This short article presents a table of new equations which can be regarded as
the generalized equations of the dispersionless limit of several nonlinear
equations. From the definition expressed in an algebraic formula, one can get
an equation for any positive numbers p and q. The equations were calculated by
using the computers and were examined by hand-calculation up to p=10. Relations
with some dispersionless hierarchies are mentioned.Comment: AmSTeX, 6 pages, amsppt.st

Son, Seung Hwan

English

arXiv

This short article presents a table of new equations which can be regarded as the generalized equations of the dispersionless limit of several nonlinear equations. From the definition expressed in an algebraic formula, one can get an equation for any positive numbers p and q. The equations were calculated by using the computers and were examined by hand-calculation up to p=10. Relations with some dispersionless hierarchies are mentioned

Son Seung Hee

CERN Document Server

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hep-th/9506172
THE EQUATIONS OF SOME DISPERSIONLESS LIMIT
Seung Hwan Son
Abstract. These equations are the generalized equations of several dispersionless
equations. A complete table for p  10 is provided.
1. Introduction
It is well-known that a lot of nonlinear solitonic equations can be transformed
into certain Hirota type bilinear equations [17]. The  -function of the KP hi-
erarchy can be characterized by the Hirota equations and the Plucker relations
are given from these equations. The dierential Fay identity which has quasi-
classical limit, is a part of the Plucker relations. The leading term of the quasi-
classical limit of the dierential Fay identity satises an identity [22] and from
the identity EQUATION(;1) is extracted.

Therefore, EQUATION(;1) is a
subset of the dispersionless KP hierarchy. EQUATION(p; q) are derived from
EQUATION(p;1). EQUATION(; 2) can be regarded as a subset of dispersion-
less KdV hierarchy. We can easily show that EQUATION(4; 3) is a dispersionless
Boussinesq equation. Therefore, EQUATION(; 3) can be regarded as a subset of
dispersionless Boussinesq hierarchy. EQUATION(; q) for q > 3 is a whole new
set of dispersionless equations which can be regarded as a subset of new hierarchy
which may have some useful application.
2. The formula
Let us use F
mn
instead of
@
2
@t
m
@t
n
(F (t
1
; : : : ; t
r
; : : : )).
Denition 2.1. EQUATION(p; q):
X
0<i
1
<<i
k
p
(i
1
+1)n
i
1
++(i
k
p
+1)n
i
k
p
=p
 

k
p
P
j=1
n
i
j
  1

!
k
p
Q
j=1
( F
1i
j
)
n
i
j
n
i
j
!
!
+
X
m+n=p
F
mn
mn
= 0:
where the terms having
@
@t
q
; : : : ;
@
@t
kq
; : : : vanish.
1991 Mathematics Subject Classication. Primary 35Q53, 35Q51.
Key words and phrases. Dispersionless limit, KdV, KP, Boussinesq equation.
* (Caution)
This notation is used only for technical simplicity of expression since the formula was found by
the author recently (Fall, 1994) [5]. This is another way of using equation numbers. Therefore,
the author strongly recommends careful use of the notation until the formula is well-known.
(Of course, given notation has no meaning elsewhere like other usual equation numbers.)
Typeset by A
M
S-T
E
X
1
2 SEUNG HWAN SON
3. EQUATION(4; q)
One can easily show that EQUATION(4,1) is a dispersionless KP and EQUA-
TION(4,2) is a dispersionless KdV.
Consider a dispersionless Boussinesq equation
(uu
x
)
x
+
1
2
u
yy
= 0: (3:1)
If we set t
1
= x and t
2
= y, then EQUATION(4,3) is
1
2
(F
xx
)
2
+
1
4
F
yy
= 0: (3:2)
Dierentiate (3.2) with respect to x. Then we get
F
xx
F
xxx
+
1
4
F
xyy
= 0: (3:3)
Setting u = 2F
xx
, (3.3) becomes

u
2
u
x
2

x
+
1
4

u
yy
2

= 0:
which is the same as (3.1).
4. Discussion
We could get the useful expression of the generalized equations of the dispersion-
less limit of KdV, KP and Boussinesq equations. And one can get a specic equation
for each (p; q). Furthermore, new hierarchies are derived from EQUATION(; q) for
q > 3. For further research, a table of equations are provided. By denition,
EQUATION(p; q) is the same as EQUATION(p;1) for q  p.
Table. Equations for (p; q).
(4,1)
1
2
F
11
2
 
1
3
F
13
+
1
4
F
22
= 0
(5,1) F
11
F
12
 
1
2
F
14
+
1
3
F
23
= 0
(6,1)
1
3
F
11
3
 
1
2
F
12
2
  F
11
F
13
+
3
5
F
15
 
1
9
F
33
 
1
4
F
24
= 0
(7,1) F
11
2
F
12
  F
12
F
13
  F
11
F
14
+
2
3
F
16
 
1
6
F
34
 
1
5
F
25
= 0
(8,1)
1
4
F
11
4
  F
11
F
12
2
  F
11
2
F
13
+
1
2
F
13
2
+ F
12
F
14
+ F
11
F
15
 
5
7
F
17
+
1
16
F
44
+
2
15
F
35
+
1
6
F
26
= 0
THE EQUATIONS OF SOME DISPERSIONLESS LIMIT 3
(9,1) F
11
3
F
12
 
1
3
F
12
3
  2F
11
F
12
F
13
  F
11
2
F
14
+ F
13
F
14
+F
12
F
15
+ F
11
F
16
 
3
4
F
18
+
1
10
F
45
+
1
9
F
36
+
1
7
F
27
= 0
(10,1)
1
5
F
11
5
 
3
2
F
11
2
F
12
2
  F
11
3
F
13
+ F
12
2
F
13
+ F
11
F
13
2
+2F
11
F
12
F
14
 
1
2
F
14
2
+ F
11
2
F
15
  F
13
F
15
  F
12
F
16
  F
11
F
17
+
7
9
F
19
 
1
25
F
55
 
1
12
F
46
 
2
21
F
37
 
1
8
F
28
= 0
(4,2)
1
2
F
11
2
 
1
3
F
13
= 0
(5,2) 0 = 0
(6,2)
1
3
F
11
3
  F
11
F
13
+
3
5
F
15
 
1
9
F
33
= 0
(7,2) 0 = 0
(8,2)
1
4
F
11
4
  F
11
2
F
13
+
1
2
F
13
2
+ F
11
F
15
 
5
7
F
17
+
2
15
F
35
= 0
(9,2) 0 = 0
(10,2)
1
5
F
11
5
  F
11
3
F
13
+ F
11
F
13
2
+ F
11
2
F
15
  F
13
F
15
  F
11
F
17
+
7
9
F
19
 
1
25
F
55
 
2
21
F
37
= 0
(4,3)
1
2
F
11
2
+
1
4
F
22
= 0
(5,3) F
11
F
12
 
1
2
F
14
= 0
(6,3)
1
3
F
11
3
 
1
2
F
12
2
+
3
5
F
15
 
1
4
F
24
= 0
(7,3) F
11
2
F
12
  F
11
F
14
 
1
5
F
25
= 0
(8,3)
1
4
F
11
4
  F
11
F
12
2
+ F
12
F
14
+ F
11
F
15
 
5
7
F
17
+
1
16
F
44
= 0
(9,3) F
11
3
F
12
 
1
3
F
12
3
  F
11
2
F
14
+ F
12
F
15
 
3
4
F
18
+
1
10
F
45
+
1
7
F
27
= 0
(10,3)
1
5
F
11
5
 
3
2
F
11
2
F
12
2
+ 2F
11
F
12
F
14
 
1
2
F
14
2
+ F
11
2
F
15
  F
11
F
17
 
1
25
F
55
 
1
8
F
28
= 0
(5,4) F
11
F
12
+
1
3
F
23
= 0
(6,4)
1
3
F
11
3
 
1
2
F
12
2
  F
11
F
13
+
3
5
F
15
 
1
9
F
33
= 0
(7,4) F
11
2
F
12
  F
12
F
13
+
2
3
F
16
 
1
5
F
25
= 0
4 SEUNG HWAN SON
(8,4)
1
4
F
11
4
 F
11
F
12
2
 F
11
2
F
13
+
1
2
F
13
2
+F
11
F
15
 
5
7
F
17
+
2
15
F
35
+
1
6
F
26
= 0
(9,4) F
11
3
F
12
 
1
3
F
12
3
  2F
11
F
12
F
13
+F
12
F
15
+F
11
F
16
 +
1
9
F
36
+
1
7
F
27
= 0
(10,4)
1
5
F
11
5
 
3
2
F
11
2
F
12
2
  F
11
3
F
13
+ F
12
2
F
13
+ F
11
F
13
2
+ F
11
2
F
15
 F
13
F
15
  F
12
F
16
  F
11
F
17
+
7
9
F
19
 
1
25
F
55
 
2
21
F
37
= 0
(6,5)
1
3
F
11
3
 
1
2
F
12
2
  F
11
F
13
 
1
9
F
33
 
1
4
F
24
= 0
(7,5) F
11
2
F
12
  F
12
F
13
  F
11
F
14
+
2
3
F
16
 
1
6
F
34
= 0
(8,5)
1
4
F
11
4
 F
11
F
12
2
 F
11
2
F
13
+
1
2
F
13
2
+F
12
F
14
 
5
7
F
17
+
1
16
F
44
+
1
6
F
26
= 0
(9,5) F
11
3
F
12
 
1
3
F
12
3
  2F
11
F
12
F
13
  F
11
2
F
14
+ F
13
F
14
+ F
11
F
16
 
3
4
F
18
+
1
9
F
36
+
1
7
F
27
= 0
(10,5)
1
5
F
11
5
 
3
2
F
11
2
F
12
2
  F
11
3
F
13
+ F
12
2
F
13
+ F
11
F
13
2
+ 2F
11
F
12
F
14
 
1
2
F
14
2
  F
12
F
16
  F
11
F
17
+
7
9
F
19
 
1
12
F
46
 
2
21
F
37
 
1
8
F
28
= 0
(7,6) F
11
2
F
12
  F
12
F
13
  F
11
F
14
 
1
6
F
34
 
1
5
F
25
= 0
(8,6)
1
4
F
11
4
 F
11
F
12
2
 F
11
2
F
13
+
1
2
F
13
2
+F
12
F
14
+F
11
F
15
 
5
7
F
17
+
1
16
F
44
+
2
15
F
35
= 0
(9,6) F
11
3
F
12
 
1
3
F
12
3
  2F
11
F
12
F
13
  F
11
2
F
14
+ F
13
F
14
+ F
12
F
15
 
3
4
F
18
+
1
10
F
45
+
1
7
F
27
= 0
(10,6)
1
5
F
11
5
 
3
2
F
11
2
F
12
2
  F
11
3
F
13
+ F
12
2
F
13
+ F
11
F
13
2
+ 2F
11
F
12
F
14
 
1
2
F
14
2
+F
11
2
F
15
 F
13
F
15
 F
11
F
17
+
7
9
F
19
 
1
25
F
55
 
2
21
F
37
 
1
8
F
28
= 0
(8,7)
1
4
F
11
4
  F
11
F
12
2
  F
11
2
F
13
+
1
2
F
13
2
+ F
12
F
14
+ F
11
F
15
+
1
16
F
44
+
2
15
F
35
+
1
6
F
26
= 0
(9,7) F
11
3
F
12
 
1
3
F
12
3
  2F
11
F
12
F
13
  F
11
2
F
14
+ F
13
F
14
+ F
12
F
15
+F
11
F
16
 
3
4
F
18
+
1
10
F
45
+
1
9
F
36
= 0
(10,7)
1
5
F
11
5
 
3
2
F
11
2
F
12
2
  F
11
3
F
13
+ F
12
2
F
13
+ F
11
F
13
2
+ 2F
11
F
12
F
14
 
1
2
F
14
2
+F
11
2
F
15
 F
13
F
15
 F
12
F
16
+
7
9
F
19
 
1
25
F
55
 
1
12
F
46
 
1
8
F
28
= 0
THE EQUATIONS OF SOME DISPERSIONLESS LIMIT 5
(9,8) F
11
3
F
12
 
1
3
F
12
3
  2F
11
F
12
F
13
  F
11
2
F
14
+ F
13
F
14
+ F
12
F
15
+F
11
F
16
+
1
10
F
45
+
1
9
F
36
+
1
7
F
27
= 0
(10,8)
1
5
F
11
5
 
3
2
F
11
2
F
12
2
  F
11
3
F
13
+ F
12
2
F
13
+ F
11
F
13
2
+ 2F
11
F
12
F
14
 
1
2
F
14
2
+ F
11
2
F
15
  F
13
F
15
  F
12
F
16
  F
11
F
17
+
7
9
F
19
 
1
25
F
55
 
1
12
F
46
 
2
21
F
37
= 0
(10,9)
1
5
F
11
5
 
3
2
F
11
2
F
12
2
  F
11
3
F
13
+ F
12
2
F
13
+ F
11
F
13
2
+ 2F
11
F
12
F
14
 
1
2
F
14
2
+ F
11
2
F
15
  F
13
F
15
  F
12
F
16
  F
11
F
17
 
1
25
F
55
 
1
12
F
46
 
2
21
F
37
 
1
8
F
28
= 0
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6 SEUNG HWAN SON
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Seung H. Son
Department of Mathematics
University of Illinois at Urbana-Champaign
E-mail address: son@math.uiuc.edu


The equations of some dispersionless limit

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