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The KdV equation has two qualitatively different types of permanent form travelling wave solution. These are referred to as cnoidal waves and solitary waves.
 
The KdV equation has two qualitatively different types of permanent form travelling wave solution. These are referred to as cnoidal waves and solitary waves.
  
== Introduction ==
+
==KdV equation in <math>(z,\tau)</math> space ==
  
 
Assume we have wave travelling with speed <math> V_0 </math> without change of form,
 
Assume we have wave travelling with speed <math> V_0 </math> without change of form,
<center><math> H(z,tau)=H(z-V_0\tau) </math></center> and substitue into KdV equation then we obtain
+
 
 +
<center><math> H(z,\tau)=H(z-V_0\tau) </math></center>  
 +
 
 +
and substitute into KdV equation then we obtain
  
 
<center><math>
 
<center><math>
-2V_oH_\xi+3HH_xi+\frac{1}{3}H_\xi\xi\xi=0</math></center>
+
-2V_oH_\xi+3HH_\xi+\frac{1}{3}H_{\xi\xi\xi}=0</math></center>
where <math>xi=z-V_0\tau </math> is the travelling wave coordinate.
+
where <math>\xi=z-V_0\tau </math> is the travelling wave coordinate.
We integrate this equation twice with respect to <math> xi</math> to give
+
 
<center><math> \frac{1}{6}H_\xi^2=V_oH^2-\frac{1}{2}H^3+D_1H+D2=f(H,V_0,D_1,D_2) </math></center>, where D_1 and D_2 are constants of integration.
+
 
 +
We rearrange and integrate this equation with respect to <math>\xi</math> to give
 +
<center><math>\frac{1}{3} \int H_{\xi\xi\xi} d\xi = \int 2V_0 H_{\xi} d\xi - 3\int H H_\xi d\xi</math></center>
 +
 
 +
<center><math> \Longrightarrow
 +
\frac{1}{3}H_{\xi\xi} = 2V_oH - \frac{3}{2}H^2 +D_1</math></center>
 +
 
 +
then multiply <math>H_\xi </math> to all terms and integrate again
 +
 
 +
 
 +
 
 +
<center><math>\Longrightarrow
 +
\frac{1}{3}H_{\xi\xi} H_\xi = 2V_0HH_\xi - \frac{3H^2}{2}H_\xi + D_1H_\xi</math></center>
 +
 
 +
 
 +
 
 +
<center><math>\Longrightarrow
 +
\frac{1}{3} \int H_{\xi\xi}H_\xi d\xi = 2V_o \int H H_\xi d\xi - \frac{3}{2} \int H^2H\xi d\xi + \int D_1 H_\xi d\xi</math></center>
 +
 
 +
 
 +
<center><math>\Longrightarrow
 +
\frac{1}{3} \cdot \frac{1}{2}H_\xi^2 = 2V_0 \frac{H^2}{2}-\frac{H^3}{2} +D_1H + D_2 </math></center>
 +
 
 +
 
 +
 
 +
<center><math> \Longrightarrow
 +
\frac{1}{6}H_\xi^2=V_oH^2-\frac{1}{2}H^3+D_1H+D_2=f(H,V_0,D_1,D_2) </math></center> where <math>D_1</math> and <math>D_2</math> are constants of integration.
 +
 
 +
==Standardization of KdV equation==
 +
 
 +
 
 +
We define <math>f(H)= V_oH^2-\frac{1}{2}H^3+D_1H+D2</math>,
 +
so <math>f(H)=\frac{1}{6}H_\xi^2 </math>
 +
 
 +
It turns out that we require 3 real roots to obtain periodic solutions.
 +
Let roots be <math> H_1 \leq H_2 \leq H_3</math>.
  
We define <math>f(H)= V_oH^2-\frac{1}{2}H^3+D_1H+D2</math>, i.e <math>f(H)=\frac{1}{6}H_\xi^2.
 
  
It turns out that we require 3 real roots to obatain periodic solutions.
 
Let roots be <math> H_1 $leq H_2 $leq H_3</math>.
 
 
We can imagine the graph of cubic function which has 3 real roots and we can now write a function
 
We can imagine the graph of cubic function which has 3 real roots and we can now write a function
 
<center><math> f(H)= \frac{-1}{2}(H-H_1)(H-H_2)(H-H_3)</math></center>
 
<center><math> f(H)= \frac{-1}{2}(H-H_1)(H-H_2)(H-H_3)</math></center>
 +
 +
 +
From the equation <math>f(H)=\frac{1}{6}H_\xi^2</math>, we require <math>f(H)>0</math>
 +
 +
We are only interested in solution for <math>H_2 < H < H_3</math> and we need <math>H_2 < H_3</math>.
 +
 +
and now solve equation in terms of the roots <math>H_i,</math>
 +
 +
We define <math>X=\frac{H}{H_3}</math>, and obtain
 +
 +
<center><math>X_\xi^2=3H_3(X-X_1)(X-X_2)(1-X)</math></center>
 +
where <math>X_i=\frac{H_i}{H}</math>
 +
 +
crest to be at <math>\xi=0</math> and <math>X(0)=0</math>
 +
 +
and a further variable Y via
 +
 +
<center><math> X = 1 +(X_2-1) \sin^2 (Y) </math></center>
 +
 +
 +
<center><math>Y_\xi^2=\frac{3}{4}H_3(1-X_1)\left\{ 1-\frac{(1-X_2)}{(1-X_1)}\sin(Y)^2 \right\}
 +
 +
...(1)</math></center>
 +
 +
so <math>Y(0)=0.</math>
 +
 +
and <center><math>\frac{dY}{d\xi}=\sqrt{l(1-k^2\sin^2(Y))},</math></center>
 +
which is separable.
 +
 +
 +
 +
 +
In order to get this into a completely standard form we define
 +
 +
<center><math>k^2=\frac{1-X_2}{1-X_1}, l=\frac{3}{4}H_3(1-X_1)
 +
 +
...(2) </math></center>
 +
 +
Clearly, <math>0 \leq k^2 \leq 1</math>
 +
and  <math>l>0.</math>
 +
 +
==Solution of the KdV equation==
 +
 +
A simple quadrature of equation (1) subject to the condition (2) the gives us
 +
 +
<center><math>\int_{0}^\bar{Y} \frac{dS}{\sqrt{l(1-k^2\sin^2(Y))}} = \int_{0}^\bar{Y} dS</math></center>
 +
 +
Jacobi elliptic function <math>y= sn(x,k)</math> can be written in the form
 +
 +
 +
<center><math> x= \int_{0}^y \frac{dt}{\sqrt{1-t^2}\sqrt{1-k^2t^2}}</math></center>
 +
for <math>0 < k^2 < 1 </math>, 
 +
 +
or equivalently
 +
 +
<center><math> x=\int_{0}^{\sin^{-1}y} \frac{dS}{\sqrt{1-k^2\sin^2(s)}} </math></center>
 +
 +
Now we can write Y with fixed values of <math>x</math>,<math>k</math> as
 +
 +
<center><math> \bar{Y}=\sin^{-1}(\mathrm {sn} (\sqrt{l}\int_0^{\bar{Y}} d\xi,k)),</math></center>
 +
 +
<center><math> \sin(Y)=\mathrm {sn}(\sqrt{l}\xi;k),</math></center>
 +
and hence
 +
<center><math>X=1+(X_2-1)\mathrm {sn}^2(\sqrt{l}\xi;k)</math></center>
 +
 +
<math>\mathrm {cn}(x;k)</math> is another Jacobi elliptic function with <math>\mathrm {cn}^2+\mathrm {sn}^2=1</math>, and waves are called "cnoidal waves".
 +
 +
Using the result <math>cn^2+sn^2=1</math>, our final result can be expressed in the form
 +
 +
<center><math>H=H_2+(H_3-H_2)\mathrm {cn}^2\left\{ \left[ \frac{3}{4}(H_3-H_1) \right]^{\frac{1}{2}}\xi;k \right\} </math></center>

Latest revision as of 07:19, 17 October 2008

Travelling Wave Solutions of the KdV Equation

The KdV equation has two qualitatively different types of permanent form travelling wave solution. These are referred to as cnoidal waves and solitary waves.

KdV equation in [math]\displaystyle{ (z,\tau) }[/math] space

Assume we have wave travelling with speed [math]\displaystyle{ V_0 }[/math] without change of form,

[math]\displaystyle{ H(z,\tau)=H(z-V_0\tau) }[/math]

and substitute into KdV equation then we obtain

[math]\displaystyle{ -2V_oH_\xi+3HH_\xi+\frac{1}{3}H_{\xi\xi\xi}=0 }[/math]

where [math]\displaystyle{ \xi=z-V_0\tau }[/math] is the travelling wave coordinate.


We rearrange and integrate this equation with respect to [math]\displaystyle{ \xi }[/math] to give

[math]\displaystyle{ \frac{1}{3} \int H_{\xi\xi\xi} d\xi = \int 2V_0 H_{\xi} d\xi - 3\int H H_\xi d\xi }[/math]
[math]\displaystyle{ \Longrightarrow \frac{1}{3}H_{\xi\xi} = 2V_oH - \frac{3}{2}H^2 +D_1 }[/math]

then multiply [math]\displaystyle{ H_\xi }[/math] to all terms and integrate again


[math]\displaystyle{ \Longrightarrow \frac{1}{3}H_{\xi\xi} H_\xi = 2V_0HH_\xi - \frac{3H^2}{2}H_\xi + D_1H_\xi }[/math]


[math]\displaystyle{ \Longrightarrow \frac{1}{3} \int H_{\xi\xi}H_\xi d\xi = 2V_o \int H H_\xi d\xi - \frac{3}{2} \int H^2H\xi d\xi + \int D_1 H_\xi d\xi }[/math]


[math]\displaystyle{ \Longrightarrow \frac{1}{3} \cdot \frac{1}{2}H_\xi^2 = 2V_0 \frac{H^2}{2}-\frac{H^3}{2} +D_1H + D_2 }[/math]


[math]\displaystyle{ \Longrightarrow \frac{1}{6}H_\xi^2=V_oH^2-\frac{1}{2}H^3+D_1H+D_2=f(H,V_0,D_1,D_2) }[/math]

where [math]\displaystyle{ D_1 }[/math] and [math]\displaystyle{ D_2 }[/math] are constants of integration.

Standardization of KdV equation

We define [math]\displaystyle{ f(H)= V_oH^2-\frac{1}{2}H^3+D_1H+D2 }[/math], so [math]\displaystyle{ f(H)=\frac{1}{6}H_\xi^2 }[/math]

It turns out that we require 3 real roots to obtain periodic solutions. Let roots be [math]\displaystyle{ H_1 \leq H_2 \leq H_3 }[/math].


We can imagine the graph of cubic function which has 3 real roots and we can now write a function

[math]\displaystyle{ f(H)= \frac{-1}{2}(H-H_1)(H-H_2)(H-H_3) }[/math]


From the equation [math]\displaystyle{ f(H)=\frac{1}{6}H_\xi^2 }[/math], we require [math]\displaystyle{ f(H)\gt 0 }[/math]

We are only interested in solution for [math]\displaystyle{ H_2 \lt H \lt H_3 }[/math] and we need [math]\displaystyle{ H_2 \lt H_3 }[/math].

and now solve equation in terms of the roots [math]\displaystyle{ H_i, }[/math]

We define [math]\displaystyle{ X=\frac{H}{H_3} }[/math], and obtain

[math]\displaystyle{ X_\xi^2=3H_3(X-X_1)(X-X_2)(1-X) }[/math]

where [math]\displaystyle{ X_i=\frac{H_i}{H} }[/math]

crest to be at [math]\displaystyle{ \xi=0 }[/math] and [math]\displaystyle{ X(0)=0 }[/math]

and a further variable Y via

[math]\displaystyle{ X = 1 +(X_2-1) \sin^2 (Y) }[/math]


[math]\displaystyle{ Y_\xi^2=\frac{3}{4}H_3(1-X_1)\left\{ 1-\frac{(1-X_2)}{(1-X_1)}\sin(Y)^2 \right\} ...(1) }[/math]

so [math]\displaystyle{ Y(0)=0. }[/math]

and

[math]\displaystyle{ \frac{dY}{d\xi}=\sqrt{l(1-k^2\sin^2(Y))}, }[/math]

which is separable.



In order to get this into a completely standard form we define

[math]\displaystyle{ k^2=\frac{1-X_2}{1-X_1}, l=\frac{3}{4}H_3(1-X_1) ...(2) }[/math]

Clearly, [math]\displaystyle{ 0 \leq k^2 \leq 1 }[/math] and [math]\displaystyle{ l\gt 0. }[/math]

Solution of the KdV equation

A simple quadrature of equation (1) subject to the condition (2) the gives us

[math]\displaystyle{ \int_{0}^\bar{Y} \frac{dS}{\sqrt{l(1-k^2\sin^2(Y))}} = \int_{0}^\bar{Y} dS }[/math]

Jacobi elliptic function [math]\displaystyle{ y= sn(x,k) }[/math] can be written in the form


[math]\displaystyle{ x= \int_{0}^y \frac{dt}{\sqrt{1-t^2}\sqrt{1-k^2t^2}} }[/math]

for [math]\displaystyle{ 0 \lt k^2 \lt 1 }[/math],

or equivalently

[math]\displaystyle{ x=\int_{0}^{\sin^{-1}y} \frac{dS}{\sqrt{1-k^2\sin^2(s)}} }[/math]

Now we can write Y with fixed values of [math]\displaystyle{ x }[/math],[math]\displaystyle{ k }[/math] as

[math]\displaystyle{ \bar{Y}=\sin^{-1}(\mathrm {sn} (\sqrt{l}\int_0^{\bar{Y}} d\xi,k)), }[/math]
[math]\displaystyle{ \sin(Y)=\mathrm {sn}(\sqrt{l}\xi;k), }[/math]

and hence

[math]\displaystyle{ X=1+(X_2-1)\mathrm {sn}^2(\sqrt{l}\xi;k) }[/math]

[math]\displaystyle{ \mathrm {cn}(x;k) }[/math] is another Jacobi elliptic function with [math]\displaystyle{ \mathrm {cn}^2+\mathrm {sn}^2=1 }[/math], and waves are called "cnoidal waves".

Using the result [math]\displaystyle{ cn^2+sn^2=1 }[/math], our final result can be expressed in the form

[math]\displaystyle{ H=H_2+(H_3-H_2)\mathrm {cn}^2\left\{ \left[ \frac{3}{4}(H_3-H_1) \right]^{\frac{1}{2}}\xi;k \right\} }[/math]