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Higher Order Symmetries of the KdV Equation Synopsis Recursion operators for the higher order symmetries (or generalized symmetries) of integrable equations such as the KdV equation are usually defined in terms of the formal inverse of the total derivative operator LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJ4RicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ==. In this worksheet we shall: construct the operator using the HorizontalHomotopy command in the JetCalculus package; construct the recursion operator 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 for the higher order symmetries of the KdV equation (Here Q is the characteristic function for the symmetry); use this recursion operator to generate the first 5 higher order symmetries of the KdV equation; and check that these higher order symmetries all commute.

Load in the required packages. with(DifferentialGeometry): with(JetCalculus): The DG software supports 2 different notational conventions for coordinates on jet spaces. It is convenient here to use to the Preferences command to change to JetNotation2. Preferences("JetNotation", "JetNotation2");

US1KZXROb3RhdGlvbjE2Ig==

The following calling sequence to DGsetup creates the 1st order jet space with 1 independent variable and 1 dependent variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Our jet notation is 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 DGsetup([x], [u], Jet, 1);

STBmcmFtZX5uYW1lOn5KZXRHNiI=

In our subsequent calculations we shall require higher order derivatives of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The order of the jet space will automatically be increased as needed. The command TotalDiff calculates the total derivative of a function on jet space, for example f := x*u[1]*u[2]^3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoKihJInhHRiYiIiImSSJ1R0YoNiNGMEYwJkYyNiMiIiMiIiQ3I0Yu

g := TotalDiff(f, x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJnR0YoLCgqJiZJInVHRig2IyIiIkYzJkYxNiMiIiMiIiRGMyomSSJ4R0YmRjNGNCIiJUYzKipGOUYzRjBGM0Y0RjYmRjE2I0Y3RjNGNzcjRi4=

k := TotalDiff(u[2], [2]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJrR0YoJkkidUdGKDYjIiIlNyNGLg==

To re-construct the function from the function 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, we use to make a type (1,0) differential bi-form and apply the horizontal homotopy operator from the variational bicomplex. HorizontalHomotopy(g*Dx);

KihJInhHJSpwcm90ZWN0ZWRHIiIiJkkidUc2IjYjRiVGJSZGJzYjIiIjIiIk

Here is a 2-line program to calculate the inverse of the total derivative. (Click on this box and then enter control-E (PC) or command-E(Mac) to execute this code). InverseTotalDiff := proc(g) local omega; omega := g &mult Tools:-DGvolume("biform"); HorizontalHomotopy(omega); end: JSFH InverseTotalDiff(g);

KihJInhHJSpwcm90ZWN0ZWRHIiIiJkkidUc2IjYjRiVGJSZGJzYjIiIjIiIk

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Define the KdV recursion operator using the InverseTotalDiff command from the previous section. (The sort command is used on the nex line to insure that the highest order term in the generalized symmetry appears last). KdVRecursion := Q -> sort(expand(TotalDiff(Q, [2]) + 2/3*u[0]*Q + 1/3*u[1]*InverseTotalDiff(Q)));

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Define the first order characteristic Q0 for the KdV equation (corresponding to the translational symmetry) and generate the higher symmetries using the recursion operator. Q0 := u[1];

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The 3rd order generalized symmetry is just the KdV flow itself: Q1 := KdVRecursion(Q0);

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The 5rd order generalized symmetry is: Q2 := KdVRecursion(Q1);

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The 7th order generalized symmetry is: Q3 := KdVRecursion(Q2);

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The 9th order generalized symmetry is: Q4 := sort(KdVRecursion(Q3));

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The command GeneralizedLieBracket can be used to show that the flows generated by the higher order symmetries all commute. Here are the evolutionary vector fields defined by the higher order symmetries calculated in the previous section. F := evalDG([Q0*D_u[0], Q1*D_u[0], Q2*D_u[0], Q3*D_u[0], Q4*D_u[0]]);

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The flows F[1] and F[2] commute: GeneralizedLieBracket(F[1], F[2]);

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We can use the Matrix command to calculate all the commutators: interface(typesetting=standard);

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Matrix(5, 5, (i,j) -> GeneralizedLieBracket(F[i], F[j]));

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Commands Illustrated JetCalculus, GeneralizedLieBracket, HorizontalHomotopy, TotalDiff Related Commands EulerLagrange, EvolutionaryVector, Prolong, Noether References P. J. Olver, Applications of Lie Groups to Differential Equations (2nd edition) pages 312-315. Release Notes The illustrated commands are available in Maple 11 and subsequent releases. Author Ian M. Anderson, Department of Mathematics and Statistics , Utah State University July 4, 2012 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc4Mzk2ODEzMzEwWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjOiImIiYtJSRfREdHNiM3JDclUSd2ZWN0b3JGJSUkSmV0RzciNyM3JDcjIiIiIiIhRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiZGJkYmRiU=