Demonstration of the REDUCE Package ConLaw for Investigating Conservation Laws of Partial Differential Equations and First Integrals of Ordinary Differential Equations

Variables and their dependences

Note that REDUCE is not (by default) case sensitive, so it is normal for all REDUCE input to be in lower case even for mathematical variables that are conventionally written in upper case.

ConLaw does not use explicit functional notation but rather works with dependent variables that have been declared to depend implicitly on other independent variables using REDUCE depend statements.  Hence, the terms function and dependent variable are used synonymously.

However, this web interface uses functional notation to specify such dependences, from which it generates the appropriate depend statement.  Specify here all variable dependences as a comma-separated list of functional forms, e.g. to declare u and v to depend on t and x use u(t,x), v(t,x)

[As a convenience, the syntax "u, v(t,x)" is interpreted here (only) as "u(t,x), v(t,x)".  Alternatively, dependence information can be included in the ConLaw input fields below.] 

Specify which version of ConLaw to run: ConLaw4 ConLaw2 ConLaw3 ConLaw1

Input to ConLaw

All versions of ConLaw take two arguments: problem and runmode. Their values are lists, the elements of which are input as separate fields below.

1. Problem

The following three input fields correspond to the three elements of the first argument to the ConLaw operator, which specifies the DEs to be investigated.  Each should be a (comma-separated) sequence of one or more elements and the third may be empty.  (The example relates to the second example in the ConLaw manual, which is also available as "canned example" 4 below.)

1. required – DEs: (partial or ordinary) differential equation(s), where each must have the form
   df(u_i,..) = ...
   with a single derivative on the left hand side (LHS), selected such that:
   • the right hand side of an equation must not include the derivative on the left nor any derivative of it;
   • the LHS of any equation must not occur in any other equation nor any derivative of the LHS;
   • each of the unknown functions occurs on the LHS of exactly one equation.

   For example:
   df(u,t) = df(u,x,3) + 6*u*df(u,x) + 2*v*df(v,x),
   df(v,t) = 2*df(u,x)*v + 2*u*df(v,x)
   

2. required – DepVars: dependent variables or functions, e.g. u, v
   [You may optionally include the independent variables on which they depend, by using functional notation here instead of in the dependence field above, e.g. u(t,x), v(t,x)]
   
3. required (unless specified in the previous input) – IndVars: independent variables, e.g. t, x
   

2. Runmode

The investigation of conservation laws is done with the conserved current P and the characteristic functions Q being allowed to depend on a successively increasing order of derivatives of the dependent variables (say u). The first parameter (minord) is the highest order of u-derivatives in the first run and the second parameter (maxord) is the highest order of u-derivatives in the last run. Depending on which version of ConLaw is used, this is the order of the highest u-derivative in the first component of the conserved current P (for ConLaw1) or the order of the highest u-derivative in any characteristic function Q (for ConLaw2/3/4).

The following five input fields specify the calculation to be done and correspond to the five elements of the second argument to the ConLaw operator (e.g. {0, 1, t, {}, {}}).

1. required – minord: the initial (minimum) value of the highest order of derivatives of u, e.g. 0
   
2. required – maxord: the final (maximum) value of the highest order of derivatives of u, e.g. 1
   
3. required – expl: whether or not the ansatz for the characteristic functions Q_j or conserved current may depend explicitly on the independent variables, e.g. ticked (or checked or shaded) meaning "yes" (or "true" or "t").
   
4. optional – flist: a sequence of unknown functions in any ansatz for P_t, Q_j, also all parameters and parametric functions in the equation that are to be calculated such that conservation laws exist:
   [You may optionally include the independent variables on which they depend, by using functional notation here instead of in the dependence field above, e.g. u(t,x), v(t,x).]
   
5. optional – inequ: a sequence of expressions none of which may be identically zero for the conservation law to be found. These are extra inequality conditions, i.e. non-vanishing conditions which must be satisfied by the unknown functions in input field 4 above.
   

Specifying an ansatz

An ansatz for a conservation law can be formulated by specifying one or more of the functions Pⁱ (input as p_t,p_x,...) for ConLaw1, one or more of the functions Q^j (input as q_1,q_2,...) for ConLaw2/4, one or more of p_t, q_j for ConLaw3. The "t" in p_t stands for a variable name, the "j" in q_j stands for an index – the number of the equation in the input system of equations with which q_j is multiplied to give the RHS of (1). There is a restriction on the structure of all the expressions for p_t, q_j that are specified, namely that they must be homogeneous and linear in some unknown function or constant. For an example see canned example 2.

All such functions and constants must be listed in flist (see above). The dependences of such functions must be defined before calling ConLawi. This is done by including them in the variable dependence field above (or by including the variable dependence in the flist field), e.g. f(t,x,u) to specify f as a function of t, x, u. Beware: If one wants to have f as a function of derivatives of u(t,x), say f depending on d³u/dtdx², then one cannot use

f(df(u,t,x,2))

but instead must use

f(u!`1!`2!`2)

if IndVars has been specified as t,x (in that order). As t is the first variable and x is the second variable in IndVars and u is differentiated once wrt. t and twice wrt. x we therefore get u!`1!`2!`2. (The character ! is the escape character to allow special characters like ` to occur in identifiers in REDUCE.) [The reason for this extra rule is that functions, like this f which are to be computed in the program CRACK must be functions of independent variables, like u!`1!`2!`2 and not of composite expressions, like df(u,t,x,2).]

Optional – Ansatz: a normal REDUCE assignment statement with a form like this: q_1 := r*df(u,x,2)
(Omit if you are copying the example input as above.) Do not use functional notation here for any unknown independent variable but use only its name.

NOTES:

1. It is not the case that for any conserved current Pⁱ satisfying (2) there must exist characteristic function(s) Q^j that satisfy (3). One can therefore not specify a known density P^t for ConLaw3 and hope to calculate the remaining Pⁱ and the corresponding Q^j with ConLaw3. What one can do is to use ConLaw1 to calculate the other components Pⁱ. But this restriction for ConLaw3 does not imply that it misses conservation laws. If (1) is a totally non-degenerate system then for each current Pⁱ satisfying (2) there exists a current Pⁱ' differing from Pⁱ only by a curl (a trivial conservation law) such that Pⁱ' satisfies (3) for some suitable Q^j.
2. The Q^j are uniquely determined only modulo the original equations (1). If one makes an ansatz for Q^j then this freedom should be removed by having the Q^j independent of the LHS's of the equations (1) and their derivatives. If the Q^j were allowed to depend on anything, (3) could be solved for one Q^j in terms of arbitrary Pⁱ and other Q^j, giving characteristic functions that are singular for solutions of the original DEs (1).

Control and run ConLaw

ConLaw provides the following control variable (flag):

• print_ – the maximum number of terms of equations that are output. The default (for this demo) is nil, which suppresses all intermediate output.  (The default value of print_ when calling ConLaw in an interactive REDUCE session is 8.)
  

Use subscripts to indicate derivatives (on dfprint) – the CRACK default.

Show all REDUCE input and output without filtering.

Output T_EX format (primarily for use with the "http://www.software.ibm.com/network/techexplorer/">IBM techexplorer or equivalent plug-in).

Please email any comments or problems to the maintainer of this demonstration.

Output from ConLaw

Each of the procedures ConLawi returns a list of conservation laws {C₁,C₂,...}; if no non-trivial conservation law is found it returns the empty list {}. Each C_i representing a conservation law has the form { {P^x, P^y, ...}, {Q¹, Q², ...} }.

For example, the first conservation law C₁ in the list returned as the solution to the earlier input example is as follows:

                                 2    2
{{{ - 4*u*v,2*( - df(u,x,2) - 3*u  - v )},

          2      3        2
  {df(u,x)  - 2*u  - 2*u*v ,

                                   2                2                2      4
    - 2*df(u,t)*df(u,x) + df(u,x,2)  + 6*df(u,x,2)*u  + 2*df(u,x,2)*v  + 9*u

          2  2    4
    + 10*u *v  + v }},

...

Canned Examples

The input fields below can be copied into the fields of the input form above and the form submitted for execution by ConLaw, or the examples can be run directly by simply clicking the appropriate "Run" button.  (Either way, the above control fields apply.) These examples are taken from the ConLaw test file (conlaw.tst).

To see more intermediate details of the following calculations set the value of print_ to 10 (see above).

1. Conservation laws of the Korteweg - de Vries equation

This example calculates all conservation laws of the KdV equation with a characteristic function of order not higher than two.

ConLaw:3
DEs: df(u,t) = -u*df(u,x) - df(u,x,3)
DepVars: u(t,x)
minord: 0
maxord: 2
expl: true

2. Conservation laws of the wave equation

This example demonstrates that one can specify an ansatz for the characteristic function of one or more equations of the PDE system. In this example all conservation laws of the wave equation, which is written as a first order system, are calculated such that the characteristic functions of the first of both equations is proportional to d²u/dx². (This will include zero as it is a multiple of d²u/dx² too.)

DEPEND:r(t,x,u,v,u!`2,v!`2)
ConLaw:2
DEs: df(u,t) = df(v,x), df(v,t) = df(u,x)
DepVars: u(t,x), v(t,x)
minord: 2
maxord: 2
expl: true
flist: r
Ansatz: q_1 := r*df(u,x,2)

3. Conservation laws of Burgers equation

For Burgers equation the following example finds all conservation laws of zero'th order in the characteristic function up to the solution of the linear heat equation. This is an example for what happens when not all conditions could be solved, but it is also an example which shows that not only characteristic functions of polynomial or rational form can be found.

ConLaw:1
DEs: df(u,t) = df(u,x,2) + df(u,x)^2/2
DepVars: u(t,x)
minord: 0
maxord: 0
expl: true

4. Conservation laws of the Ito system

In this example all conservation laws of the Ito system are calculated that have a characteristic function of order not higher than one. This is a further example of non-polynomial conservation laws.

ConLaw:4
DEs: df(u,t) = df(u,x,3) + 6*u*df(u,x) + 2*v*df(v,x),
df(v,t) = 2*df(u,x)*v + 2*u*df(v,x)
DepVars: u(t,x), v(t,x)
minord: 0
maxord: 1
expl: true

5. Conservation laws of the 5th order Korteweg - de Vries equation

In this example the 5th order Korteweg - de Vries equation is investigated concerning conservation laws of order 0 and 1 in the conserved density P_t. Parameters a,b,c in the PDE are determined such that conservation laws exist. This complicates the problem by making it non-linear with a number of cases to be considered. Some of the subcases below can be combined to reduce their number which currently is not done automatically.

ConLaw:1
DEs: df(u,t) = -df(u,x,5) - a*u^2*df(u,x) - b*df(u,x)*df(u,x,2) - c*u*df(u,x,3)
DepVars: u(t,x)
minord: 0
maxord: 1
expl: true
flist: a,b,c

BEWARE: This example currently takes 5 to 10 minutes to run!

6. First integrals of the Lorentz ODE system

ConLawi can also be used to determine first integrals of ODEs. The generality of the ansatz is not just specified by the order. For example, the Lorentz system below is a first order system therefore any first integrals are zero order expressions. The ansatz to be investigated below looks for first integrals of the form a₁(x, t) + a₂(y, t) + a₃(z, t) = constant and determines parameters s, b, r such that first integrals exist.

DEPEND:a1(x,t), a2(y,t), a3(z,t)
ConLaw:1
DEs: df(x,t) = - s*x + s*y,
df(y,t) = x*z + r*x - y,
df(z,t) = x*y - b*z
DepVars: x(t), y(t), z(t)
minord: 0
maxord: 0
expl: true
flist: a1,a2,a3,s,r,b
Ansatz: p_t := a1 + a2 + a3