KdV modified and complex modified from AKNS

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In summary, the Korteweg-de Vries (KdV) equation is a nonlinear partial differential equation that describes long, shallow water waves. In the AKNS system, it is modified by adding an additional term to account for the effects of dispersion and nonlinearity. The KdV modified equation differs from the original by including a "modified dispersion term" crucial for accurately modeling wave behavior. The complex modified KdV equation is obtained by applying a complex transformation to the AKNS system, allowing for the study of soliton solutions. These equations have various applications in physics, including the study of nonlinear wave phenomena and soliton solutions. However, there are still many open research questions related to their existence, stability, and behavior
  • #1
Dustinsfl
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Let \(
L_R =
\begin{pmatrix}
\partial - 2q\partial^{-1}r & 2q\partial^{-1}q\\
-2r\partial^{-1}r & -\partial + 2r\partial^{-1}q
\end{pmatrix}
\).
We know that
\[
\begin{pmatrix}
q\\
-r
\end{pmatrix}_t = -iL_R^n
\begin{pmatrix}
q\\
r
\end{pmatrix}
\]
When \(n = 2\), the RHS is
\[
-
\begin{pmatrix}
q_{xx} & -2q^2r\\
r_{xx} & -2r^2q
\end{pmatrix}
\]
For \(n = 3\), the RHS is
\[
i
\begin{pmatrix}
q_{xxx} - 2q\partial^{-1}rq_{xx} + 2q\partial^{-1}qr_{xx} & -2(q^2)_xr - 2q^2r_x\\
-r_{xxx} - 2r\partial^{-1}rq_{xx} + 2r\partial^{-1}qr_{xx} & 2(r^2)_xq + 2r^2q_x
\end{pmatrix}
\]
Thus,
\[
q_t = i[q_{xxx} - 6qq_xr].
\]
I am supposed to be able to obtain the modified and complex modified KdV equations by the substitutions \(r = -q\) and \(r = -q^*\), respectively. However, I get
\begin{align}
q_t - iq_{xxx} - 6iq^2q_x &= 0\\
q_t - iq_{xxx} - 6i\lvert q\rvert^2q_x &= 0
\end{align}
but the equations should be
\begin{align}
q_t + q_{xxx} + 6q^2q_x &= 0\\
q_t + q_{xxx} + 6\lvert q\rvert^2q_x &= 0
\end{align}
Where should I have picked up another multiple of \(i\)?
 
Last edited:
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  • #2


It seems that you may have missed a factor of \(i\) when calculating the RHS for \(n = 2\). The correct expression should be:
\[
-
\begin{pmatrix}
iq_{xx} & -2iq^2r\\
ir_{xx} & -2ir^2q
\end{pmatrix}
\]
This would result in the correct modified KdV equation when substituting \(r = -q\) or \(r = -q^*\). I hope this helps!
 

FAQ: KdV modified and complex modified from AKNS

What is the KdV equation and how is it modified in the AKNS system?

The Korteweg-de Vries (KdV) equation is a nonlinear partial differential equation that describes the evolution of long, shallow water waves. It is modified in the AKNS system by adding an additional term that takes into account the effects of dispersion and nonlinearity.

How does the KdV modified equation differ from the original KdV equation?

The KdV modified equation includes an additional term known as the "modified dispersion term," which accounts for the effects of dispersion and nonlinearity on the propagation of waves. This term is crucial in accurately modeling the behavior of long, shallow water waves.

What is the complex modified KdV equation and how is it related to the AKNS system?

The complex modified KdV equation is a version of the KdV modified equation that includes an additional complex term. This equation is obtained by applying a complex transformation to the AKNS system, which allows for the study of soliton solutions.

What are the applications of the KdV modified and complex modified equations in physics?

The KdV modified and complex modified equations have various applications in physics, including the study of nonlinear wave phenomena in fluid dynamics, plasma physics, and condensed matter physics. They are also used in the study of soliton solutions, which have important implications in fields such as optical communications and quantum computing.

Are there any open research questions related to the KdV modified and complex modified equations?

Yes, there are still many open research questions related to the KdV modified and complex modified equations. Some of these include the existence and stability of soliton solutions, the effects of higher-order nonlinearities on the equations, and the behavior of the equations in multi-dimensional systems.

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