In space plasma physics many situations arise in which one will wish to examine the behavior of a solitary wave - say in a particular plasma environment. One such is for nonlinear ion-acoustic waves.

The solitary wave (soliton) is just a traveling wave of form:

u(x, t) = f(x - ct)

for some smooth function f that decays rapidly at infinity and under certain conditions can travel without change of shape (see graphic).

Consider the partial differential equation:

u_{tt } - u_{xx } = u ( 2 u** ^{2}** - 1)

Which has a family of solitary wave solutions:

u(x,t) = sech (x cos q + t sinh q)

We do not expect, for mathematical reasons, the sum of two such solutions (superposition) to also be a solution. There is, however, a class of soliton equations which does exhibit a form of nonlinear superposition. An n-soliton solution then is a solution that is asymptotic to a nontrivial sum of n solitary waves, e.g.

å ^{n}_{t=1} f _{i} (x - c_{i} t) as t -> - oo and to the sum of the same waves:

å ^{n}_{t=1} f _{i} (x - c_{i} t + r_{i})

with non-zero phase shifts r_{i} as t -> oo. I.e. after nonlinear interaction the individual solitary waves pass through each other, keeping their velocities and shapes - but with phase shifts.

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 such a soliton equation admitting nonlinear superposition. Here c_{ s} is the ion sound speed, and v_{ o} the electron thermal speed. In the form shown we note the appearance of the dissipative term (m ¶ ^{2 }v / ¶x^{2}) and that 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 one will obtain a wave form such a shown in the top graphic. We focus more on this case now by adjusting the KdV equation.

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' ]

## No comments:

Post a Comment