Hi there, I'm having some difficulty in understanding how the change of variables by considering a retarded time frame can be obtained for this particular eqn I have.(adsbygoogle = window.adsbygoogle || []).push({});

Say I have this original equation,

[itex]

\frac{\partial A}{\partial z} + \beta_1 \frac{\partial A}{\partial t}+ \frac{i \beta_2}{2} \frac{\partial^2 A}{\partial t^2}+\frac{\alpha}{2} A = i \gamma |A|^2 A

[/itex]

Considering a frame of reference moving with speed [itex]\nu_g[/itex] (i.e. group velocity), a new variable T can be defined as follows

[itex]T=t-\frac{z}{\nu_g} = t-\beta_1 z[/itex]

where [itex]\beta_1=\frac{1}{\nu_g}[/itex]

Apparently, the differential equation can be reduced to

[itex]

\frac{\partial A}{\partial z} -\frac{i \beta_2}{2} \frac{\partial^2 A}{\partial T^2}+\frac{\alpha}{2} A = i \gamma |A|^2 A

[/itex]

----------------------------------------------------------------------------

From my understanding, the first order differential term [itex]\beta_1 \frac{\partial A}{\partial t} [/itex] disappears because

[itex]

\frac{\partial A}{\partial t} = \frac{\partial A}{\partial T} \times \frac{\partial T}{\partial t}

[/itex]

Since

[itex]

T =t-\frac{z}{\nu_g} = t-\beta_1 z \\

\therefore \frac{\partial T}{\partial t} = 1-\frac{\partial}{\partial t} (\beta_1 z) \\

= 1-\beta_1 \nu_g = 0 \phantom{0} (\because \beta_1=\frac{1}{\nu_g})

[/itex]

Hence,

[itex]

\frac{\partial A}{\partial t} = 0

[/itex]

However, I can't wrap my head around how to transform the second derivative [itex] \frac{\partial^2 A}{\partial t^2}[/itex]?

[itex]

\frac{\partial^2 A}{\partial t^2} = \frac{\partial}{\partial t}(\frac{\partial A}{\partial t})

= \frac{\partial}{\partial t}(\frac{\partial A}{\partial T} \times \frac{\partial T}{\partial t})

[/itex]

Which seems to give me zero following from the above argument for the first order differential term [itex]\frac{\partial A}{\partial t}[/itex]...

Am I missing something here? I would really appreciate if anyone could help. Thanks in advance!

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# Change of variables for 2nd order differential

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