常数，且 $\mu +\eta \ne 0$ 。易知

$\frac{\partial }{\partial t}=-c\frac{\text{d}}{\text{d}\xi },\text{\hspace{0.17em}}\frac{\partial }{\partial x}=\mu \frac{\text{d}}{\text{d}\xi },\text{\hspace{0.17em}}\frac{\partial }{\partial {x}^{3}}={\mu }^{3}\frac{{\text{d}}^{3}}{\text{d}{\xi }^{3}},\text{\hspace{0.17em}}\frac{\partial }{\partial y}=\sigma \frac{\text{d}}{\text{d}\xi },\text{\hspace{0.17em}}\frac{\partial }{\partial {y}^{3}}={\sigma }^{3}\frac{{\text{d}}^{3}}{\text{d}{\xi }^{3}}.$ (6)

$\left(a\mu -c\right)\frac{\text{d}u}{\text{d}\xi }+b\left(\mu +\eta \right)\frac{\text{d}{u}^{n}}{\text{d}\xi }+\left({\mu }^{3}+{\eta }^{3}\right)\frac{{\text{d}}^{3}{u}^{n}}{\text{d}{\xi }^{3}}=0$ (7)

1) 当 $n=1$ 时，方程(7)变形为

$\frac{{\text{d}}^{3}u}{\text{d}{\xi }^{3}}+\frac{1}{\rho }\left(b+\frac{a\mu -c}{\mu +\eta }\right)\frac{\text{d}u}{\text{d}\xi }=0,$ (8)

$b+\frac{a\mu -c}{\mu +\eta }\ge 0$ ，解得紧孤子解

$u\left(x,y,t\right)=\left\{\begin{array}{ll}{\left({\lambda }_{1}^{2}+{\lambda }_{2}^{2}\right)}^{-\frac{1}{2}}\mathrm{sin}\left(\sqrt{\frac{1}{\rho }\left(b+\frac{a\mu -c}{\mu +\eta }\right)}\xi +\theta \right)+A,\hfill & |\phi |\le \text{π}\hfill \\ 0,\hfill & 其它\hfill \end{array}$

$b+\frac{a\mu -c}{\mu +\eta }>0$ ，我们得到下列形式的孤立波相似解(下式中 ${\lambda }_{3},{\lambda }_{4}$ 都是常数)

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(x,y,t\right)={\lambda }_{3}{\mathrm{cosh}}_{\left(\frac{{\lambda }_{4}}{{\lambda }_{3}}\right)}\left(\sqrt{-\frac{1}{\rho }\left(b+\frac{a\mu -c}{\mu +\eta }\right)}\xi +A\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}当\frac{{k}_{4}}{{k}_{3}}>0;\\ 或\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(x,y,t\right)={\lambda }_{3}{\mathrm{sinh}}_{\left(-\frac{{\lambda }_{4}}{{\lambda }_{3}}\right)}\left(\sqrt{-\frac{1}{\rho }\left(b+\frac{a\mu -c}{\mu +\eta }\right)}\xi +A\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}当\frac{{k}_{4}}{{k}_{3}}<0.\end{array}$

2) 当 $n\ge 2$ 时，我们考虑下面两种情况：

$\left(a\mu -c\right)u+b\left(\mu +\eta \right){u}^{n}+\left({\mu }^{3}+{\eta }^{3}\right)\frac{{\text{d}}^{2}{u}^{n}}{\text{d}{\xi }^{2}}=0.$ (9)

$\frac{\text{d}{u}^{n}}{\text{d}\xi }=Z,\frac{{\text{d}}^{2}{u}^{n}}{\text{d}{\xi }^{2}}=Z\frac{\text{d}Z}{\text{d}{u}^{n}}$ ，代入式(9)中分离变量得

$\left({\mu }^{3}+{\eta }^{3}\right)Z\text{d}Z=n\left[\left(c-a\mu \right){u}^{n}-b\left(\mu +\eta \right){u}^{2n-1}\right]\text{d}u.$ (10)

$\frac{{\mu }^{3}+{\eta }^{3}}{2}{Z}^{2}=n\left[\frac{c-a\mu }{n+1}{u}^{n+1}-\frac{b\left(\mu +\eta \right)}{2n}{u}^{2n}\right].$ (11)

$\frac{n{u}^{\frac{n-3}{2}}\text{d}u}{\sqrt{\frac{2n\left(c-a\mu \right)}{\left(n+1\right)\left(\mu +\eta \right)}-b{u}^{n-1}}}=±\frac{\text{d}\xi }{\sqrt{\rho }}.$ (12)

$V=\sqrt{\frac{2n\left(c-a\mu \right)}{\left(n+1\right)\left(\mu +\eta \right)}-b{u}^{n-1}}$

$\frac{\text{d}V}{\sqrt{\frac{2n\left(c-a\mu \right)}{\left(n+1\right)\left(\mu +\eta \right)}-{V}^{2}}}=±\frac{n-1}{2n}\sqrt{\frac{b}{\rho }}\text{d}\xi .$ (13)

$\frac{a\mu -c}{\mu +\eta }\ne 0$ 时，对式(13)积分解得如下紧孤子解

$u\left(x,y,t\right)=\left\{\begin{array}{ll}{\left[\frac{2n\left(c-a\mu \right)}{b\left(n+1\right)\left(\mu +\eta \right)}{\mathrm{cos}}^{2}\left(\frac{n-1}{2n}\sqrt{\frac{b}{\rho }}\xi ±r\right)\right]}^{\frac{1}{n-1}},\hfill & |\varphi |\le \frac{\text{π}}{2}\hfill \\ 0,\hfill & 其它\hfill \end{array}$

$u\left(x,y,t\right)={\left[\frac{2n\left(c-a\mu \right)}{b\left(n+1\right)\left(\mu +\eta \right)}{\mathrm{cosh}}^{2}\left(\frac{n-1}{2n}\sqrt{-\frac{b}{\rho }}\xi ±ir\right)\right]}^{\frac{1}{n-1}},i=\sqrt{-1}.$

$b\frac{\text{d}{u}^{n}}{\text{d}\xi }+\left({\mu }^{2}-\mu \eta +{\eta }^{2}\right)\frac{{\text{d}}^{3}{u}^{n}}{\text{d}{\xi }^{3}}=0.$ (14)

$b>0$ 时，解方程(14)得 ${u}^{n}={\left({k}_{1}^{2}+{k}_{2}^{2}\right)}^{\frac{1}{2}}\text{sin}\left(\sqrt{\frac{b}{\rho }}\xi +\alpha \right)+A$ ，其中 ${k}_{1},{k}_{2}$ 为任意常数，且当 ${k}_{1}={k}_{2}=0$ 时， $\alpha =0$ ，否则 $\alpha$ 满足 $\mathrm{sin}\alpha ={k}_{1}{\left({k}_{1}^{2}+{k}_{2}^{2}\right)}^{-\frac{1}{2}}$$\mathrm{cos}\alpha ={k}_{2}{\left({k}_{1}^{2}+{k}_{2}^{2}\right)}^{-\frac{1}{2}}$

$u\left(x,y,t\right)=\left\{\begin{array}{ll}{\left[{\left({k}_{1}^{2}+{k}_{2}^{2}\right)}^{\frac{1}{2}}\mathrm{sin}\left(\sqrt{\frac{b}{\rho }}\xi +\alpha \right)+A\right]}^{\frac{1}{n}},\hfill & |\psi |\le \text{π}\hfill \\ 0,\hfill & 其它\hfill \end{array}$

$b<0$ ，我们有解

$u\left(x,y,t\right)={\left[{k}_{3}\mathrm{exp}\left(\sqrt{-\frac{b}{\rho }}\xi \right)+{k}_{4}\mathrm{exp}\left(-\sqrt{-\frac{b}{\rho }}\xi \right)+A\right]}^{\frac{1}{n}}$${k}_{3},{k}_{4}$ 为任意常数。

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(x,y,t\right)={\left[2{k}_{3}{\mathrm{cosh}}_{\left(\frac{{k}_{4}}{{k}_{3}}\right)}\left(\sqrt{-\frac{b}{\rho }}\xi \right)+A\right]}^{\frac{1}{n}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}当\frac{{k}_{4}}{{k}_{3}}>0;\\ 或\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(x,y,t\right)={\left[2{k}_{3}{\mathrm{cosh}}_{\left(-\frac{{k}_{4}}{{k}_{3}}\right)}\left(\sqrt{-\frac{b}{\rho }}\xi \right)+A\right]}^{\frac{1}{n}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}当\frac{{k}_{4}}{{k}_{3}}<0.\end{array}$

2.2. 具有负指数的KdV型方程

$\left(a\mu -c\right)\frac{\text{d}u}{\text{d}\xi }+b\left(\mu +\eta \right)\frac{\text{d}{u}^{-n}}{\text{d}\xi }+\left({\mu }^{3}+{\eta }^{3}\right)\frac{{\text{d}}^{3}{u}^{-n}}{\text{d}{\xi }^{3}}=0,$ (15)

$\left(a\mu -c\right)u+b\left(\mu +\eta \right){u}^{-n}+\left({\mu }^{3}+{\eta }^{3}\right)\frac{{\text{d}}^{2}{u}^{-n}}{\text{d}{\xi }^{2}}=0.$ (16)

$\frac{-n{u}^{\frac{-n-\text{3}}{2}}\text{d}u}{\sqrt{\frac{2n\left(c-a\mu \right)}{\left(n-1\right)\left(\mu +\eta \right)}-b{u}^{-n-1}}}=±\frac{\text{d}\xi }{\sqrt{\rho }},$ (17)

$b>0$ 时，解得周期解

$u\left(x,y,t\right)={\left[\frac{b\left(n-1\right)\left(\mu +\eta \right)}{2n\left(c-a\mu \right)}{\mathrm{sec}}^{2}\left(\frac{n+1}{2n}\sqrt{\frac{b}{\rho }}\xi ±r\right)\right]}^{\frac{1}{n+1}}.$

$b<0$ 时，解得孤立子解

$u\left(x,y,t\right)={\left[\frac{b\left(n-1\right)\left(\mu +\eta \right)}{2n\left(c-a\mu \right)}{\text{sech}}^{2}\left(\frac{n+1}{2n}\sqrt{-\frac{b}{\rho }}\xi ±ir\right)\right]}^{\frac{1}{n+1}}.$

$b\frac{\text{d}{u}^{-n}}{\text{d}\xi }+\left({\mu }^{2}-\mu \eta +{\eta }^{2}\right)\frac{{\text{d}}^{3}{u}^{-n}}{\text{d}{\xi }^{3}}=0.$ (18)

$b>0$ 时，解之得周期解

$u\left(x,y,t\right)={\left[{\left({l}_{1}^{2}+{l}_{2}^{2}\right)}^{\frac{1}{2}}\mathrm{sin}\left(\sqrt{\frac{b}{\rho }}\xi +\beta \right)+A\right]}^{-\frac{1}{n}},$

$b<0$ 时，解之得一般解

$u\left(x,y,t\right)={\left[{l}_{3}\mathrm{exp}\left(\sqrt{-\frac{b}{\rho }}\xi \right)+{l}_{4}\mathrm{exp}\left(-\sqrt{-\frac{b}{\rho }}\xi \right)+A\right]}^{-\frac{1}{n}}.$

$A=0$ 时则有两个特殊的孤立子解

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(x,y,t\right)={\left[\frac{1}{2{l}_{3}}{\text{sech}}_{\left(\frac{{l}_{4}}{{l}_{3}}\right)}\left(\sqrt{-\frac{b}{\rho }}\xi \right)\right]}^{\frac{1}{n}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}当\frac{{l}_{4}}{{l}_{3}}>0;\\ 或\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(x,y,t\right)={\left[\frac{1}{2{l}_{3}}\mathrm{csc}{h}_{\left(-\frac{{l}_{4}}{{l}_{3}}\right)}\left(\sqrt{-\frac{b}{\rho }}\xi \right)\right]}^{\frac{1}{n}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}当\frac{{l}_{4}}{{l}_{3}}<0.\end{array}$

3. 结论

Table 1. The physical structures of solutions for the generalized dispersive equations

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