integrability - 2025 ACF Regionals

Question

Lax pairs can be used to solve for the equations of motion in systems with this property. For 10 points each:

[10m] Name this property of dynamical systems that have as many constants of motion as degrees of freedom, thus preventing chaotic orbits. Systems with this property, as their name implies, tend to be solvable analytically using quadratures.

ANSWER: integrability [or integrable; accept complete integrability]

[10h] Lax pairs were introduced to solve this standard integrable PDE. This equation is the continuum limit of the Fermi–Pasta–Ulam problem and can be solved analytically using a squared hyperbolic secant function.

ANSWER: KdV equation [or Korteweg–De Vries equation]

[10e] The KdV equation is solved by applying an “inverse transform” named for this process, which is also the basis for solving the Lax equation. Classical examples of this process include ones named for Rayleigh and Compton.

ANSWER: scattering [accept inverse scattering transform; accept Rayleigh scattering or Compton scattering]

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Summary


California	2025-02-01	Y	3	13.33	100%	33%	0%
Florida	2025-02-01	Y	3	13.33	100%	0%	33%
Lower Mid-Atlantic	2025-02-01	Y	6	10.00	83%	17%	0%
Midwest	2025-02-01	Y	6	15.00	100%	17%	33%
North	2025-02-01	Y	3	13.33	100%	33%	0%
Northeast	2025-02-01	Y	5	16.00	100%	40%	20%
Overflow	2025-02-01	Y	5	12.00	100%	0%	20%
South Central	2025-02-01	Y	2	20.00	100%	50%	50%
Southeast	2025-02-01	Y	4	12.50	100%	25%	0%
UK	2025-02-01	Y	10	14.00	100%	30%	10%
Upper Mid-Atlantic	2025-02-01	Y	8	11.25	88%	25%	0%
Upstate NY	2025-02-01	Y	3	6.67	67%	0%	0%

Data


Claremont A	Claremont B	0	0	10	10
UC Berkeley A	UC Berkeley B	10	0	10	20
UCLA	UCSD	0	0	10	10
Florida B	Valencia A	0	0	10	10
Florida A	Florida State A	0	10	10	20
UCF B	UCF C	0	0	10	10
Duke	Virginia Tech A	0	0	10	10
Liberty A	Wake Forest	0	0	10	10
UNC B	Liberty B	10	0	10	20
Virginia A	South Carolina	0	0	10	10
Liberty C	UNC C	0	0	0	0
William & Mary	UNC D	0	0	10	10
Illinois A	Chicago A	10	10	10	30
Chicago D	Illinois B	0	0	10	10
Illinois C	Chicago C	0	0	10	10
Indiana B	Indiana A	0	0	10	10
WashU B	Missouri	0	10	10	20
WashU A	Missoui S&T	0	0	10	10
Carleton	Minnesota B	0	0	10	10
Iowa State	Wisconsin A	10	0	10	20
Winona State	Wisconsin B	0	0	10	10
BU	UMass Boston	10	0	10	20
Harvard B	Brandeis B	0	0	10	10
Brown A	Dartmouth A	0	0	10	10
MIT A	Tufts A	10	0	10	20
Purdue	Georgetown B	0	10	10	20
Iowa	Georgetown A	0	0	10	10
Northwestern A	Notre Dame C	0	0	10	10
Notre Dame B	Northwestern B	0	0	10	10
Stanford B	Notre Dame A	0	0	10	10
ASU	TAMU	0	10	10	20
Texas A	Texas B	10	0	10	20
Louisville A	Alabama A	0	0	10	10
Georgia Tech C	Georgia Tech A	10	0	10	20
Vanderbilt A	Tennessee A	0	0	10	10
Georgia Tech D	Bruin A	0	0	10	10
Harvard A	Brandeis A	0	10	10	20
Cambridge A	Cambridge E	10	0	10	20
Cambridge B	Warwick B	10	0	10	20
Warwick A	Cambridge C	0	0	10	10
Cambridge D	Bristol	0	0	10	10
Oxford B	Durham	0	0	10	10
Imperial A	NYU C	0	0	10	10
Sheffield	Imperial B	0	0	10	10
Oxford A	Southampton A	10	10	10	30
Manchester	Southampton B	0	0	10	10
Columbia B	NYU A	0	0	10	10
Haverford A	Columbia C	0	0	10	10
Maryland A	John Jay College	10	0	10	20
Johns Hopkins A	Vassar A	0	0	10	10
Penn A	Maryland B	10	0	10	20
NYU B	Yale A	0	0	10	10
Penn B	George Washington B	0	0	10	10
George Washington A	Rutgers A	0	0	0	0
Cornell D	Binghamton A	0	0	10	10
ESF A	RIT A	0	0	0	0
Cornell C	RIT B	0	0	10	10
LSE	Oxford C	0	0	10	10