Introduction to KdV - Revision history

Levi: /* Formula for the solitary wave */

2025-09-30T05:55:39Z

Formula for the solitary wave

← Older revision		Revision as of 05:55, 30 September 2025
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	Therefore		Therefore
	<center><math>		<center><math>
	f=\frac{1}{2}c\,\mathrm{sech}^{2}\left( \frac{\sqrt{c}}{2}\left( \zeta+a\right) \right)		f(\zeta)=\frac{1}{2}c\,\mathrm{sech}^{2}\left( \frac{\sqrt{c}}{2}\left( \zeta+a\right) \right)
	</math></center>		</math></center>
	Of course we assumed that <math>x=x-ct</math> so the formula for the solitary wave is		Of course we assumed that <math>\zeta=x-ct</math> so the formula for the solitary wave is
	given by		given by
	<center><math>		<center><math>

2025-09-24T08:27:30Z

Travelling Wave Solution

← Older revision		Revision as of 08:27, 24 September 2025
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	</math></center>		</math></center>
	There is a		There is a
	homoclinic connection which connects the equilibrium point at the origin. This ~~holoclinic~~		homoclinic connection which connects the equilibrium point at the origin. This homoclinic
	connection represents the solitary wave. Within this homoclinic connection		connection represents the solitary wave. Within this homoclinic connection
	lie periodic orbits which represent the cnoidal waves.		lie periodic orbits which represent the cnoidal waves.

2025-09-21T10:17:50Z

Travelling Wave Solution

← Older revision		Revision as of 10:17, 21 September 2025
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	-cf+3f^{2}+\frac{\mathrm{d}^{2}f}{\mathrm{d}\zeta^{2}}=A_{1}		-cf+3f^{2}+\frac{\mathrm{d}^{2}f}{\mathrm{d}\zeta^{2}}=A_{1}
	</math></center>		</math></center>
	where <math>A</math> is a constant of integration.		where <math>A_1</math> is a constant of integration.

	If we think about this equation as Newton's second law in a potential well <math>		If we think about this equation as Newton's second law in a potential well <math>

2025-09-21T10:15:54Z

Travelling Wave Solution

← Older revision		Revision as of 10:15, 21 September 2025
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	[http://en.wikipedia.org/wiki/Gustav_de_Vries de Vries] in 1895.		[http://en.wikipedia.org/wiki/Gustav_de_Vries de Vries] in 1895.

	We begin with the assumption that the wave travels with ~~contant~~ form, i.e.		We begin with the assumption that the wave travels with constant form, i.e.
	is of the form		is of the form
	<center><math>		<center><math>

2025-08-14T05:32:41Z

Travelling Wave Solution

← Older revision		Revision as of 05:32, 14 August 2025
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	\end{matrix}</math></center>		\end{matrix}</math></center>
	If we chose <math>A_{1}=0</math> then we obtain two equilibria at <math>(f,v)=\left(		If we chose <math>A_{1}=0</math> then we obtain two equilibria at <math>(f,v)=\left(
	0,0\right) </math> and <math>(3/c,0).</math> If we analysis these equilibria we find the		0,0\right) </math> and <math>(c/3,0).</math> If we analysis these equilibria we find the
	first is a saddle and the second is a nonlinear center (it is neither repelling nor		first is a saddle and the second is a nonlinear center (it is neither repelling nor
	attracting).		attracting).

2023-08-10T03:57:26Z

Travelling Wave Solution

← Older revision		Revision as of 03:57, 10 August 2023
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	where <math>A</math> is a constant of integration.		where <math>A</math> is a constant of integration.

	If think about this equation as Newton's second law in a potential well <math>		If we think about this equation as Newton's second law in a potential well <math>
	V(f) </math> ~~for which~~ the equation is		V(f) </math> then the equation is
	<center><math>		<center><math>
	\frac{\mathrm{d}^{2}f}{\mathrm{d}\zeta^{2}}= -\frac{\mathrm{d}V}{\mathrm{d}f}		\frac{\mathrm{d}^{2}f}{\mathrm{d}\zeta^{2}}= -\frac{\mathrm{d}V}{\mathrm{d}f}

2022-08-12T02:31:03Z

Formula for the solitary wave

2020-08-31T02:46:59Z

Lecture Videos

← Older revision		Revision as of 02:46, 31 August 2020
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	{{#ev:youtube\|NictSlSgRbM}}		{{#ev:youtube\|NictSlSgRbM}}




	~~[[Category:Simple Nonlinear Waves]]~~
	~~[[Category:Nonlinear Water-Wave Theory]]~~
	~~[[Category:Shallow Depth]]~~

2020-08-24T03:55:14Z

Formula for the cnoidal wave

← Older revision		Revision as of 03:55, 24 August 2020
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	In the limit the two solutions agree. We also obtain a sinusoidal solution in the limit of		In the limit the two solutions agree. We also obtain a sinusoidal solution in the limit of
	small amplitude.		small amplitude.

			== Lecture Videos ==

			=== Part 1 ===

			{{#ev:youtube\|SzZ-KhvvPio}}

			=== Part 2 ===

			{{#ev:youtube\|QltlSQQBtrs}}

			=== Part 3 ===

			{{#ev:youtube\|NJ7h3Z9QtvU}}

			=== Part 4 ===

			{{#ev:youtube\|NictSlSgRbM}}




			[[Category:Simple Nonlinear Waves]]
			[[Category:Nonlinear Water-Wave Theory]]
			[[Category:Shallow Depth]]

2020-08-24T02:53:48Z

Travelling Wave Solution

← Older revision		Revision as of 02:53, 24 August 2020
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	We can also integrate the equation		We can also integrate the equation
	<center><math>		<center><math>
	-cf+3f^{2}+\frac{\mathrm{d}^{2}f}{\mathrm{d}x^{2}}=A_{1}		-cf+3f^{2}+\frac{\mathrm{d}^{2}f}{\mathrm{d}\zeta^{2}}=A_{1}
	</math></center>		</math></center>
	by multiplying by <math>f^{\prime }</math>and integrating. This gives us		by multiplying by <math>f^{\prime }</math>and integrating. This gives us
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	This is a separable equation and the only challenge is to integrate		This is a separable equation and the only challenge is to integrate
	<center><math>		<center><math>
	\int \frac{1}{\sqrt{-V(f)}} \mathrm{d}f.		\int \frac{1}{\sqrt{-2V(f)}} \mathrm{d}f.
	</math></center>		</math></center>