# 贝克隆变换与几类KdV型方程Bäcklund Transformation and Several Types of KdV Equations

DOI: 10.12677/PM.2018.82015, PDF, HTML, XML, 下载: 830  浏览: 1,140  科研立项经费支持

Abstract: In this paper, we investigate Bӓcklund transformations of several types of KdV equations. We obtain the solitary wave solutions of a new type KdV equation and a fifth-order KdV equation with the help of Bӓcklund transformations.

1. 引言

2. 贝克隆变换的定义

${u}_{t}=K\left(u\right)$

${v}_{t}=G\left(v\right)$ ，对固定的 $t$$u\left(t,x\right)\in S\left(ℝn\right)$

${B\left(u\left(t,x\right),v\left(t,x\right)\right)|}_{t=0}=0$ ，则有 ${B\left(u\left(t,x\right),v\left(t,x\right)\right)|}_{t={t}_{0}}=0,\text{\hspace{0.17em}}\forall {t}_{0}>0$

${u}_{t}=K\left(u\right)\stackrel{B}{↔}{v}_{t}=G\left(v\right)$

3. KdV型方程的贝克隆变换和孤立子解

${u}_{t}-{u}_{xxx}+6u{u}_{x}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(KdV\right)$

${v}_{t}-{v}_{xxx}+6{v}^{2}{v}_{x}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(mKdV\right)$

${Q}_{t}-{Q}_{xxx}+3\frac{{Q}_{x}{Q}_{xx}}{Q}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{New}\text{\hspace{0.17em}}\text{KdV}\right)$

$u=-{v}_{x}-{v}^{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{Miura}\right)$

$v=\frac{{Q}_{x}}{Q}={\left(\mathrm{ln}Q\right)}^{\prime }$ (1)

$u=-\frac{{Q}_{xx}}{Q}$ (2)

${\left(\frac{{Q}_{x}}{Q}\right)}_{t}-{\left(\frac{{Q}_{x}}{Q}\right)}_{xxx}+6{\left(\frac{{Q}_{x}}{Q}\right)}^{2}{\left(\frac{{Q}_{x}}{Q}\right)}_{x}=0$

${\left(\frac{{Q}_{t}}{Q}\right)}_{x}-{\left[\frac{{Q}_{xxx}}{Q}-3\frac{{Q}_{x}{Q}_{xx}}{{Q}^{2}}+2{\left(\frac{{Q}_{x}}{Q}\right)}^{3}-2{\left(\frac{{Q}_{x}}{Q}\right)}^{3}\right]}_{x}=0$

$u=-{\left(\frac{{Q}_{x}}{Q}\right)}_{x}-{\left(\frac{{Q}_{x}}{Q}\right)}^{2}=-\frac{{Q}_{xx}Q-{Q}_{x}^{2}}{{Q}^{2}}-{\left(\frac{{Q}_{x}}{Q}\right)}^{2}=-\frac{{Q}_{xx}}{Q}$

$v\left(t,x\right)=R\left(x-ct\right)=R\left(\xi \right)$ ，其中 $c>0$$\xi =x-ct$

$-c{R}^{\prime }-{R}^{‴}+6{R}^{2}{R}^{\prime }=-c{R}^{\prime }-{R}^{‴}+{\left(2{R}^{3}\right)}^{\prime }=0$

$v\left(x-ct\right)=\sqrt{\frac{c}{2}}\mathrm{tanh}\left[\frac{\sqrt{c}\left(x-ct\right)}{\sqrt{2}}\right]=\sqrt{\frac{c}{2}}\mathrm{tanh}\left(\sqrt{\frac{c}{2}}\xi \right)$

$v\left(\xi \right)=\sqrt{\frac{c}{2}}\mathrm{tanh}\left(\sqrt{\frac{c}{2}}\xi \right)={\left[\mathrm{ln}\mathrm{cosh}\left(\sqrt{\frac{c}{2}}\xi \right)\right]}^{\prime }={\left(\mathrm{ln}Q\right)}^{\prime }$

$Q\left(x-ct\right)=\mathrm{cosh}\left(\sqrt{\frac{c}{2}}\left(x-ct\right)\right)$

4. 五阶KdV型方程的贝克隆变换和孤立子解

${u}_{t}+{u}_{xxx}+6u{u}_{x}-\mu {\left({u}_{xxxx}+5{u}_{x}^{2}+10u{u}_{x}+10{u}^{3}\right)}_{x}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(5KdV\right)$

${v}_{t}+{v}_{xxx}-\frac{3}{2}{v}^{2}{v}_{x}-\mu \left({v}_{xxxx}+\frac{5}{2}{v}_{x}^{3}+10v{v}_{x}{v}_{xx}+\frac{5}{2}{v}_{}^{3}{v}_{xx}+\frac{15}{8}{u}^{4}{u}_{x}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{5mKdV}\right)$

$u=\frac{1}{2}{v}_{x}-\frac{1}{4}{v}^{2}$ (3)

${P}_{\mu ,u}={u}_{t}+{u}_{xxx}+6u{u}_{x}-\mu {\left({u}_{xxxx}+5{u}_{x}^{2}+10u{u}_{x}+10{u}^{3}\right)}_{x}=0$

${W}_{\mu ,v}={v}_{t}+{v}_{xxx}-\frac{3}{2}{v}^{2}{v}_{x}-\mu \left({v}_{xxxx}+\frac{5}{2}{v}_{x}^{3}+10v{v}_{x}{v}_{xx}+\frac{5}{2}{v}^{3}{v}_{xx}+\frac{15}{8}{u}^{4}{u}_{x}\right)=0$

$\frac{1}{2}\left({\partial }_{x}-v\right){W}_{\mu ,v}={P}_{\mu ,u}$ (4)

${W}_{\mu ,v}=0⇒{P}_{\mu ,u}=0$

$v\left(t,x\right)=\sqrt{2c}\mathrm{tanh}\left[\sqrt{\frac{c}{2}}\left(x-\beta t\right)\right],\text{\hspace{0.17em}}\beta =c+\mu {c}^{2}$

$u\left(t,x\right)=\frac{c}{2}{\mathrm{sech}}^{2}\left(\sqrt{\frac{c}{2}}\left(x-\beta t\right)\right),\text{\hspace{0.17em}}\beta =c+\mu {c}^{2}$

[1] Calogero, F. and Degasperis, A. (1980) Spectral Transform and Solitons I, Studies in Mathematics and Its Application. Vol. 13, North Holland, Amsterdam.

[2] Fokas, A.S. and Fuchssteiner, B. (1981) Bӓcklund Transformation for Hereditary Symmetries. Nonlinear Analysis: Theory, Methods & Applications, 5, 423-432. https://doi.org/10.1016/0362-546X(81)90025-0

[3] Gu, C., Hu, H. and Zhou, Z. (2005) Darboux Transformations in Integrable Systems. Springer, Dordrecht. https://doi.org/10.1007/1-4020-3088-6

[4] 郭柏灵, 庞小峰. 孤立子[M]. 北京: 科学出版社, 1987.

[5] Miura, R.M. (Ed.) (1976) Bӓcklund Transformations, I.S.T. Method and Their Applications. Lecture Notes in Math., 515, Springer, Berlin.

[6] Rogers, C. and Schief, W.K. (2002) Bӓcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511606359

[7] Alejo, M.A. and Munoz, C. (2015) Dynamics of Complex-Valued Modified KdV Solitons with Applications to the Stability of Breathers. Analysis and PDE, 8, 629-674. https://doi.org/10.2140/apde.2015.8.629

 [1] Calogero, F. and Degasperis, A. (1980) Spectral Transform and Solitons I, Studies in Mathematics and Its Application. Vol. 13, North Holland, Amsterdam. [2] Fokas, A.S. and Fuchssteiner, B. (1981) Bӓcklund Transformation for Hereditary Symmetries. Nonlinear Analysis: Theory, Methods & Applications, 5, 423-432. https://doi.org/10.1016/0362-546X(81)90025-0 [3] Gu, C., Hu, H. and Zhou, Z. (2005) Darboux Transformations in Integrable Systems. Springer, Dordrecht. https://doi.org/10.1007/1-4020-3088-6 [4] 郭柏灵, 庞小峰. 孤立子[M]. 北京: 科学出版社, 1987. [5] Miura, R.M. (Ed.) (1976) Bӓcklund Transformations, I.S.T. Method and Their Applications. Lecture Notes in Math., 515, Springer, Berlin. [6] Rogers, C. and Schief, W.K. (2002) Bӓcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory. Cam-bridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511606359 [7] Alejo, M.A. and Munoz, C. (2015) Dy-namics of Complex-Valued Modified KdV Solitons with Applications to the Stability of Breathers. Analysis and PDE, 8, 629-674. https://doi.org/10.2140/apde.2015.8.629