(* This the old readme file for condens2004 and earlier codes *) (* Needs to be updated! *) (* Read Condens--Documentation.nb for instructions *) (* New code: condens2007.m replaces condens2004.m *) (* ************************************************************************* *) (* *) (* Documentation file D_README.TXT *) (* *) (* ************************************************************************* *) (* *) (* Documentation for installing and using condens.m *) (* Updated: May 5, 1998 at 23:20 *) (* *) (*****************************************************************************) (* *) (* *** M A T H E M A T I C A P R O G R A M *** *) (* *) (* SYMBOLIC COMPUTATION of CONSERVED DENSITIES for SYSTEMS of *) (* NONLINEAR EVOLUTION EQUATIONS *) (* *) (* program name: condens.m *) (* *) (* purpose : computation of conserved densities and corresponding fluxes, *) (* with compatibility conditions if the system has parameters, *) (* and verification of the conservation law. *) (* *) (* input to condens.m : system of nonlinear evolution equations of any order,*) (* any degree, polynomial type, in variables x and t, *) (* only constant parameters, functions in x and t are *) (* NOT allowed as parameters *) (* *) (* u[i]_t = f(u[1],...,u[N],u[1]_{nx},...,u[N]_{nx}) *) (* with i=1,...,N; and n=1,2,... *) (* *) (* output : density and flux of desired rank (if it exists), *) (* and compatibility conditions for parameters, if applicable *) (* *) (* tested on: IBM RISC 6000, IBM Compatible PC 486, SGI Indigo2 XL *) (* *) (* language : Mathematica 3.0 and 2.2 (also versions 2.0 and 2.1) *) (* *) (* authors : U. Goktas and W. Hereman *) (* Department of Mathematical and Computer Sciences *) (* Colorado School of Mines *) (* Golden, CO 80401-1887, USA *) (* *) (* Written in 1995-1996 as part of Goktas Master's Thesis project *) (* *) (* Version 3.2: May 4, 1998 *) (* Previous Version 3.0: February 24, 1997 *) (* Original Version: February 29, 1996 *) (* *) (* Copyright 1998 *) (* *) (*****************************************************************************) Copyright by Unal Goktas and Willy Hereman (1998): No part of the conserved density program condens.m may be reproduced or sold without written consent of the authors. MATHEMATICA copyright and trademark of Wolfram Research, Inc., Urbana-Champaign, Illinois, USA. We are glad to offer you the possibility to carry out the tedious calculations of conserved densities for systems of evolution equations by computer. -------------------------------------- The main sources of information are: 1) The research paper JSC1997.TEX: Unal Goktas and Willy Hereman, ``Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations", Journal of Symbolic Computation, vol. 24, pp. 591-621 (1997). Describes the algorithm, the code, scope, limitations, usage, etc. Has many worked examples of systems with and without parameters. Has two major applications. Serves as manual for the program condens.m Reviews other software packages, and work in progress. 2) The Master's thesis: Unal Goktas, ``Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations". Master of Science Thesis, Colorado School of Mines, Golden, CO, U.S.A. (1996). Available in PostScript form as file condens.ps at FTP site given below. Has much more details. 3) Other related papers: * The research paper CPC1998.TEX: W. Hereman, U. Goktas, M. Colagrosso, and A. Miller ``Algorithmic integrability tests for nonlinear differential and lattice equations", Special Issue on `Computer Algebra in Physics Research,' Computer Physics Communications, vol. 101 (1998) submitted. * The research paper WESTER98.TEX: Unal Goktas and W. Hereman, ``Symbolic algorithms to investigate nonlinear differential and lattice equations", in: Comparative Computer Algebra, Ed.: M. Wester, Wiley, New York (1998) submitted. * The proceedings paper WP1998.TEX: Unal Goktas and W. Hereman, ``Invariants and symmetries for partial differential equations and lattices", Proc. Fourth International Conference on Mathematical and Numerical Aspects of Wave Propagation, Ed.: J. A. DeSanto, Colorado School of Mines, Golden, Colorado, June 1-5, 1998, SIAM, Philadelphia (1998) pp 403-407. ---------------------------------- Related Software A comprehensive package, called InvariantsSymmetries.m, for the computation of invariants (conservation laws) and higher-order symmetries has been added to MathSource, the program data base of Wolfram Research, Inc. The computation of conservation laws and higher-order symmetries for lattices is also possible via the code InvariantsSymmetries.m. The code be obtained free of cost from: U. Goktas and W. Hereman, InvariantsSymmetries.m: A Mathematica Integrability Package for the Computation of Invariants and Symmetries (1997). Available from MathSource (Item: 0208-932, Applications/Mathematics) via FTP: mathsource.wolfram.com or via Internet URL: http://www.mathsource.com/cgi-bin/MathSource/Applications/0208-932. The computation of conservation laws and higher-order symmetries for lattices is also possible via the code diffdens.m. See URL: Internet URL: http://www.mines.edu/fs_home/whereman/ or FTP to CARTAN.MINES.EDU, login with anonymous, use your email address or name as password. Then change to the directory pub/software/mathematica/diffdens. ---------------------------------- Theoretical Background A discussion of the computation of conserved densities and some examples may be found in the papers: File kdv95.tex (on the floppy in the subdirectory condens\document): W. Hereman and W. Zhuang, ``Symbolic software for soliton theory", Proceedings of Conference KdV '95, April 1995, Amsterdam, The Netherlands, Eds.: M. Hazewinkel, H. W. Capel and E. M. de Jager, Kluwer Academic Publishers, pp. 361-378 (1995). Also: Acta Applicandae Mathematicae, vol 39, pp. 361-378 (1995) File schamel.tex (on the floppy in the subdirectory condens\document): F. Verheest and W. Hereman, ``Conservation laws and solitary wave solutions for generalized Schamel equations", Physica Scripta, vol. 50, pp. 611-614 (1994) File mvdnls.tex (on the floppy in the subdirectory condens\document): R. Willox, W. Hereman and F. Verheest, ``Complete integrability of a modified vector derivative nonlinear Schrodinger equation", Physica Scripta, vol. 51, pp. 21-26 (1995) ------------------------ Where to get the Code Condens.m The software is also available via anonymous FTP from our two FTP sites: CARTAN.MINES.EDU and FTP.MINES.EDU (or MINES.EDU for short) (1) FTP to CARTAN.MINES.EDU, login with anonymous, use your email address or name as password. Then change to the directory pub/software/mathematica/condens. or (2) FTP to FTP.MINES.EDU (MINES.EDU for short), login with anonymous, use your email address or name as password. Then change to the directory pub/papers/math_cs_dept/software/condens. Alternatively, go to the World Wide Web home page of Willy Hereman Internet URL: http://www.mines.edu/fs_home/whereman/ There is a hot link to the FTP site. ---------------------------------- What you Should Have In the directory CONDENS on this floppy, you should have the following files: In the subdirectory PROGRAM you will find - condens.m : the Mathematica source code for the program [For development only: - char.m : the Mathematica source code to compute characteristics of conserved densities [the latter is not really used in the program condens.m, a description on how to compute characteristics with char.m is given below] In the subdirectory DOCUMENT you will find - d_readme.txt : the file you are reading now and the - JSC1997.TEX, CPC1998.TEX, WP1998, WESTER98.TEX, KDV95.TEX, SCHAMEL.TEX and MVDNLS.TEX: the papers listed above In the subdirectory TESTDECK you will find, for example: - d_kdv.m: the data file to run the Korteweg-de Vries equation, - d_kdvr6.o: the output for the Korteweg-de Vries equation (density of rank 6). and about 30 other d_xxx.m files corresponding to other examples, all for use with condens.m. The data files contain either one evolution equation or a system of evolution equations (usually two or three coupled equations, but there are more elaborate examples). Examples of data files with single evolution equations contain the data for the various fifth-order KdV equations, the Burgers' equation, the generalized KdV equation, the Schamel equation, and many more. For some of them the output files are also available. The data files d_hirsat.m, d_ito.m, d_dnls.m and d_mvdnls.m, etc., provide examples of systems of two evolution equations. Examples are the systems due to Hirota-Satsuma, Ito, Broer, etc. More complicated examples deal with the derivative nonlinear Schrodinger equation, the modified vector derivative nonlinear Schrodinger equation, and others. Again, for some of them the output files are available. Examples of data files with more than two evolution equations are also there, e.g. for a three-component KdV equation, a multi-component NLS equation. These data files are called d_3ckdv.m and d_2cnls.m The file d_kdv.m show how to test a conserved density (known from the literature). It is intended to demonstrate how to enter a given form of the density for the Korteweg-de-Vries equation. That data file also shows how to enter the rank of rho, the name of the output file, and if so desired, the weights for the dependent variables. Entering the extra information in the data file allows the user to use our code less interactively. The various data files (d_xxx.m) and the program file (condens.m) need to be in the same directory on your mainframe or PC, where Mathematica can find them. ----------------------------------------- Installing the Program How to install and use the program. Also, how to run examples: After the files are in place on your system (that should have Mathematica, version 2.0 or higher, we used versions 2.2 and 3.0), you should try to compute a set of conserved densities for a simple evolution equation (for example, the Korteweg-de Vries equation). The data file is available. For comparison the output file is also given. Our program condens.m is a MENU DRIVEN program, which makes its use very easy and transparent. Once Mathematica comes up, just read in the file condens.m as follows: In[1]:= < No trace of internal computations, otherwise, set it True. eq[1][x,t] = D[u[1][x,t],t]+u[1][x,t]*D[u[1][x,t],x]+D[u[1][x,t],{x,3}]; ---> eq[k][x,t] = ... Give the kth equation of the system in Mathematica notation. Note that there is no `== 0' at the end. Constant parameters in the equation are allowed. Note that if you request to compute a conservation law of certain rank which will only exist if parameters in the equation satisfy certain constraints, then the program will compute these constraints. Run the d_5kdv.m, d_7kdv.m, or the d_phrsat.m cases to see what happens! noeqs = 1; ---> Specify the number of equations in the system. name = "KdV Equation"; ---> Pick a short name for the system. The quotes are essential. (* bla bla bla ..... *) ---> Anything within (* and *) are comments, they are ignored by Mathematica. parameters = {}; ---> Give the list of the parameters in the system. If there are none, set parameters = {}. weightpars = {}; ---> Give the list of the parameters that are assumed to have weights. Note that weighted parameters are not listed in parameters, which is the list of parameters without weight. See d_bous.m and d_mvdnls.m for examples. (* rhorank = 6; *) ---> Optional, give the desired rank of the density. Useful if you want to work with the program less interactively (in batch mode). (* myfile = "d_kdvr6.o *) ---> Optional, give a name of the output file. Useful for less interactive use of the program. (* weightu[1]=2; weight[t]=3; *) ---> Optional, specify a choice for some or all of the weights. The program then skips the computation of the weights, but still checks for consistency. Particularly useful if there are several free weights and non-interactive use is preferred. formrho[x,t] = {}; (* or alternatively formrho[x,t] = 0 *) ---> Since the form of the density rho is not given, the program will automatically compute it. (* density of rank 6: *) (* formrho[x,t] = {c[1]*u[1][x,t]^3+c[2]*D[u[1][x,t],x]^2}; *) (* density of rank 8: *) (* formrho[x,t] = {c[1]*u[1][x,t]^4+c[2]*u[1][x,t]*D[u[1][x,t],x]^2+ c[3]*D[u[1][x,t],{x,2}]^2+c[4]*D[u[1][x,t],x]*D[u[1][x,t],{x,3}]}; *) ---> Alternatively, one could give a form of the density (rank 6 and 8). The density must be given in expanded form and with coefficients c[i]. The braces are essential. If rho is given, the program skips both the computation of the weights and the form of the density. Instead, the code uses what is given and computes the coefficients c[i]. This option allows one, for example, to test densities from the literature. ---------------------------------- Another Example of a Data File Here is a typical data file (called d_phrsat.m) for use with condens.m. It involves a parameter: aa. (* start of data file d_phrsat.m *) debug = False; (* Hirota-Satsuma System with one parameter *) eq[1][x,t]=D[u[1][x,t],t]-aa*D[u[1][x,t],{x,3}]-6*aa*u[1][x,t]*D[u[1][x,t],x]+ 6*u[2][x,t]*D[u[2][x,t],x]; eq[2][x,t]=D[u[2][x,t],t]+D[u[2][x,t],{x,3}]+3*u[1][x,t]*D[u[2][x,t],x]; noeqs = 2; name = "Hirota-Satsuma System (parameterized)"; parameters = {aa}; weightpars = {}; (**** user can supply the rhorank and/or the name for the output file ****) (* rhorank = 6; *) (* myfile = "phrsatr6.o"; *) (**** user can give the weights of u[1], u[2], and partial t, make ****) (**** sure they are correct! If not, you will see! ****) (* weightu[1] = 2; *) (* weightu[2] = 2; *) (* weight[t] = 3; *) formrho[x,t] = {}; (**** user can supply the form of rho ****) (* formrho[x,t]={c[1]*u[1][x,t]^3+c[2]*u[1][x,t]*u[2][x,t]^2+ c[3]*D[u[1][x,t],x]^2+c[4]*D[u[2][x,t],x]^2}; *) (* end of data file d_phrsat.m *) -------------------------------------- Running your own Examples If you want to run your own example you will have to prepare a data file and give its name when prompted in the program menu. The form of the density is computed based on the choice of the rank, rather than on the degree of rho. During the execution of the program, the user must enter the desired rank of rho. The rank of rho should be an integer multiple of the lowest weight of the dependent variables (and parameters). However, a fractional rank are allowed (if the lowest weights is fractional). Also note that the form of the densities rho is not unique. Densities can always be integrated by parts to obtain equivalent forms, modulo total derivatives. To compute equivalent forms, with Mathematica 2.2, type Integrate[rho[x,t],x]. The non-integrable part in the resulting expression is an equivalent density. After the computation is finished, the density rho[x,t] and the flux j[x,t] are both available. Type rho[x,t] and j[x,t] to see there explicit forms. The can also be seen in a nicer subscript notation via pdeform[rho[x,t]] and pdeform[j[x,t]]. The `system' for the unknown coefficients c[i] is also available. The flux j[x,t] is not automatically shown if it contains more than 20 terms. If the system for the c[i] is free of parameters and more than 50 equations it is also suppressed. If rho[x,t] depends on arbitrary coefficients c[i], it can be further split into independent pieces via the function split. Type either split[rho[x,t]] or split[pdeform[rho[x,t]]. The computation of the flux is computed based on integration by parts (with the standard Integrate). This is the only place in the program where we use the function Integrate. ----------------------------- Free Weights (i) If there is one free weight, the program will automatically pick a positive (most likely integer) weight. (ii) If for some reason that free weight could not be determined automatically, the program would still prompt you for a value. Once you know what the missing information is, it can be added to the data file (see e.g. d_fairl1.m), like this: weightu[4] = 1; (* determination of a free weight for d_fairl1.m *) Make sure that for the given equation or system all the weights can be determined automatically if you want to use the program in batch mode. (ii) If there two or more free weights, the program will ask for information. ---------------------------------------- Other Interesting Examples To see an example where the form of a density rho needs to be tested, look at the data files d_kdv.m and d_phrsat.m, where the form of rho is explicitly given. The braces around the form of rho are essential. To learn how the program determines the parameters (given in equations) for which certain conservation laws exist, work with the data file d_5kdv.m To learn about cases where parameters have scaling weights (dimensions), run the case d_mvdnls.m. To see how one can deal with situations where dependent variables have a free scale, run the case d_nodisp.m. ------------------------------------- Extra Information: Scope and Limitations Our program can handle PDEs that can be written as systems of evolution equations. The evolution equations must be polynomials in the dependent variables (no integral terms). Only two independent variables x and t are allowed. No terms in the evolution equations should explicitly depend on x and t. Theoretically, there is no limit on the number of evolution equations. In practice, for large systems, the computations may take a long time or require a lot of memory. The computational speed depends primarily on the amount of memory. The program only computes polynomial conserved densities in the dependent variables and their derivatives, without explicit dependencies on x and/or t. By design, the program constructs only densities that are uniform in rank. The uniform rank assumption for the monomials in the density allows one to compute independent conserved densities piece by piece, without having to split linear combinations of conserved densities. Due to the superposition principle, a linear combination of conserved densities of unequal rank is still a conserved density. This situation arises frequently when parameters with weight are introduced in the PDEs. The input systems may have one or more parameters, which are assumed to be nonzero. If a system has parameters, the program will attempt to compute the compatibility conditions for these parameters such that densities (of a given rank) might exist. The assumption that all parameters in the given evolution equation must be nonzero is necessary. As a result of setting parameters to zero in a given evolution equation, the weights and therefore the rank of the density might change. In general, the compatibility conditions for the parameters could be highly nonlinear, and there is no general algorithm to solve them. The program automatically generates the compatibility conditions, and solves them for parameters that occur linearly. Groebner basis techniques could be used to reduce complicated nonlinear systems into equivalent, yet simpler, non-linear systems. The assumption that the evolution equations are uniform in rank is not very restrictive. If the uniform rank condition is violated, the user can introduce one or more parameters with weights. This also allows some flexibility in the form of the densities. Although built up with terms that are uniform in rank, the densities do not have to be uniform in rank with respect to the dependent variables alone. Conversely, when the uniform rank condition is fulfilled, the introduction of extra parameters (with weights) in the given PDE leads to a linear combination of conservation laws, not to new ones. In cases where it is not clear whether or not parameters with weight should be introduced, one should start with parameters without weight. If this causes incompatibilities in the assignment of weights (due to non-uniformity), the program may provide a suggestion. Quite often, it recommends that one or more parameters be moved from the list of `parameters' into the list `weightpars' of weighted parameters. For systems with two or more free weights, the user will be prompted to enter values for the free weights. If only one weight is free, the program will automatically compute some choices for the free weight, and eventually continue with the lowest integer or fractional value. Negative weights are not allowed, except for weight[t]. Zero weights are allowed, but at least one of the dependent variables must have positive weight. The code checks these conditions, and if they are violated the computations are aborted. Note that fractional weights and densities of fractional rank are permitted. Our program is a tool in the search of the first half-dozen conservation laws. An existence proof (showing that there are indeed infinitely many conservation laws) must be done analytically, e.g. by explicitly constructing the recursion operator that connects conserved densities, or by computing high-order symmetries with Lie symmetry software. If our program succeeds finding a large set of independent conservation laws, there is a good chance that the system of PDEs has infinitely many conserved densities and that the recursion operator could be constructed explicitly. If the number of conservation laws is 3 or less, most likely the PDEs are not integrable, at least not in that coordinate representation. -------------------------- The file WORKLOG.M For PDEs with parameters and when the system for the coefficients c[i] is complicated, the program saves in a `worklog.m' file the following: - the name of the output file, - the given system of PDEs, - the rank of rho, - the form of rho, - the system for the coefficients c[i], - the coefficient matrix of that system, - the lists of unknown coefficients c[i] and parameters p[i] Independent from the program, the worklog files can later be analyzed with Mathematica functions. If there are parameters in the problem, then the worklog files are automatically saved if the number of equations, the number of unknowns and parameters adds up to 20 or more. ---------------------------------------- Example of WORKLOG.M file This example of a WORKLOG.M file, corresponds to the fifth-order KdV equation (d_5kdv.m), with three parameters, and the computation of density of rank 12. The file is called sampwork.m on the floppy. myfile = "d_5kdv12.o" eqlist = {Derivative[0, 1][u[1]][x, t] + aa*u[1][x, t]^2*Derivative[1, 0][u[1]][x, t] + bb*Derivative[1, 0][u[1]][x, t]*Derivative[2, 0][u[1]][x, t] + cc*u[1][x, t]*Derivative[3, 0][u[1]][x, t] + Derivative[5, 0][u[1]][x, t]} rhorank = 12 formrho[x, t] = c[1]*u[1][x, t]^6 + c[2]*u[1][x, t]^3*Derivative[1, 0][u[1]][x, t]^2 + c[3]*Derivative[1, 0][u[1]][x, t]^4 + c[4]*u[1][x, t]^2*Derivative[2, 0][u[1]][x, t]^2 + c[5]*Derivative[2, 0][u[1]][x, t]^3 + c[6]*u[1][x, t]*Derivative[3, 0][u[1]][x, t]^2 + c[7]*Derivative[4, 0][u[1]][x, t]^2 system = {15*bb*c[1] - 90*cc*c[1] - 2*aa*c[2] == 0, -360*c[1] - 3*bb*c[2] + 9*cc*c[2] - 12*aa*c[3] + 4*aa*c[4] == 0, -150*c[1] - bb*c[2] + 5*cc*c[2] + 5*aa*c[4] == 0, -3600*c[1] - 27*bb*c[2] + 105*cc*c[2] - 60*aa*c[3] + 80*aa*c[4] == 0, 375*c[2] - 36*bb*c[3] + 54*cc*c[3] + 28*bb*c[4] - 60*cc*c[4] + 66*aa*c[5] - 216*aa*c[6] == 0, 210*c[2] + 18*cc*c[3] + 23*bb*c[4] - 57*cc*c[4] + 24*aa*c[5] - 174*aa*c[6] == 0, 195*c[2] + 36*cc*c[3] + 16*bb*c[4] - 39*cc*c[4] + 48*aa*c[5] - 138*aa*c[6] == 0, 135*c[2] + 18*cc*c[3] + 13*bb*c[4] - 32*cc*c[4] + 24*aa*c[5] - 104*aa*c[6] == 0, 135*c[2] - 12*bb*c[3] + 6*cc*c[3] + 14*bb*c[4] - 32*cc*c[4] + 6*aa*c[5] - 96*aa*c[6] == 0, 120*c[2] - 12*bb*c[3] + 24*cc*c[3] + 7*bb*c[4] - 14*cc*c[4] + 30*aa*c[5] - 60*aa*c[6] == 0, 15*c[2] + 2*bb*c[4] - 5*cc*c[4] - 14*aa*c[6] == 0, -5*c[4] - bb*c[6] + 9*aa*c[7] == 0, -30*c[4] - 3*cc*c[5] - 7*bb*c[6] - cc*c[6] + 56*aa*c[7] == 0, 60*c[3] - 90*c[4] + 6*bb*c[5] + 3*cc*c[5] - 8*bb*c[6] + 252*aa*c[7] == 0, -130*c[4] - 27*cc*c[5] - 35*bb*c[6] - 9*cc*c[6] + 252*aa*c[7] == 0, 60*c[3] - 230*c[4] + 6*bb*c[5] + 3*cc*c[5] - 36*bb*c[6] + 504*aa*c[7] == 0, 180*c[3] - 160*c[4] + 18*bb*c[5] + 6*cc*c[5] - 3*bb*c[6] - cc*c[6] + 560*aa*c[7] == 0, 120*c[3] - 240*c[4] + 12*bb*c[5] - 21*cc*c[5] - 37*bb*c[6] - 9*cc*c[6] + 630*aa*c[7] == 0, 180*c[3] - 400*c[4] + 18*bb*c[5] - 18*cc*c[5] - 59*bb*c[6] - 9*cc*c[6] + 1008*aa*c[7] == 0, 720*c[3] - 790*c[4] + 72*bb*c[5] + 9*cc*c[5] - 47*bb*c[6] - 9*cc*c[6] + 2520*aa*c[7] == 0, 5*c[6] + 2*bb*c[7] + 5*cc*c[7] == 0, -15*c[5] + 10*c[6] + 18*bb*c[7] + 12*cc*c[7] == 0, -30*c[5] + 35*c[6] + 42*bb*c[7] + 39*cc*c[7] == 0, -15*c[5] + 55*c[6] + 36*bb*c[7] + 57*cc*c[7] == 0} coefmatrix = {{15*bb - 90*cc, -2*aa, 0, 0, 0, 0, 0}, {-360, -3*bb + 9*cc, -12*aa, 4*aa, 0, 0, 0}, {-150, -bb + 5*cc, 0, 5*aa, 0, 0, 0}, {-3600, -27*bb + 105*cc, -60*aa, 80*aa, 0, 0, 0}, {0, 375, -36*bb + 54*cc, 28*bb - 60*cc, 66*aa, -216*aa, 0}, {0, 210, 18*cc, 23*bb - 57*cc, 24*aa, -174*aa, 0}, {0, 195, 36*cc, 16*bb - 39*cc, 48*aa, -138*aa, 0}, {0, 135, 18*cc, 13*bb - 32*cc, 24*aa, -104*aa, 0}, {0, 135, -12*bb + 6*cc, 14*bb - 32*cc, 6*aa, -96*aa, 0}, {0, 120, -12*bb + 24*cc, 7*bb - 14*cc, 30*aa, -60*aa, 0}, {0, 15, 0, 2*bb - 5*cc, 0, -14*aa, 0}, {0, 0, 0, -5, 0, -bb, 9*aa}, {0, 0, 0, -30, -3*cc, -7*bb - cc, 56*aa}, {0, 0, 60, -90, 6*bb + 3*cc, -8*bb, 252*aa}, {0, 0, 0, -130, -27*cc, -35*bb - 9*cc, 252*aa}, {0, 0, 60, -230, 6*bb + 3*cc, -36*bb, 504*aa}, {0, 0, 180, -160, 18*bb + 6*cc, -3*bb - cc, 560*aa}, {0, 0, 120, -240, 12*bb - 21*cc, -37*bb - 9*cc, 630*aa}, {0, 0, 180, -400, 18*bb - 18*cc, -59*bb - 9*cc, 1008*aa}, {0, 0, 720, -790, 72*bb + 9*cc, -47*bb - 9*cc, 2520*aa}, {0, 0, 0, 0, 0, 5, 2*bb + 5*cc}, {0, 0, 0, 0, -15, 10, 18*bb + 12*cc}, {0, 0, 0, 0, -30, 35, 42*bb + 39*cc}, {0, 0, 0, 0, -15, 55, 36*bb + 57*cc}} unknownlist = {c[1], c[2], c[3], c[4], c[5], c[6], c[7]} parameters = {aa, bb, cc} ------------------------------------- Conclusions and Future Work A prototype of condens.m was used to compute conserved densities of a generalized KdV equation due to Schamel, and a modified vector derivative NLS equation. Based on the results obtained with the software, integrability questions for these equations could be answered. Details are the papers mentioned higher. We offer the scientific community a Mathematica package to carry out the tedious calculations of conserved densities for systems of nonlinear evolution equations. Extensions of the algorithm presented in this paper towards PDEs with more than one spatial variable, dynamical systems, are under development. Code for systems of difference-differential equations (lattice equations) has been designed. ------------------------------------- Contact Information To learn about new updates of the program, or in case of trouble, contact me. By phone: (303) 273-3881 (office, with voice mail) or 3860 (secretary), or (303) 440-6089 (home, with answering service); By fax: (303) 273-3875 (mention for Dr. Hereman) By email: whereman@mines.edu By mail: Dr. Willy Hereman Associate Professor Department of Mathematical and Computer Sciences Colorado School of Mines Golden, Colorado 80401-1887, USA Any comments about the program are welcomed by the authors. Good luck! Willy Hereman Golden, May 5, 1998. (* ************************* END of D_README.TXT ******************** *)