2.4. COMPATIBILITY FOR SYSTEMS OF PDES 41
which has the solution
Q D G.p; q/ C H.u/ (2.238)
for arbitrary functions G and H . From (2.227), we find that G satisfies G
pp
C G
qq
D 0; from
(2.223d), that H satisfies H
00
D 0 giving that H D cu where c is an arbitrary constant, noting
that we have suppressed the second constant of integration due to translational freedom. is
leads to our main result. Equations of the form
u
t
D u
xx
C u
yy
C cu C G
u
x
; u
y
are compatible with the first-order equations
u
t
D cu C G
u
x
; u
y
;
where c is an arbitrary constant and G.p; q/, a function satisfying G
pp
C G
qq
D 0.
2.4 COMPATIBILITY FOR SYSTEMS OF PDES
We now extend the idea of compatibility to systems of PDEs. In this section we consider the
Cubic Schrödinger equation
i
t
C
xx
C kj j
2
D 0: (2.239)
If we assume that D u C iv then (2.239) becomes
v
t
C u
xx
C ku.u
2
C v
2
/ D 0;
u
t
C v
xx
C kv.u
2
C v
2
/ D 0:
(2.240)
Here, we seek compatibility with the pair of first-order PDEs
u
t
C A.t; x; u; v/u
x
C B.t; x; u; v/v
x
D U.t; x; u; v/;
u
t
C C.t; x; u; v/u
x
C D.t; x; u; v/v
x
D V .t; x; u; v/;
(2.241)
for some functions A; B; C; D; U and V to be determined. We solve (2.240) and (2.241) for
u
t
; v
t
; u
xx
and v
xx
and require compatibility, that is
.u
t
/
xx
D .u
xx
/
t
and .v
t
/
xx
D .v
xx
/
t
: (2.242)
Isolating the coefficients of u
x
and v
x
gives rise to two sets of determining equations for the
unknowns A; B; C; D; U and V . Each set contains ten determining equations. We only list six
of each ten as they are the only ones needed in our preliminary analysis. Also, they are the smaller
42 2. COMPATIBILITY
equations. ese are:
A
uu
D 0; (2.243a)
2A
uv
C B
uu
D 0; (2.243b)
A
vv
C 2B
uv
D 0; (2.243c)
B
vv
D 0; (2.243d)
U
uu
2A
xu
C .2C B/A
u
AA
v
2AB
u
DC
u
C C
v
D 0; (2.243e)
U
vv
2B
xv
C 2DA
v
C DB
u
.C C 4B/B
v
BD
u
C .A 2D/D
v
D 0; (2.243f)
2U
uv
2A
xv
2B
xu
C 3DA
u
C .C 2B/A
v
3BB
u
3AB
v
BC
u
C .A 2D/C
v
DD
u
CD
v
D 0; (2.243g)
and
C
uu
D 0; (2.244a)
2C
uv
C D
uu
D 0; (2.244b)
C
vv
C 2D
uv
D 0; (2.244c)
D
vv
D 0; (2.244d)
V
uu
2C
xu
C .2A D/A
u
C CA
v
C .B C 4C /C
u
AC
v
2AD
u
D 0; (2.244e)
V
vv
2D
xv
C BB
u
C AB
v
C 2DC
v
C DD
u
C .C 2B/D
v
D 0; (2.244f)
2V
uv
2C
xv
2D
xu
C BA
u
C AA
v
C .2A D/B
u
C CB
v
C3DC
u
C 3C C
v
C .2C B/D
u
3AD
v
D 0: (2.244g)
We easily solve (2.243a)–(2.243d) and (2.244a)–(2.244d) giving
A D A
1
.t; x/uv B
1
.t; x/v
2
C A
2
.t; x/u C A
3
.t; x/v C A
4
.t; x/; (2.245a)
B D A
1
.t; x/u
2
C B
1
.t; x/uv C B
2
.t; x/u C B
3
.t; x/v C B
4
.t; x/; (2.245b)
C D C
1
.t; x/uv D
1
.t; x/v
2
C C
2
.t; x/u C C
3
.t; x/v C C
4
.t; x/; (2.245c)
D D C
1
.t; x/u
2
C D
1
.t; x/uv C D
2
.t; x/u C D
3
.t; x/v C D
4
.t; x/; (2.245d)
where A
i
; B
i
; C
i
and D
i
; i D 1; 2; 3; 4 are arbitrary functions of integration. We impose (2.245)
on the remaining equations in (2.243) and (2.244) and then require the resulting equations to
be compatible in the sense that
.U
uu
/
v
D .U
uv
/
u
; .U
uv
/
v
D .U
vv
/
u
; .V
uu
/
v
D .V
uv
/
u
; and .V
uv
/
v
D .V
vv
/
u
: (2.246)
We performing this calculation and set the coefficients with respect to u; v; u
2
; uv and v
2
to
zero. e coefficients of u
2
; uv and v
2
are presented below:
A
2
1
D 0; (2.247a)
2A
1
B
1
A
1
C
1
2C
1
D
1
D 0; (2.247b)
3A
1
D
1
B
1
C
1
C B
2
1
C 4D
2
1
D 0; (2.247c)
2.4. COMPATIBILITY FOR SYSTEMS OF PDES 43
B
1
D
1
D 0; (2.248a)
6A
1
B
1
3A
1
C
1
C 4C
1
D
1
D 0; (2.248b)
3B
2
1
C 2D
2
1
A
1
D
1
3C
1
B
1
D 0; (2.248c)
A
1
C
1
D 0; (2.249a)
6C
1
D
1
3B
1
D
1
C 4A
1
B
1
D 0; (2.249b)
2A
2
1
3C
2
1
C
A
1
D
1
C 3B
1
C
1
D 0 D 0;
(2.249c)
and
D
2
1
D 0; (2.250a)
2A
1
B
1
C B
1
D
1
2C
1
D
1
D 0; (2.250b)
4A
2
1
C C
2
1
C 3A
1
D
1
B
1
C
1
D 0: (2.250c)
From (2.247)–(2.250) we see that A
1
D D
1
D 0 and
B
1
.B
1
C
1
/ D 0; C
1
.B
1
C
1
/ D 0; (2.251)
from which we deduce that C
1
D B
1
. We now return to (2.246) and isolate coefficients with
respect to u and v. ese are:
B
1
.7A
3
11B
2
3C
2
C 3D
3
/ D 0; (2.252a)
B
1
.3A
2
C 2C
3
C D
2
/ D 0; (2.252b)
B
1
.5B
3
C
3
C 2D
2
/ D 0; (2.252c)
B
1
.5A
3
C 9B
2
C C
2
9D
3
/ D 0; (2.252d)
B
1
.9A
2
B
3
9C
3
5D
2
/ D 0; (2.252e)
B
1
.2A
3
C B
2
5C
2
/ D 0; (2.252f)
B
1
.A
3
2B
2
C 3D
3
/ D 0; (2.252g)
B
1
.3A
2
C 3B
3
C 11C
3
C 7D
2
/ D 0; (2.252h)
from which we see two cases emerge: (i) B
1
¤ 0 and (ii) B
1
D 0.
Case (i) B
1
¤ 0
Solving (2.252) gives
A
2
D 0; A
3
D 2B
2
; C
1
D B
1
; C
2
D B
2
; C
3
D B
3
; D
2
D 2B
3
; D
3
D 0; (2.253a)
and from the remaining compatibility conditions (2.246), we obtain
B
4
D
B
2
B
3
B
1
3
2
B
1
x
B
1
; C
4
D
B
2
B
3
B
1
C
3
2
B
1
x
B
1
; D
4
D A
4
C
B
2
2
B
2
3
B
1
: (2.254)
44 2. COMPATIBILITY
Now that we have consistency, we solve (2.243e)–(2.243g) and (2.244e)–(2.244g) for U and V .
As these are fairly complicated (with 20 terms each), we suppress the output. We now return
to (2.242) and consider the remaining 4 and 4 terms. As we are able to isolate terms involving
u and v, we do so. is gives rise to a total of 114 new determining equations. It is left as an
exercise to the read to show that the final forms of A; B; C; D; U; and V are:
A D c
1
v
2
C 2c
2
t C 2c
3
; B D C D c
1
uv; D D c
1
u
2
C 2c
2
t C 2c
3
;
U D c
1
.
c
2
t C c
3
/
.u
2
C v
2
/v .c
2
x C c
4
/v; (2.255)
V D c
1
.
c
2
t C c
3
/
.u
2
C v
2
/u C .c
2
x C c
4
/u;
giving rise to the following compatible equations:
u
t
C .c
1
v
2
C 2c
2
t C 2c
3
/ u
x
C c
1
uvv
x
D c
1
.c
2
t C c
3
/v.u
2
C v
2
/ .c
2
x C c
4
/v;
v
t
C c
1
uvu
x
C .c
1
u
2
C 2c
2
t C 2c
3
/v
x
D c
1
.c
2
t C c
3
/u.u
2
C v
2
/ C .c
2
x C c
4
/u;
(2.256)
where c
1
c
4
are arbitrary constants.
Case (ii) B
1
D 0
In order to facilitate finding a solution to (2.243) and (2.244) with the forms of A; B; C; and D
given in (2.245) (with A
1
D B
1
D C
1
D D
1
D 0), we note that the original system (2.240) is
invariant under the transformation
u ! v; v ! u: (2.257)
We would expect the augmented pair (2.241) to also be invariant under the same transformation.
In this case system (2.241) is
u
t
C .A
2
u C A
3
v CA
4
/u
x
C .B
2
u C B
3
v CB
4
/v
x
D U.t; x; u; v/;
v
t
C .C
2
u C C
3
v CC
4
/u
x
C .D
2
u C D
3
v CD
4
/v
x
D V .t; x; u; v/;
(2.258)
which becomes under (2.257)
u
t
C .D
3
u D
2
v CD
4
/u
x
C .C
3
u C C
2
v C
4
/v
x
D V .t; x; v; u/;
v
t
C .B
3
u C B
2
v B
4
/u
x
C .A
3
u A
2
v CA
4
/v
x
D U.t; x; v; u/:
(2.259)
Comparing (2.258) and (2.259) gives
A
2
D D
3
; A
3
D D
2
; A
4
D D
4
; B
2
D C
3
; B
3
D C
2
; B
4
D C
4
;
C
2
D B
3
; C
3
D B
2
; C
4
D B
4
; D
2
D A
3
; D
3
D A
2
; D
4
D A
4
;
(2.260)
and from (2.260) we have
A
2
D B
2
D C
2
D D
2
D A
3
D B
3
D C
3
D D
3
D 0; and C
4
D B
4
; D
4
D A
4
: (2.261)
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