Graphic and Symbolic Experiments with the Harry Dym Equation

Graphic and Symbolic Experiments with the Harry Dym Equation
By Shalosh B. Ekhad and Doron Zeilberger
.pdf .tex
Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger

Written: Aug. 2, 2024

ל ז כ ר נ ח ו ם צ ב י ב ן י צ ח ק ה ל ו י ז" ל

In fond memory of our academic father (for DZ) and grandfather (for SBE), Harry Dym (Jan 26, 1938 - July 18, 2024).

We use both symbolic and numeric computations to experiment with the Harry Dym equation, of central importance in soliton theory.

Maple package

HDequation.txt, a Maple package to investigate, graphically and symbolically, the Harry Dym Equation of Soliton Theory fame

Harry Dym Equation Picture Gallery.

Sample Input and Output for HDequation.txt

If you want to see the first 31 terms in the the variable x, in the "series expansion" of the general (generic, symbolic) solution, u(x,t), of the Harry Dym Equation
u_t= u u_xxx
with general initial conditions
u(0,t)=a₀(t) u_x(0,t)=a₁(t) u_xx(0,t)=a₂(t)
where, for the sake of clarity we abberviate (d^r/ d t^r)a_i(t) by b_i,r for i=0,1,2
the input gives the output.
BTW, this took 74 seconds.
If you want to see the first 41 terms in the the variable x, in the "series expansion" of the general (generic, symbolic) solution, u(x,t), of the Harry Dym Equation
u_t= u u_xxx
with general initial conditions
u(0,t)=1 u_x(0,t)=a₁(t) u_xx(0,t)=a₂(t)
where, for the sake of clarity we abberviate (d^r/ d t^r)a_i(t) by b_i,r for i=1,2
the input gives the output.
BTW, this took 134 seconds.

If you want to see the first 31 terms in the the variable x, in the "series expansion" of the general (generic, symbolic) solution, u(x,t), of the KdV Equation
u_t+u_xxx-6u_x=0
with general initial conditions
u(0,t)=a₀(t) u_x(0,t)=a₁(t) u_xx(0,t)=a₂(t)
where, for the sake of clarity we abberviate (d^r/ d t^r)a_i(t) by b_i,r for i=0,1,2
the input gives the output.
BTW, this took 97 seconds.
If you want to see the first 41 terms in the the variable x, in the "series expansion" of the general (generic, symbolic) solution, u(x,t), of the KdV Equation
u_t+u_xxx-6u_x=0
with general initial conditions
u(0,t)=1 u_x(0,t)=a₁(t) u_xx(0,t)=a₂(t)
where, for the sake of clarity we abberviate (d^r/ d t^r)a_i(t) by b_i,r for i=1,2
the input gives the output.
BTW, this took 311 seconds.

Personal Journal of Shalosh B. Ekhad and Zeilberger