Example Calculations for the KdV and IST - Revision history

Levi: /* Example 2: Hat Function Potential */

2025-09-30T06:01:51Z

Example 2: Hat Function Potential

← Older revision		Revision as of 06:01, 30 September 2025
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	We cannot work with a hat function numerically, because the jump in <math>u</math> leads		We cannot work with a hat function numerically, because the jump in <math>u</math> leads
	to high frequencies which dominate the response.. We can smooth our function		to high frequencies which dominate the response. We can smooth our function
	by a number of methods. We use here the function <math>\tanh\left( x\right) </math> so		by a number of methods. We use here the function <math>\tanh\left( x\right) </math> so
	we write		we write
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	\|-		\|-
	! Animation		! Animation
	! Three-dimensional plot.		! Three-dimensional plot
	\|-		\|-
	\| [[Image:Wide_function.gif\|thumb\|right\|500px\|Evolution of <math>u(x,t)</math>.]]		\| [[Image:Wide_function.gif\|thumb\|right\|500px\|Evolution of <math>u(x,t)</math>.]]
	\| [[Image:Widefunction.jpg\|thumb\|right\|500px\|Evolution of <math>u(x,t)</math> ]]		\| [[Image:Widefunction.jpg\|thumb\|right\|500px\|Evolution of <math>u(x,t)</math>. ]]

	\|}		\|}

2020-10-01T03:38:19Z

← Older revision		Revision as of 03:38, 1 October 2020
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	sufficiently rapidly but not too rapidly.		sufficiently rapidly but not too rapidly.

	~~== Lecture Videos ==~~

	~~=== Part 1 ===~~

	~~{{#ev:youtube\|K0zeheguRKo}}~~


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	\|}		\|}

			== Lecture Videos ==

			=== Part 1 ===

			{{#ev:youtube\|K0zeheguRKo}}

2020-10-01T03:32:02Z

Example 2: Hat Function Potential

← Older revision		Revision as of 03:32, 1 October 2020
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	where <math>\nu</math> is an appropriate constant to make the function increase in value		where <math>\nu</math> is an appropriate constant to make the function increase in value
	sufficiently rapidly but not too rapidly.		sufficiently rapidly but not too rapidly.

			== Lecture Videos ==

			=== Part 1 ===

			{{#ev:youtube\|K0zeheguRKo}}


	{\| class="wikitable"		{\| class="wikitable"

2015-01-28T04:09:51Z

Example 2: Hat Function Potential

← Older revision		Revision as of 04:09, 28 January 2015
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	<center><math>		<center><math>
	u\left( x\right) =\frac{20}{2}\left( \tanh\left( \nu\left( x+1\right)		u\left( x\right) =\frac{20}{2}\left( \tanh\left( \nu\left( x+1\right)
	\right) -\tanh\left( \nu\left( x+1\right) \right) \right)		\right) -\tanh\left( \nu\left( x-1\right) \right) \right)
	</math></center>		</math></center>
	where <math>\nu</math> is an appropriate constant to make the function increase in value		where <math>\nu</math> is an appropriate constant to make the function increase in value

2010-09-24T04:23:40Z

← Older revision		Revision as of 04:23, 24 September 2010
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	We have already calculated the scattering data for the delta function		We have already calculated the scattering data for the delta function
	potential. The scattering data is		potential in [[Properties of the Linear Schrodinger Equation]]. The scattering data is
	<center><math>		<center><math>
	S\left( \lambda,0\right) =\left( k_{1},\sqrt{k_{1}},\frac{u_{0}}{2ik-u_{0}		S\left( \lambda,0\right) =\left( k_{1},\sqrt{k_{1}},\frac{u_{0}}{2ik-u_{0}
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	\right.		\right.
	</math></center>		</math></center>
			We have already solved this case in [[Properties of the Linear Schrodinger Equation]].
	For the even solutions we need to solve		For the even solutions we need to solve
	<center><math>		<center><math>

2010-09-24T03:17:26Z

← Older revision		Revision as of 03:17, 24 September 2010
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	}}		}}

	==Example1 ~~Delta~~ function potential.==		We consider here the two examples we treated in [[Properties of the Linear Schrodinger Equation]].

			==Example1: <math>\delta</math> function potential==

	We have already calculated the scattering data for the delta function		We have already calculated the scattering data for the delta function

2010-09-24T03:15:00Z

Example 2 Hat Function Potential

← Older revision		Revision as of 03:15, 24 September 2010
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	</math></center>		</math></center>

	==Example 2 Hat Function Potential==		==Example 2: Hat Function Potential==

	We solve for the case when		We solve for the case when

2010-09-24T03:14:51Z

Example 2 A Hat Function

← Older revision		Revision as of 03:14, 24 September 2010
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	</math></center>		</math></center>

	==Example 2 A Hat Function==		==Example 2 Hat Function Potential==

	We solve for the case when		We solve for the case when

2010-09-24T01:47:40Z

← Older revision		Revision as of 01:47, 24 September 2010
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	{{nonlinear waves course		{{nonlinear waves course
	\| chapter title = Example Calculations for the KdV and IST		\| chapter title = Example Calculations for the KdV and IST
	\| next chapter = [[~~Connection betwen KdV and the Schrodinger Equation~~]]		\| next chapter = [[Reaction-Diffusion Systems]]
	\| previous chapter = [[~~Reaction-Diffusion Systems~~]]		\| previous chapter = [[Connection betwen KdV and the Schrodinger Equation]]
	}}		}}

2010-09-24T00:18:25Z

Example 2 A Hat Function

← Older revision		Revision as of 00:18, 24 September 2010
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	u\left( x\right) =\left\{		u\left( x\right) =\left\{
	\begin{matrix}		\begin{matrix}

	0, & x\notin\left[ -1,1\right] \\		0, & x\notin\left[ -1,1\right] \\
	20, & x\in\left[ -1,1\right]		20, & x\in\left[ -1,1\right]
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	\right.		\right.
	</math></center>		</math></center>
	For ~~this case~~ we need to solve		For the even solutions we need to solve
	<center><math>		<center><math>
	\tan\kappa=\frac{k}{\kappa}		\tan\kappa=\frac{k}{\kappa}
	</math></center>		</math></center>
	~~which has solution~~ and		where <math>\kappa=\sqrt{b-k^{2}}</math>.

			For the odd solutions we need to solve and
	<center><math>		<center><math>
	\tan\kappa=-\frac{\kappa}{k}		\tan\kappa=-\frac{\kappa}{k}
	</math></center>		</math></center>
	~~which has solution.~~

	Recall that the solitons have amplitude <math>2k_{n}^{2}</math> or <math>-2\lambda_{n}</math>. This		Recall that the solitons have amplitude <math>2k_{n}^{2}</math> or <math>-2\lambda_{n}</math>. This