Conservation Laws for the KdV - Revision history

Levi: /* Proof of an Infinite Number of Conservation Laws */ fix

2025-10-04T09:26:05Z

Proof of an Infinite Number of Conservation Laws: fix

← Older revision		Revision as of 09:26, 4 October 2025
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	Note that each of the odd conservation laws (<math>w_1, w_3</math> etc.) are just an <math>\partial _{x}</math> of some <math>X(u)</math> and therefore does not actually provide a conservation law.		Note that each of the odd conservation laws (<math>w_1, w_3</math> etc.) are just an <math>\partial _{x}</math> of some <math>X(u)</math> and therefore does not actually provide a conservation law.

	As <math>\int_{-\infty }^{\infty }\partial_{x} X(u) \mathrm{d} x = 0</math>, <math>w_0</math>, <math>w_2</math>, and <math>w_4</math> correspond to conservation of <math>u</math> (mass), <math>u^2</math> (momentum), and <math>2 u^3 - (\partial_{x} u)^2</math> (energy).		As <math>\int_{-\infty }^{\infty }\partial_{x} X(u) \mathrm{d} x = 0</math>, <math>w_0</math>, <math>w_2</math>, and <math>w_4</math> correspond to conservation of <math>\int_{-\infty }^{\infty }u\mathrm{d} x</math> (mass), <math>\int_{-\infty }^{\infty }u^2\mathrm{d} x</math> (momentum), and <math>\int_{-\infty }^{\infty }(2 u^3 - (\partial_{x} u)^2)\mathrm{d} x</math> (energy).

	== Lecture Videos ==		== Lecture Videos ==

Levi at 06:19, 28 September 2025

2025-09-28T06:19:49Z

← Older revision		Revision as of 06:19, 28 September 2025
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	}}		}}

	One of the most interesting ~~freatures~~ of the KdV is the existence of		One of the most interesting features of the KdV is the existence of
	infinitely many conservation laws. Let's begin with some basics of		infinitely many conservation laws. Let's begin with some basics of
	conservation laws. If we can write our equation of the form		conservation laws. If we can write our equation of the form

Levi at 06:38, 27 September 2025

2025-09-27T06:38:52Z

← Older revision		Revision as of 06:38, 27 September 2025
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	One of the most interesting freatures of the KdV is the existence of		One of the most interesting freatures of the KdV is the existence of
	infinitely many conservation laws. ~~Lets~~ begin with some basics of		infinitely many conservation laws. Let's begin with some basics of
	conservation laws. If we can write our equation of the form		conservation laws. If we can write our equation of the form
	<center><math>		<center><math>

Levi: /* Proof of an Infinite Number of Conservation Laws */ some changes

2025-09-24T19:17:33Z

Proof of an Infinite Number of Conservation Laws: some changes

← Older revision		Revision as of 19:17, 24 September 2025
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	^{n}w_{n}\left( x,t\right)		^{n}w_{n}\left( x,t\right)
	</math></center>		</math></center>
	Since the equation ~~is in~~ conservation form then		Since we can rewrite the equation into the conservation form <math>\partial _{t}w+\partial_{x}\left(3w^2-2\varepsilon ^{2}w^{3}+\partial _{x}^{2}w\right)=0</math>, then
	<center><math>		<center><math>
	\int_{-\infty }^{\infty }w\left( x,t,\varepsilon \right) \mathrm{d} x = \mathrm{constant}		\int_{-\infty }^{\infty }w\left( x,t,\varepsilon \right) \mathrm{d} x = \mathrm{constant}
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	</math></center>		</math></center>
	<center><math>		<center><math>
	w_{3} = \partial _{x}^3 u + 4 u \partial_{x} u		w_{3} = \partial _{x}^3 u + 4 u \partial_{x} u = \partial _{x}(\partial _{x}^2 u + 2 u^2)
	</math></center>		</math></center>
	<center><math>		<center><math>
	w_{4} = \partial _{x}^4 u + 6 u \partial_{x}^2 u + 5 (\partial_{x} u)^2 + 2 u^3 = \partial _{x}^4 u + 6 ~~\partial_{x} (~~u \partial_{x} u) + 2 u^3 - (\partial_{x} u)^2		w_{4} = \partial _{x}^4 u + 6 u \partial_{x}^2 u + 5 (\partial_{x} u)^2 + 2 u^3 = \partial _{x}(\partial _{x}^3 u + 6 u \partial_{x} u) + 2 u^3 - (\partial_{x} u)^2
	</math></center>		</math></center>
	Note that each of the odd conservation laws (<math>w_1, w_3</math> etc.) are just ~~the derivative~~ (~~with some modification~~) ~~of the previous law~~ and		Note that each of the odd conservation laws (<math>w_1, w_3</math> etc.) are just an <math>\partial _{x}</math> of some <math>X(u)</math> and therefore does not actually provide a conservation law.
	therefore does not actually provide a ~~new~~ conservation law.

	As <math>\int_{-\infty }^{\infty }\partial_{x} X(u) \mathrm{d} x = 0</math>, <math>w_0</math>, <math>w_2</math>, and <math>w_4</math> correspond to conservation of <math>u</math> (mass), <math>u^2</math> (momentum), and <math>2 u^3 - (\partial_{x} u)^2</math> (energy).		As <math>\int_{-\infty }^{\infty }\partial_{x} X(u) \mathrm{d} x = 0</math>, <math>w_0</math>, <math>w_2</math>, and <math>w_4</math> correspond to conservation of <math>u</math> (mass), <math>u^2</math> (momentum), and <math>2 u^3 - (\partial_{x} u)^2</math> (energy).

Levi: /* Proof of an Infinite Number of Conservation Laws */ add w_4 and mass-momentum-energy

2025-09-24T18:04:10Z

Proof of an Infinite Number of Conservation Laws: add w_4 and mass-momentum-energy

← Older revision		Revision as of 18:04, 24 September 2025
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	<center><math>		<center><math>
	0=w_{3}-\partial _{x}w_{2}-2w_{0}w_{1}		0=w_{3}-\partial _{x}w_{2}-2w_{0}w_{1}
			</math></center>
			<center><math>
			0=w_{4}-\partial _{x}w_{3}-2w_{0}w_{2}-w_{1}^2
	</math></center>		</math></center>

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	<center><math>		<center><math>
	w_{3} = \partial _{x}^3 u + 4 u \partial_{x} u		w_{3} = \partial _{x}^3 u + 4 u \partial_{x} u
			</math></center>
			<center><math>
			w_{4} = \partial _{x}^4 u + 6 u \partial_{x}^2 u + 5 (\partial_{x} u)^2 + 2 u^3 = \partial _{x}^4 u + 6 \partial_{x} (u \partial_{x} u) + 2 u^3 - (\partial_{x} u)^2
	</math></center>		</math></center>
	Note that each of the odd conservation laws (<math>w_1, w_3</math> etc.) are just the derivative (with some modification) of the previous law and		Note that each of the odd conservation laws (<math>w_1, w_3</math> etc.) are just the derivative (with some modification) of the previous law and
	therefore does not actually provide a new conservation law.		therefore does not actually provide a new conservation law.

			As <math>\int_{-\infty }^{\infty }\partial_{x} X(u) \mathrm{d} x = 0</math>, <math>w_0</math>, <math>w_2</math>, and <math>w_4</math> correspond to conservation of <math>u</math> (mass), <math>u^2</math> (momentum), and <math>2 u^3 - (\partial_{x} u)^2</math> (energy).

	== Lecture Videos ==		== Lecture Videos ==

Levi: some fixes

2025-09-24T17:51:22Z

some fixes

← Older revision		Revision as of 17:51, 24 September 2025
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	\int_{-\infty }^{\infty }u\mathrm{d} x		\int_{-\infty }^{\infty }u\mathrm{d} x
	</math></center>		</math></center>
	is conserved. This corresponds to conservation of ~~momentum~~. We can also		is conserved. This corresponds to conservation of mass. We can also
	write the KdV equation as		write the KdV equation as
	<center><math>		<center><math>
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	\int_{-\infty }^{\infty }u^{2}\mathrm{d} x		\int_{-\infty }^{\infty }u^{2}\mathrm{d} x
	</math></center>		</math></center>
	must be conserved. This corresponds to conservation of ~~energy~~. It turns out		must be conserved. This corresponds to conservation of momentum. It turns out
	that there is an infinite number of conserved quantities and we give here		that there is an infinite number of conserved quantities and we give here
	the proof of this.		the proof of this.
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	</math></center>		</math></center>

	We can solve ~~recurrsively~~ to obtain		We can solve recursively to obtain
	<center><math>		<center><math>
	w_{0}=u		w_{0}=u

Levi at 17:44, 24 September 2025

2025-09-24T17:44:57Z

← Older revision		Revision as of 17:44, 24 September 2025
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	}^{\infty }\partial _{x}X\left( u\right) \mathrm{d} x		}^{\infty }\partial _{x}X\left( u\right) \mathrm{d} x
	</math></center>		</math></center>
	The second integral will be zero if <math>u\rightarrow 0</math> as <math>x\rightarrow \pm		The second integral will be zero if <math>X(u)\rightarrow 0</math> as <math>x\rightarrow \pm
	\infty .</math> Therefore		\infty .</math> Therefore
	<center><math>		<center><math>

Levi at 16:59, 24 September 2025

2025-09-24T16:59:25Z

← Older revision		Revision as of 16:59, 24 September 2025
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	}}		}}

	One of the most interesting freatures of the KdV is the ~~exisitence~~ of		One of the most interesting freatures of the KdV is the existence of
	infinitely many conservation laws. Lets begin with some basics of		infinitely many conservation laws. Lets begin with some basics of
	conservation laws. If we can write our equation of the form		conservation laws. If we can write our equation of the form

Meylan: /* Part 2 */

2020-09-07T02:59:17Z

Part 2

← Older revision		Revision as of 02:59, 7 September 2020
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	=== Part 2 ===		=== Part 2 ===

	{{#ev:youtube\|~~mGxDHfYIs6I~~}}		{{#ev:youtube\|QQjh14uguvg}}

Meylan: /* Proof of an Infinite Number of Conservation Laws */

2020-09-07T02:54:49Z

Proof of an Infinite Number of Conservation Laws

← Older revision		Revision as of 02:54, 7 September 2020
Line 138:		Line 138:
	Note that each of the odd conservation laws (<math>w_1, w_3</math> etc.) are just the derivative (with some modification) of the previous law and		Note that each of the odd conservation laws (<math>w_1, w_3</math> etc.) are just the derivative (with some modification) of the previous law and
	therefore does not actually provide a new conservation law.		therefore does not actually provide a new conservation law.

			== Lecture Videos ==

			=== Part 1 ===

			{{#ev:youtube\|9w3-IWX2b5g}}

			=== Part 2 ===

			{{#ev:youtube\|mGxDHfYIs6I}}