Ultradiscretization with parity variables for nonlinear oscillator andits conserved quantity
Shin Isojima (Hosei University, Japan)

Collaboration with Hirotaka Toyama(traditional) Ultradiscretization (UD)

Discrete hard spring equation

For a given difference equation for ,1. Replacewith().
2. Take the limit.

Discrete Hard spring equation

Solution (2) ---non-periodic but `summarizable’---From the initial values→ unique→is not unique:
We consider the following cases.

From the formula
we obtain a piecewise-linear equation for.

Again,

 Under proper restriction,
the resulting equation is regarded a Cellular Automaton (CA).
 This CA (may) inherits some essential properties of the original equation.

is not unique.

(ii)

Conserved quantityUnique→ introduce new cases(iii)

The box-and-ball system, andwhereandUnique
UD,

(i)
Note:

First and typical example:
Discrete Lotka--Volterra equation

andIndependent ofis not unique.

, andis not unique.

andThis is an integrable discretization!

Negative Difficulty in Ultradiscretization

P-ultradiscrete hard spring equation (forward schime)

The difference equation must be subtraction-free.

Notation:

Summary by diagram (2) ---Bounded type---

Negative
Difficulty

Its solutions must be definite-sign.

(i)(ii),For
negative term, and apply UD:, we assume

and move the(iii),If,
For these parameters,,,

has no solution.
should be negative:

We call this situation as`
is the quasi-conserved quantity.’: branching
point

Notation:

Ultradiscretization with parity variables (p-UD)

P-ultradiscrete analogue of the conserved quantity

Introduce parity variable,
Amplitude
by replacing

Notation:

That is,by(the sign of.

), and.

ForSolution (3) ---non-periodic and complicated---，,,,, we consider four cases,,,,From the initial values→ unique→is not unique:

,(i)

(i)
Again,(ii)(ii)
UniqueAll terms
arepositive(iii)(iii)

We consider this set of equations as UD analogue ofBehavior of solution and conserved quantity(i)Both equations have solutions,,,

is given.
or (iii)

Only (iii) has a solution, which means
That is, we obtain the unique solution:
 Assume the pair

Summary by diagram (3) ---Unbounded type--,Solution (1) ---eventually periodic---.

(i)

is not unique.

.

Time evolution of p-UD equation

 Assume the pairFrom
, we have, andwhere
The successive solution depends on .
(We have to introduce further cases.)(vi)

We putis not unique.

.
.

is given. We haveor (iii), which means.
.

We write
We obtain the unique solution as follows.

・・・・・

Notation:・・・・

That is, we obtain indeterminate solutions: We do not meet indeterminate-type evolution.
 The solution is eventually four-periodic.

Our study

Summary by diagram (1) ---periodic type---.

: branchingpointConclusion

We have studied:
all initial values for

Hard spring equation

andfor.

The behavior of solutions and ultradiscretized conserved quantity
is categorized as follows.

We calculate the p-ultradiscretized conserved quantity
from possible pairs
We obtain

Their integrable discrete analogue is given in a Japanese book,`Discrete and Ultradiscrete Systems’ by Prof. Hirota and Prof. Takahashi.

shows the behavior:(i)
and is not always equal to(ii)holds.

We constructed p-ultradiscrete analogue of the hard-spring equation andits conserved quantity.

We give the p-UD analogue of a nonlinear equation with conserved quantity (CQ)
and examine the behavior of ultradiscretized solutions and CQ.

Conserved quantity,Hence,is actually the conserved quantity for this soution.

Solutions

Ultradiscretized CQ

Exactly or eventually
four-periodic

Conserved

Non-periodic but summarizedby `bounded diagram’`quasi-conserved’

Non-periodic with`unbounded diagram’Not always conserved

We need…
Further analysis for non-periodic cases.
Study on solutions of the original difference equation.

Reference
Shin Isojima, Hirotaka Toyama: Ultradiscrete analogues of the hard-spring equation and its conserved quantity,Jpn. J. Ind. Appl. Math., to appear (DOI: 10.1007/s13160-018-0329-5)