变形KdV-Burgers(Modified KdV-Burgers equation)是一个非线性偏微分方程:[1]
解析解
![{\displaystyle u(x,t)=-(1/6)*\beta *{\sqrt {(}}6)/{\sqrt {(}}\alpha )-{\sqrt {(}}6)*_{C}2*cot(_{C}1+_{C}2*x+(-2*_{C}2^{3}+(1/6)*\beta ^{2}*_{C}2)*t)/{\sqrt {(}}\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28101cb25577dc73b3440c4be0e210ecc83ba88f)
![{\displaystyle u(x,t)=-(1/6)*\beta *{\sqrt {(}}6)/{\sqrt {(}}\alpha )-{\sqrt {(}}6)*_{C}2*coth(_{C}1+_{C}2*x+(2*_{C}2^{3}+(1/6)*\beta ^{2}*_{C}2)*t)/{\sqrt {(}}\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b48cfd81cac022606a5c915d978507eff1944d)
![{\displaystyle u(x,t)=-(1/6)*\beta *{\sqrt {(}}6)/{\sqrt {(}}\alpha )+{\sqrt {(}}6)*_{C}2*tan(_{C}1+_{C}2*x+(-2*_{C}2^{3}+(1/6)*\beta ^{2}*_{C}2)*t)/{\sqrt {(}}\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06e061d44c26fea9f5bc8587b96543a3573fb4a5)
![{\displaystyle u(x,t)=-(1/6)*\beta *{\sqrt {(}}6)/{\sqrt {(}}\alpha )-{\sqrt {(}}6)*_{C}2*tanh(_{C}1+_{C}2*x+(2*_{C}2^{3}+(1/6)*\beta ^{2}*_{C}2)*t)/{\sqrt {(}}\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0105aa0ad29a9e9e4d8692bbe04070e88286701)
![{\displaystyle u(x,t)=(1/6)*\beta *{\sqrt {(}}6)/{\sqrt {(}}\alpha )+{\sqrt {(}}6)*_{C}2*cot(_{C}1+_{C}2*x+(-2*_{C}2^{3}+(1/6)*\beta ^{2}*_{C}2)*t)/{\sqrt {(}}\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76a854f9c3a23599c54cf543a078e7a20b1ba98d)
![{\displaystyle u(x,t)=(1/6)*\beta *{\sqrt {(}}6)/{\sqrt {(}}\alpha )+{\sqrt {(}}6)*_{C}2*coth(_{C}1+_{C}2*x+(2*_{C}2^{3}+(1/6)*\beta ^{2}*_{C}2)*t)/{\sqrt {(}}\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1204ed1402bf0a5f678dfcf6361fbaddb3a4f701)
![{\displaystyle u(x,t)=(1/6)*\beta *{\sqrt {(}}6)/{\sqrt {(}}\alpha )-{\sqrt {(}}6)*_{C}2*tan(_{C}1+_{C}2*x+(-2*_{C}2^{3}+(1/6)*\beta ^{2}*_{C}2)*t)/{\sqrt {(}}\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b5714462cefb9c2f7852e2fa53b8543c645246c)
![{\displaystyle u(x,t)=(1/6)*\beta *{\sqrt {(}}6)/{\sqrt {(}}\alpha )+{\sqrt {(}}6)*_{C}2*tanh(_{C}1+_{C}2*x+(2*_{C}2^{3}+(1/6)*\beta ^{2}*_{C}2)*t)/{\sqrt {(}}\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31005bd5917e044853fc9a906d8c33dbaca7e3d9)
行波图
参考文献
- ^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p1041 CRC PRESS
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