KdV-mKdV方程

KdV-mKdV方程是一個非線性偏微分方程：^[1]

${\displaystyle u_{t}+6*\alpha *u*u_{x}+6*\beta *u^{2}*u_{x}+\gamma *U_{xxx}=0}$

解析解

${\displaystyle u(x,t)=-1/(2*\beta )-{\sqrt {(}}\beta *\gamma *(-1+_{C}1^{2}))*_{C}3*JacobiNC(-_{C}2-_{C}3*x+(1/2)*_{C}3*(-2*\beta *\gamma *_{C}3^{2}+4*\beta *_{C}3^{2}*\gamma *_{C}1^{2}-3)*t/\beta ,_{C}1)/\beta }$

${\displaystyle u(x,t)=-1/(2*\beta )+{\sqrt {(}}\beta *\gamma *(-1+_{C}1^{2}))*_{C}3*JacobiNC(-_{C}2-_{C}3*x+(1/2)*_{C}3*(-2*\beta *\gamma *_{C}3^{2}+4*\beta *_{C}3^{2}*\gamma *_{C}1^{2}-3)*t/\beta ,_{C}1)/\beta }$

${\displaystyle u(x,t)=-1/(2*\beta )-{\sqrt {(}}-\beta *\gamma *(-1+_{C}1^{2}))*_{C}3*JacobiND(_{C}2+_{C}3*x+(1/2)*_{C}3*(2*\beta *_{C}3^{2}*\gamma *_{C}1^{2}-4*\beta *\gamma *_{C}3^{2}+3)*t/\beta ,_{C}1)/\beta }$

${\displaystyle u(x,t)=-1/(2*\beta )+{\sqrt {(}}-\beta *\gamma *(-1+_{C}1^{2}))*_{C}3*JacobiND(_{C}2+_{C}3*x+(1/2)*_{C}3*(2*\beta *_{C}3^{2}*\gamma *_{C}1^{2}-4*\beta *\gamma *_{C}3^{2}+3)*t/\beta ,_{C}1)/\beta }$

${\displaystyle u(x,t)=-1/(2*\beta )-\gamma *_{C}2*sech(_{C}1+_{C}2*x-(1/2)*_{C}2*(2*\beta *\gamma *_{C}2^{2}-3)*t/\beta )/{\sqrt {(}}\beta *\gamma )}$

${\displaystyle u(x,t)=-1/(2*\beta )-\gamma *_{C}3*JacobiDN(_{C}2+_{C}3*x+(1/2)*_{C}3*(2*\beta *_{C}3^{2}*\gamma *_{C}1^{2}-4*\beta *\gamma *_{C}3^{2}+3)*t/\beta ,_{C}1)/{\sqrt {(}}\beta *\gamma )}$

${\displaystyle u(x,t)=-1/(2*\beta )-\gamma *_{C}2*cot(_{C}1+_{C}2*x-(1/2)*_{C}2*(4*\beta *\gamma *_{C}2^{2}-3)*t/\beta )/{\sqrt {(}}-\beta *\gamma )}$

${\displaystyle u(x,t)=-1/(2*\beta )-\gamma *_{C}2*coth(_{C}1+_{C}2*x+(1/2)*_{C}2*(4*\beta *\gamma *_{C}2^{2}+3)*t/\beta )/{\sqrt {(}}-\beta *\gamma )}$

${\displaystyle u(x,t)=-1/(2*\beta )-\gamma *_{C}2*csch(_{C}1+_{C}2*x-(1/2)*_{C}2*(2*\beta *\gamma *_{C}2^{2}-3)*t/\beta )/{\sqrt {(}}-\beta *\gamma )}$

${\displaystyle u(x,t)=-1/(2*\beta )-\gamma *_{C}2*tan(_{C}1+_{C}2*x-(1/2)*_{C}2*(4*\beta *\gamma *_{C}2^{2}-3)*t/\beta )/{\sqrt {(}}-\beta *\gamma )}$

${\displaystyle u(x,t)=-1/(2*\beta )-\gamma *_{C}2*tanh(_{C}1+_{C}2*x+(1/2)*_{C}2*(4*\beta *\gamma *_{C}2^{2}+3)*t/\beta )/{\sqrt {(}}-\beta *\gamma )}$

${\displaystyle u(x,t)=-1/(2*\beta )-\gamma *_{C}3*JacobiNS(_{C}2+_{C}3*x+(1/2)*_{C}3*(2*\beta *\gamma *_{C}3^{2}+2*\beta *_{C}3^{2}*\gamma *_{C}1^{2}+3)*t/\beta ,_{C}1)/{\sqrt {(}}-\beta *\gamma )}$

行波圖

Kdv-mKdv equation traveling wave plot

Kdv-mKdv equation traveling wave plot

Kdv-mKdv equation traveling wave plot

Kdv-mKdv equation traveling wave plot

Kdv-mKdv equation traveling wave plot

Kdv-mKdv equation traveling wave plot

Kdv-mKdv equation traveling wave plot

Kdv-mKdv equation traveling wave plot

Kdv-mKdv equation traveling wave plot

Kdv-mKdv equation traveling wave plot

Kdv-mKdv equation traveling wave plot

Kdv-mKdv equation traveling wave plot

參考文獻

1. 李志斌編著 《非線性數學物理方程的行波解》 181頁 科學出版社 2008

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3. 李志斌編著 《非線性數學物理方程的行波解》 科學出版社
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