MORE ON RSOLVE--Discrete Painleve Equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg43806] MORE ON RSOLVE--Discrete Painleve Equations*From*: Peter Szabo <peterszabo20022003 at yahoo.co.uk>*Date*: Mon, 6 Oct 2003 02:07:57 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Dear Colleagues, I sent a message today about my failure to use RSolve to solve a set of discrete/difference equations: y[n+1]+(a_1 -b*n*n -2)*y[n] + y[n-1]==0. y[m+1]+(a_2 -b*m*m -2)*y[m] + y[m-1]==0. The problem I faced when using RSolve was either it said that the "Out" was the same as the "In" and hence no operation was performed, or that it said that the expression was not a (discrete) equation. This happened for both a single equation and for the system given above. However, the package worked for the ALL test examples in the Mathematica book. Thus, the possibility of a software error/erroneous loading is ruled out. To recapitulate, these are ordinary difference representations (lattice equations) for the 2-D time independent Schroedinger equation with harmonic potential. The condition is a_1 + a_2=a, which is the coefficient of the partial difference equation (combined case). Also, "n" and "m" are the iteration indices (independent variables or lattice variables) for the 2 dimensions respectively. I tried RSolve with a simple form of the first discrete Painleve equation: y[n+1]+y[n]+y[n-1]+((a*n +b)/(1+y[n])) +mu==0. Here, "a", "b" and "mu" are constants. As you very well know, this is a simple modification of the example given by Eq. (3.3.1) in B. Grammaticos, F. W. Nijhoff and A. Ramani, "Discrete Pailleve Equations", Lecture Notes for the Cargese School, (1996). The only modifications were that the third constant "gamma" is set to zero and the translation y[n]->y[n+1] is done in the denominator of the fourth term above, that is associated with the constants. This is done fo the case of siplification. Here too, RSolve gives NO ANSWER!. The same also occurs for the original form of the first discrete Painleve eqation given in the above citation. Could anyone PLEASE help me out in this? Most Respectfully Yours Peter Szabo ________________________________________________________________________ Want to chat instantly with your online friends? Get the FREE Yahoo! Messenger http://mail.messenger.yahoo.co.uk

**lattice Schroedinger equation**

**RSolve**

**lattice Schroedinger equation**

**RSolve**