Sinh-Gordon: Difference between revisions

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The sinh-Gordon equation is a nonlinear partial differential equation^[1]

\varphi _{tt}-\varphi _{xx}=\sinh \varphi

that has applications in physics and hydrodynamics. It is known for its soliton solutions and arises as a special case of the Toda lattice equation.^[2]

Exact solutions

{\begin{aligned}f_{1}&={\frac {2}{\lambda }}\ln \left(\tan \left({\frac {\lambda b(kx+\mu t+\theta )}{2{\sqrt {b\lambda \left(\mu ^{2}-ak^{2}\right)}}}}\right)\right)\\f_{2}&=-{\frac {2}{\lambda }}\ln \left(\tan \left({\frac {\lambda b(kx+\mu t+\theta )}{2{\sqrt {b\lambda \left(\mu ^{2}-ak^{2}\right)}}}}\right)\right)\\f_{3}&={\frac {2}{\lambda }}\arctan \left(\exp \left({\frac {\lambda b(kx+\mu t+\theta )}{2{\sqrt {b\lambda \left(\mu ^{2}-ak^{2}\right)}}}}\right)\right)\\f_{4}&=-{\frac {2}{\lambda }}\arctan \left(\exp \left({\frac {\lambda b(kx+\mu t+\theta )}{2{\sqrt {b\lambda \left(\mu ^{2}-ak^{2}\right)}}}}\right)\right)\end{aligned}}

where $k, μ$ , and $θ$ are arbitrary constants and it is assumed that

b\lambda \left(\mu ^{2}-ak^{2}\right)>0.

Gallery

References

^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p485 CRC PRESS
^ Yuanxi, Xie (February 2006). Il Nuovo Cimento B. 121 (2): 115–121. doi:10.1393/ncb/i2005-10164-6. {{cite journal}}: Missing or empty |title= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)

[1] Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p485 CRC PRESS

[Yuanxi-2] Yuanxi, Xie (February 2006). Il Nuovo Cimento B. 121 (2): 115–121. doi:10.1393/ncb/i2005-10164-6. {{cite journal}}: Missing or empty |title= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)

[1]

[2]

@@ Line 1: / Line 1: @@
 The '''sinh-Gordon equation''' is a nonlinear [[partial differential equation]]<ref>Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p485 CRC PRESS</ref>
-<math>\varphi_{tt}- \varphi_{xx} = \sinh\varphi </math>
+:<math>\varphi_{tt}- \varphi_{xx} = \sinh\varphi </math>
 that has applications in [[physics]] and [[hydrodynamics]]. It is known for its [[soliton]] solutions and arises as a special case of the [[Toda lattice]] equation.<ref name=Yuanxi>{{cite journal|last=Yuanxi|first=Xie|coauthors=Tang, Jiashi|journal=Il Nuovo Cimento B|date=February 2006|volume=121|issue=2|pages=115–121|doi=10.1393/ncb/i2005-10164-6}}</ref>
 ==Exact solutions==
+:<math>\begin{align}
-:<math> f1 :=\frac{  2}{\lambda}\ln\left(    \tan\left(\frac{\lambda b(kx+\mu t+\theta)}{\left(2\sqrt{(b\lambda(\mu^2-ak^2))}\right)}\right)\right)  </math>
-:<math> f2 :=\frac{ -2}{\lambda}\ln\left(    \tan\left(\frac{\lambda b(kx+\mu t+\theta)}{\left(2\sqrt{(b\lambda(\mu^2-ak^2))}\right)}\right)\right)  </math>
+f_1 &= \frac{2}{\lambda}\ln\left(\tan\left(\frac{\lambda b(kx+\mu t+\theta)}{2\sqrt{b\lambda \left (\mu^2-ak^2 \right )}}\right)\right) \\
-:<math> f3 :=\frac{  2}{\lambda}\arctan\left(\exp\left(\frac{\lambda b(kx+\mu t+\theta)}{\left(2\sqrt{(b\lambda(\mu^2-ak^2))}\right)}\right)\right)  </math>
+f_2 &= -\frac{2}{\lambda}\ln\left(\tan\left(\frac{\lambda b(kx+\mu t+\theta)}{2\sqrt{b\lambda \left (\mu^2-ak^2 \right )}}\right)\right) \\
-:<math> f4 :=\frac{ -2}{\lambda}\arctan\left(\exp\left(\frac{\lambda b(kx+\mu t+\theta)}{\left(2\sqrt{(b\lambda(\mu^2-ak^2))}\right)}\right)\right)  </math>
+f_3 &= \frac{2}{\lambda}\arctan\left(\exp\left(\frac{\lambda b(kx+\mu t+\theta)}{2\sqrt{b\lambda \left (\mu^2-ak^2 \right )}}\right)\right) \\
+f_4 &= -\frac{2}{\lambda}\arctan\left(\exp\left(\frac{\lambda b(kx+\mu t+\theta)}{2\sqrt{b\lambda \left (\mu^2-ak^2 \right )}}\right)\right)  \end{align}</math>
-where <math>k</math>, <math>\mu</math>, and <math>\theta</math> are arbitrary constants and it is assumed that <math>b\lambda(\mu^2-ak^2)>0</math>.
+where {{math|''k'', ''μ''}}, and {{math|''θ''}} are arbitrary constants and it is assumed that
+:<math>b\lambda \left (\mu^2-ak^2 \right )>0.</math>
 ==Gallery==