Schamel equation

The Schamel equation is [2]

${\displaystyle \phi _{t}+(1+b{\sqrt {\phi }})\phi _{x}+\phi _{xxx}=0,}$

where ${\displaystyle \phi _{(t,x)}}$ stands for ${\displaystyle \partial _{(t,x)}\phi }$. In the case of ion-acoustic solitary waves, the parameter ${\displaystyle b}$ reflects the effect of electrons trapped in the trough of the electrostatic potential ${\displaystyle \phi }$. It is given by ${\displaystyle b={\frac {1-\beta }{\sqrt {\pi }}}}$, where ${\displaystyle \beta }$, the trapping parameter, reflects the status of the trapped electrons, ${\displaystyle \beta =0}$ representing a flat-topped stationary trapped electron distribution, ${\displaystyle \beta <0}$ a dip or depression. It holds ${\displaystyle 0\leq \phi \leq \psi \ll 1}$, where ${\displaystyle \psi }$ is the wave amplitude. All quantities are normalized: the potential energy by electron thermal energy, the velocity by ion sound speed, time by inverse ion plasma frequency and space by electron Debye length. Note that for a KdV equation ${\displaystyle b{\sqrt {\phi }}}$ is replaced by ${\displaystyle \phi }$ such that the nonlinearity becomes bilinear (see later).

The steady state solitary wave solution, ${\displaystyle \phi (x-v_{0}t)}$, is given in the comoving frame by:

${\displaystyle \phi (x)=\psi \operatorname {sech} ^{4}\left({\sqrt {\frac {b{\sqrt {\psi }}}{30}}}x\right)}$

${\displaystyle v_{0}=1+{\frac {8}{15}}b{\sqrt {\psi }}.}$

The speed of the structure is supersonic, ${\displaystyle v_{0}>1}$, since ${\displaystyle b}$ has to be positive, ${\displaystyle 0<b}$, which corresponds in the ion acoustic case to a depressed trapped electron distribution ${\displaystyle \beta <1}$.[2][7]

The proof of this solution uses the analogy to classical mechanics via
${\displaystyle \phi _{xx}=:-{\mathcal {V}}'(\phi )}$ with ${\displaystyle {\mathcal {V}}(\phi )}$, being the corresponding pseudo-potential. From this we get by an integration: ${\displaystyle {\frac {\phi _{x}^{2}}{2}}+{\mathcal {V(\phi )}}=0}$, which represents the pseudo-energy, and from the Schamel equation: ${\displaystyle -{\mathcal {V}}(\phi )={\frac {(v_{0}-1)}{2}}\phi ^{2}-{\frac {4b}{15}}\phi ^{5/2}}$. Through the obvious demand, namely that at potential maximum, ${\displaystyle \phi =\psi }$, the slope ${\displaystyle \phi _{x}}$ of ${\displaystyle \phi }$ vanishes we get: ${\displaystyle {\mathcal {V}}(\psi )=0}$. This is a nonlinear dispersion relation (NDR) because it determines the phase velocity ${\displaystyle v_{0}}$ given by the second expression. The canonical form of ${\displaystyle {\mathcal {V}}(\phi )}$ is obtained by replacing ${\displaystyle v_{0}}$ with the NDR. It becomes:

${\displaystyle -{\mathcal {V}}(\phi )={\frac {4}{15}}b\phi ^{2}({\sqrt {\psi }}-{\sqrt {\phi }}).}$

The use of this expression in ${\displaystyle x(\phi )=\int _{\phi }^{\psi }{\frac {d\xi }{\sqrt {-2{\mathcal {V}}(\xi )}}}}$, which follows from the pseudo-energy law, yields by integration:

${\displaystyle x(\phi )={\sqrt {\frac {30}{b{\sqrt {\psi }}}}}\tanh ^{-1}\left({\sqrt {1-{\sqrt {\frac {\phi }{\psi }}}}}\right).}$

This is the inverse function of ${\displaystyle \phi (x)}$ as given in the first equation. Note that the integral in the denominator of ${\displaystyle x(\phi )}$ exists and can be expressed by known mathematical functions. Hence ${\displaystyle \phi (x)}$ is a mathematically disclosed function. However, the structure often remains mathematically undisclosed, i.e. it cannot be expressed by known functions (see for instance Sect. Logarithmic Schamel equation). This generally happens if more than one trapping scenarios are involved, as e.g. in driven intermittent plasma turbulence.[8]

In contrast to the KdV equation, the Schamel equation is an example of a non-integrable evolution equation. It only has a finite number of (polynomial) constants of motion [9] and does not pass a Painlevé test.[4][10] Since a so-called Lax pair (L,P) does not exist,[11] it is not integrable by the inverse scattering transform.[12]

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