The *Korteweg de Vries* (KdV) equation:

(- v_{ o} + c_{ s} + v) ¶ ** ^{ }**v / ¶x’ - m ¶

^{2 }v / ¶x

^{2}’ + a ¶

^{3 }v / ¶x

^{3}= 0

is a well-known example of a soliton equation admitting nonlinear superposition, see e.g. a graphical representation of a soliton solution here:

In the KdV equation c_{ s} is the ion sound speed, and v_{ o} the electron thermal speed. In the form shown, note the appearance of the *dissipative term:*

(m ¶ ^{2 }v / ¶x^{2})

and for a soliton to evolve into a shock a dissipative mechanism is needed. In the more common situation steepening of the wave balances dispersion and we obtain a wave form such as shown in the graphic

On integrating once, and excluding the shock evolution term (in m) , we obtain:

^{2 }V/ dx

^{2 }- (v

_{ o }-

**c**

_{ }_{ s}) v - v

^{2}/ 2 = 0

^{2 }V/ dx

^{2 }= - ¶ v V' (x) =

_{ s}

**- v**

_{ o}) v

^{2}/ 2 + v

^{3}/ 6 ]

*pseudo*

*potential*and designate it:

_{ s}

**- v**

_{ o}) v

^{2}/ 2 + v

^{3}/ 6 ]

*Sagdeev potential*. It is left to the industrious reader to do a simple plot of F (v) vs. v, with v

_{ max }shown on v -axis.

*soliton*solution we have:

_{v= 0}

**= 0; ¶**

^{2}F/ ¶ v

^{2}< 0

__<__v

__<__v

_{ max}

_{v= v max}

**> 0**

_{ s}

**- v**

_{ o }> 0, the other for c

_{ s}

**- v**

_{ o }< 0. Since we demand only a localized wave form then it will always be the latter form used, i.e. in further analyses. One such is to obtain a soliton solution for the KdV equation:

^{2 }V/ dx

^{2 }- (v

_{ o }-

**c**

_{ }_{ s}) v - v

^{2}/ 2 = 0

_{ s}

**- v**

_{ o }< 0. The procedure is then to multiply the KdV by v' and then integrate to obtain:

^{2}) = (v

_{ o }-

**c**

_{ }_{ s}) v

^{2}/ 2 - v

^{3}/ 6

_{ o }-

**c**

_{ }_{ s}) sech

^{2}[(v

_{ o }-

**c**

_{ }_{ s}/ 4 a) ½ x' ]

**Suggested Problems**:

_{ s}

**- v**

_{ o }< 0 and indicate the position of v

_{ max}on the graph.

^{2 }V/ dx

^{2 }- (v

_{ o }-

**c**

_{ }_{ s}) v - v

^{2}/ 2 = 0

_{ o }-

**c**

_{ }_{ s}) sech

^{2}[(v

_{ o }-

**c**

_{ }_{ s}/ 4 a) ½ x' ]