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{\bf 
Graphic and Symbolic Experiments with the Harry Dym Equation
}
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\centerline
{\it Shalosh B. EKHAD and Doron ZEILBERGER}
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\qquad \qquad \qquad 
{\it In fond memory of Harry Dym (Jan. 26, 1938 - July 18, 2024)}
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{\bf Abstract}:  
We use both symbolic and numeric computations to experiment with the Harry Dym equation, of central importance in soliton theory.



{\bf Preface}

On July 18, 2024, our beloved {\it academic father}, Harry Dym, suddenly died. That was such a shock, since Harry was so much alive, both
physically and mathematically. This modest tribute to his memory is about the {\bf Harry Dym equation}, one of the
most important differential equations in {\it soliton theory}. Curiously, Harry never published anything about it,
and the name was coined by the great soliton expert Martin Kruskal. In Harry's own words ([D1], p.10):

{\it
``Martin Kruskal spent the academic year 1973-1974 on sabbatical at the Weizmann Institute. In the course of the year he gave a number of interesting lectures
on isospectral problems for the Schroedinger equation and the connections with the
KdV equation. After so many years of living with the string equation, it seemed
natural to explore analogous questions in this setting also. A few hours of calculations (and miscalculations) led to the conclusion that if the density of the
string is parametrized by t and allowed to evolve according to the partial
differential equation $r_t-r^3r_{xxx}=0$, then the eigenvalues of the string equation
$-r^{-1}y''=\lambda y$. with appropriate side conditions, would stay fixed in time. In 1974,
Martin reported on these calculations in a series of lectures at the Batelle Institute
and called the PDE the Harry Dym equation. The name stuck, even though I never
wrote any papers on the subject. (Actually a draft of a paper which explored a
number of questions related to the theory of such equations was prepared in collaboration with Martin. But Martin took it back with him to Princeton, where it
is presumably still collecting dust in his office.)''
}


Since we like {\it symbolic} computation, we were wondering whether we can try to find as many as possible terms in the `series expansion'
of a generic solution of the Harry Dym equation subject to general {\it initial conditions} (in $x$).
We were also curious how a typical solution looks near the origin of the $(x,t)$-plane. To that end one, of us (DZ) wrote
a Maple package {\tt HDequation.txt}, available from:

{\tt https://sites.math.rutgers.edu/\~{}zeilberg/tokhniot/HDequation.txt} \quad .

Here we will describe some of the output produced by the {\it other `one of us'} (SBE).

{\bf Series Expansions for the general solution of the Harry Dym Equation}

{\bf Theorem 1}: Let $a_0(t)$, $a_1(t)$, and $a_2(t)$ be  {\it any} functions, or formal power series, in $t$. We are interested in the
{\it series expansion}, in $x$, of the {\it general solution} $u(x,t)$, of the celebrated Harry Dym equation
$$
u_t=u^3 u_{xxx}  \quad,
$$
subject to the initial conditions
$$
u(0,t)=a_0(t) \quad, \quad
u_x(0,t)=a_1(t) \quad, \quad
u_{xx}(0,t)=a_2(t) \quad.
$$
Write
$$
u(x,t)= \sum_{i=0}^{\infty} A_i(t) x^i \quad.
$$
For $i \in \{0,1,2\}$, and any $r \geq 0$,  let's abbreviate:
$$
b_{i,r} := \frac{d^r}{dt^r} a_i(t) \quad .
$$
We have:
$$
A_0(t)=b_{0,0} \quad, \quad A_1(t)=b_{1,0} \quad, \quad A_2(t)=b_{2,0} \quad, \quad A_3(t)= \frac{b_{0,1}}{6 b_{0,0}^{3}} \quad, \quad
A_4(t)= \frac{b_{0,0} b_{1,1}-3 b_{0,1} b_{1,0}}{24 b_{0,0}^{4}} \quad,
$$
$$
A_5(t)=\frac{b_{0,0}^{2} b_{2,1}-3 b_{0,0} b_{0,1} b_{2,0}-3 b_{0,0} b_{1,0} b_{1,1}+6 b_{0,1} b_{1,0}^{2}}{60 b_{0,0}^{5}} \quad ,
$$
$$
A_6(t)=\frac{-18 b_{0,0}^{3} b_{1,0} b_{2,1}-18 b_{0,0}^{3} b_{1,1} b_{2,0}+72 b_{0,0}^{2} b_{0,1} b_{1,0} b_{2,0}+36 b_{0,0}^{2} b_{1,0}^{2} b_{1,1}-60 b_{0,0} b_{0,1} b_{1,0}^{3}+b_{0,0} b_{0,2}-6 b_{0,1}^{2}}{720 b_{0,0}^{7}} \quad .
$$
$$
A_7(t)=\frac{1}{5040 b_{0,0}^{8}} \cdot
$$
$$
(-72 b_{0,0}^{4} b_{2,0} b_{2,1}+144 b_{0,0}^{3} b_{0,1} b_{2,0}^{2}+144 b_{0,0}^{3} b_{1,0}^{2} b_{2,1}+288 b_{0,0}^{3} b_{1,0} b_{1,1} b_{2,0}-720 b_{0,0}^{2} b_{0,1} b_{1,0}^{2} b_{2,0}
$$
$$
-240 b_{0,0}^{2} b_{1,0}^{3} b_{1,1}+360 b_{0,0} b_{0,1} b_{1,0}^{4}+b_{0,0}^{2} b_{1,2}-21 b_{0,0} b_{0,1} b_{1,1}-15 b_{0,0} b_{0,2} b_{1,0}+105 b_{0,1}^{2} b_{1,0}) \quad ,
$$
$$
A_8(t)= \frac{1}{40320 b_{0,0}^{9}} \cdot
$$
$$
(1440 b_{0,0}^{4} b_{1,0} b_{2,0} b_{2,1}+720 b_{0,0}^{4} b_{1,1} b_{2,0}^{2}-3600 b_{0,0}^{3} b_{0,1} b_{1,0} b_{2,0}^{2}-1200 b_{0,0}^{3} b_{1,0}^{3} b_{2,1}-3600 b_{0,0}^{3} b_{1,0}^{2} b_{1,1} b_{2,0}
$$
$$
+7200 b_{0,0}^{2} b_{0,1} b_{1,0}^{3} b_{2,0}+1800 b_{0,0}^{2} b_{1,0}^{4} b_{1,1}-2520 b_{0,0} b_{0,1} b_{1,0}^{5}+2 b_{0,0}^{3} b_{2,2}-78 b_{0,0}^{2} b_{0,1} b_{2,1}-66 b_{0,0}^{2} b_{0,2} b_{2,0}-21 b_{0,0}^{2} b_{1,0} b_{1,2}
$$
$$
-21 b_{0,0}^{2} b_{1,1}^{2}+462 b_{0,0} b_{0,1}^{2} b_{2,0}+501 b_{0,0} b_{0,1} b_{1,0} b_{1,1}+177 b_{0,0} b_{0,2} b_{1,0}^{2}-1416 b_{0,1}^{2} b_{1,0}^{2}) \quad .
$$

To see $A_i(t)$ for $i \leq 30$, see the output file:

{\tt https://sites.math.rutgers.edu/\~{}zeilberg/tokhniot/oHDequation1.txt} \quad .

If we take $a_0(t)=1$, i.e. $u(0,t)=1$ for all $t$, things simplify, and we have the following theorem.

{\bf Theorem 1'}: Let $a_1(t)$, and $a_2(t)$ be  {\it any} functions, or formal power series, in $t$. We are interested in the
{\it series expansion}, in $x$, of the {\it general solution} $u(x,t)$, of the celebrated Harry Dym equation
$$
u_t=u^3 u_{xxx}  \quad ,
$$
subject to the initial conditions
$$
u(0,t)=1 \quad, \quad
u_x(0,t)=a_1(t) \quad, \quad
u_{xx}(0,t)=a_2(t) \quad.
$$
Write
$$
u(x,t)= \sum_{i=0}^{\infty} B_i(t) x^i \quad.
$$
For $i \in \{1,2\}$, and any $r \geq 0$,  let's abbreviate:
$$
b_{i,r}:= \frac{d^r}{dt^r} a_i(t) \quad .
$$
We have:
$$
B_0(t)=1 \quad, \quad B_1(t)=b_{1,0} \quad, \quad B_2(t)=b_{2,0} \quad, \quad B_3(t)=0  \quad, \quad
B_4(t)=\frac{b_{1,1}}{24} \quad, \quad B_5(t)= -\frac{b_{1,0} b_{1,1}}{20}+\frac{b_{2,1}}{60} \quad ,
$$
$$
B_6(t)= \frac{1}{20} b_{1,1} b_{1,0}^{2}-\frac{1}{40} b_{1,1} b_{2,0}-\frac{1}{40} b_{2,1} b_{1,0} \quad ,
$$
$$
B_7(t)=\frac{1}{5040} b_{1,2}-\frac{1}{21} b_{1,1} b_{1,0}^{3}+\frac{2}{35} b_{1,1} b_{1,0} b_{2,0}+\frac{1}{35} b_{2,1} b_{1,0}^{2}-\frac{1}{70} b_{2,1} b_{2,0} \quad,
$$
$$
B_8(t)=
$$
$$
-\frac{1}{1920} b_{1,1}^{2}-\frac{1}{1920} b_{1,0} b_{1,2}+\frac{1}{20160} b_{2,2}+\frac{5}{112} b_{1,1} b_{1,0}^{4}-\frac{5}{56} b_{1,1} b_{1,0}^{2} b_{2,0}-\frac{5}{168} b_{2,1} b_{1,0}^{3}+\frac{1}{28} b_{1,0} b_{2,1} b_{2,0}+\frac{1}{56} b_{1,1} b_{2,0}^{2}
\quad,
$$
$$
B_9(t)=
\frac{1}{1120} b_{1,2} b_{1,0}^{2}-\frac{1}{6720} b_{2,2} b_{1,0}+\frac{1}{560} b_{1,1}^{2} b_{1,0}-\frac{1}{3360} b_{1,2} b_{2,0}-\frac{1}{2240} b_{1,1} b_{2,1}-\frac{1}{24} b_{1,1} b_{1,0}^{5}
$$
$$
+\frac{5}{168} b_{2,1} b_{1,0}^{4}+\frac{1}{84} b_{2,1} b_{2,0}^{2}+\frac{5}{42} b_{1,1} b_{1,0}^{3} b_{2,0}-\frac{5}{84} b_{1,0}^{2} b_{2,1} b_{2,0}-\frac{5}{84} b_{1,0} b_{1,1} b_{2,0}^{2} \quad,
$$
$$
B_{10}(t)=
-\frac{1}{11200} b_{2,1}^{2}+\frac{1}{3628800} b_{1,3}+\frac{37}{22400} b_{1,1} b_{1,0} b_{2,1}+\frac{73}{67200} b_{1,2} b_{1,0} b_{2,0}-\frac{1}{24} b_{1,0} b_{2,1} b_{2,0}^{2}-\frac{7}{48} b_{1,1} b_{1,0}^{4} b_{2,0}
$$
$$
+\frac{1}{12} b_{1,0}^{3} b_{2,1} b_{2,0}+\frac{1}{8} b_{1,0}^{2} b_{1,1} b_{2,0}^{2}-\frac{2}{1575} b_{1,2} b_{1,0}^{3}+\frac{73}{67200} b_{1,1}^{2} b_{2,0}
$$
$$
-\frac{2}{525} b_{1,1}^{2} b_{1,0}^{2}+\frac{19}{67200} b_{2,2} b_{1,0}^{2}-\frac{1}{11200} b_{2,2} b_{2,0}+\frac{7}{180} b_{1,1} b_{1,0}^{6}-\frac{7}{240} b_{2,1} b_{1,0}^{5}-\frac{1}{72} b_{1,1} b_{2,0}^{3} \quad.
$$

For expressions for $B_i(t)$ for all $i \leq 40$, see the output file:

{\tt https://sites.math.rutgers.edu/\~{}zeilberg/tokhniot/oHDequation1S.txt} \quad .


For the analogous expansions for the solutions of the  Korteweg–De Vries (KdV) equation see the output files

{\tt https://sites.math.rutgers.edu/\~{}zeilberg/tokhniot/oHDequation2.txt} \quad ,

and

{\tt https://sites.math.rutgers.edu/\~{}zeilberg/tokhniot/oHDequation2S.txt} .


{\bf Pictures}

It turns out that for most specific, random, choices of initial conditions $a_0(t)$, $a_1(t)$, $a_2(t)$, the above {\it formal power series} {\bf converge}
near the origin. So by truncating the series up to $30$ terms, we got nice plots.

See

{\tt https://sites.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimhtml/HDequationPics.html} \quad .


{\bf Comment}: Surpsingly (and perhaps not, we are not experts), the plots of the solutions for random initial conditions for the KdV and the HD
equations look very similar, at least near the origin.

{\bf Appendix: Harry Dym's favorite quotes}

Harry Dym was not {\it just} a great researcher, he was an amazing {\it teacher}, and {\it expositor}, and had a great sense of humor.
Each of the chapters of his brilliant textbook {\it Linear Algebra in Action} [D2], that we {\bf strongly recommend}, has a witty
and sometimes funny {\it motto}. In order to encourage you to read that masterpiece, here the first eleven of them.
For the remaining $15$ read the book!

$\bullet$ Preface: {\it A foolish consistency is the hobgoblin of little minds, $\dots$ } \quad - \quad Ralph Waldo Emerson, {\it Self Reliance}

$\bullet$ Chapter 1: 

{\it The road to wisdom? Well it's plain and simple to express.

Err and err and err again, but less and less and less}. (cited in {\it Mathematical Writing} by D.E. Knuth, T. Larrabee, and P.M. Roberts.)

$\bullet$ Chapter 2: 

\quad {\it ... People can tell you... do it like this, But that ain't the way to learn, You got to do it all by yourself.} --- Willie Mays

$\bullet$ Chapter 3:  {\it I was working on the proof of one of my poems all morning, and took out a comma, In the afternoon I put it back again}. \quad - Oscar Wilde.

$\bullet$ Chapter 4:  {\it Can you imagine a mathematician writing {\bf Moby Dick}? Let my name be Ishmael, let the captain's name be Ahab, let the boats name be Pequod,
and let the whale's name be as in the title} \quad - \quad B. A. Cipra


$\bullet$ Chapter 5:  {\it Look at him, he doesn't drink, he doesn't smoke, he doesn't chew, he doesn't stay out late, and he still can't hit.} \quad - \quad Casey Stengel.

$\bullet$ Chapter 6:  {\it  Some people believe that football is a matter of life and death. I'm very disappointed with that attitude, I can assure you its much, much
more important than that.} \quad - \quad Bill Shankly, former manager of Liverpool.

$\bullet$ Chapter 7 :  

{\it I give you now Professor Twist, A conscientious scientist, }

$\dots$ $\dots$

Camped on a tropic riverside, One day he missed his loving bride.

She had, the guide informed him later, Been eaten by an alligator.

Professor Twist could not but smile. ``You mean," he said, `` a crocodile.''

\quad\quad\quad\quad\quad\quad\quad\quad The Purist, by Ogden Nash.

$\bullet$ Chapter 8 :  {\it A proof should be as simple as possible, but not simpler}. \quad - \quad Paraphrase of Albert Einstein's remark on deep truths.

$\bullet$ Chapter 9 :  {\it Not everything that one thinks, should one say; not everything that one says, should one write, and not everything that
one writes, should one publish} \quad - \quad Dictum of the Soloveitchick family.


$\bullet$ Chapter 10 :  {\it Let's throw everything away. Then there will be room for what's left} \quad - \quad Irene Dym.



{\bf References}

[D1] Harry Dym, {\it Looking back}, in: {\it ``Interpolation Theory, Systems Theory and Related Topics''}, The Harry Dym Anniversary volume,
Operator Theory Advances and Applications {\bf 134} (2002), (Daniel Alpay, Israel Gohberg, and Victor Vinnikov, editors), 1-19. \hfill\break
{\tt https://sites.math.rutgers.edu/\~{}zeilberg/akherim/HarryDym60.pdf} \quad .


[D2] Harry Dym, {\it ``Linear Algebra in Action''} (Graduate Studies in Mathematics {\bf 78}), American Mathematical Society, 2007.

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\bigskip
Shalosh B. Ekhad and Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Email: {\tt  ShaloshBEkhad at gmail dot com} \quad, \quad {\tt DoronZeil] at gmail dot com}   \quad .


Written: {\bf Aug. 2, 2024}.


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