Ablowitz M.A. Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons
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8.7 More general classes of nonlinear evolution equations 205
8.7 More general classes of nonlinear evolution equations
A natural question is: what is a general class of “solvable” nonlinear evolution
equations?Hereweusethe2× 2 scattering problem
v
1,x
= −ikv
1
+ q(x, t)v
2
, (8.45a)
v
2,x
= −ikv
2
+ r(x, t)v
1
, (8.45b)
to investigate this issue. Ablowitz, Kaup, Newell and Segur (1974) answered
this question by considering the equations (8.45) associated with the functions
A, B and C in (8.34). Under certain restrictions, a relation was found that gives
a class of solvable nonlinear equations. This relationship depends upon the dis-
persion relation of the linearized form of the nonlinear equations and a certain
integro-differential operator. Suppose that q and r vanish sufficiently rapidly as
|x|→∞: then a general evolution equation can be shown to be
r
−q
t
+ 2A
0
(L)
r
q
= 0 (8.46a)
where A
0
(k) = lim
|x|→∞
A(x, t, k) (we may think of A
0
(k) as the ratio of two
entire functions), and L is the integro-differential operator given by
L =
1
2i
∂
x
− 2r(I
−
q)2r(I
−
r)
−2q(I
−
q) −∂
x
+ 2q(I
−
r)
, (8.46b)
where ∂
x
≡ ∂/∂x and
(I
−
f )(x) ≡
x
−∞
f (y)dy. (8.46c)
Note that L operates on (r, q), and I
−
operates both on the functions immedi-
ately to its right and also on the functions to which L is applied. Equation (8.46)
may be written in matrix form
σ
3
u
t
+ 2A
0
(L)u = 0,
where
σ
3
=
10
0 −1
, u =
r
q
.
It is significant that A
0
(k) is closely related to the dispersion relation of the
associated linearized problem. If f (x) and g(x) vanish sufficiently rapidly as
|x|→∞, then in the limit |x|→−∞,
f (x)(I
−
g)(x) ≡ f (x)
x
−∞
g(y)dy → 0,
206 Solitons and integrable equations
and so for infinitesimal q and r, we have that L is the diagonal differential
operator
L =
1
2i
∂
∂x
10
0 −1
.
Hence in this limit (8.46) yields
r
t
+ 2A
0
−
1
2
i∂
x
r = 0,
−q
t
+ 2A
0
1
2
i∂
x
q = 0.
The above equations are linear (decoupled) partial differential equations solv-
able by Fourier transform methods. Substituting the wave solutions r(x, t) =
exp(i(kx − ω
r
(2k)t)), q(x, t) = exp(i(kx −ω
q
(−2k)t)), into the above equations,
we obtain the relationship
A
0
(k) =
1
2
iω
r
(2k) = −
1
2
iω
q
(−2k). (8.47)
Therefore a general evolution equation is given by
r
−q
= −iω(2L)
r
q
(8.48)
with ω
r
(k) = ω(k); in matrix form this equation takes the form
σ
3
u
t
+ iω(2L)u = 0.
Hence, the form of the nonlinear evolution equation is characterized by
the dispersion relation of its associated linearized equations and an integro-
differential operator; further details can be found in Ablowitz et al. (1974).
For the nonlinear Schr
¨
odinger equation
iq
t
= q
xx
± 2q
2
q
∗
, (8.49)
the associated linear equation is
iq
t
= q
xx
.
Therefore the dispersion relation is ω
q
(k) = −k
2
, and so from (8.47), A
0
(k)is
given by
A
0
(k) = 2ik
2
. (8.50)
The evolution is obtained from either (8.46)or(8.48); therefore we have
r
−q
t
= −4iL
2
r
q
= −2L
r
x
q
x
= i
r
xx
− 2r
2
q
q
xx
− 2q
2
r
.
8.7 More general classes of nonlinear evolution equations 207
When r = ∓q
∗
, both of these equations are equivalent to the nonlinear
Schr
¨
odinger equation (8.49). Note that (8.50) is in agreement with (8.41) with
a = 2i (that is lim
|x|→∞
A = 2ik
2
, since lim
|x|→∞
r(x, t) = 0). This explains
´a posteriori why the expansion of A, B and C in powers of k are related so
closely to the dispersion relation. Indeed, now in retrospect, the fact that the
nonlinear Schr
¨
odinger equation is related to (8.50) implies that an expansion
starting at k
2
will be a judicious choice. Similarly, the modified KdV equation
and sine–Gordon equation can be obtained from the operator equation (8.48)
using the dispersion relations ω(k) = −k
3
and ω(k) = k
−1
, respectively. These
dispersion relations suggest expansions commencing in powers of k
3
and k
−1
,
respectively, that indeed we saw to be the case in the earlier section.
The derivation of (8.48) required that q and r tend to 0 as |x|→0 and
therefore we cannot simply set r = −1 in order to obtain the equivalent result
for the Schr
¨
odinger scattering problem
v
xx
+ (k
2
+ q)v = 0. (8.51)
However the essential ideas are similar and Ablowitz, Kaup, Newell and Segur
(1974) also showed that a general evolution equation in this case is
q
t
+ γ(L)q
x
= 0, (8.52a)
where
L ≡−
1
4
∂
2
x
− q +
1
2
q
x
I
+
, (8.52b)
I
+
f (x) ≡
∞
x
f (y)dy. (8.52c)
γ(k
2
) =
ω(2k)
2k
, (8.52d)
with ∂
x
= ∂/∂x and ω(k), is the dispersion relation of the associated linear
equation with q = exp(i(kx − ω(k)t)).
For the KdV equation
q
t
+ 6qq
x
+ q
xxx
= 0,
the associated linear equation is
q
t
+ q
xxx
= 0.
Therefore ω(k) = −k
3
and so γ(k
2
) = −4k
2
, thus γ(L) = −4L. Hence (8.52)
yields
q
t
− 4Lq
x
= 0,
208 Solitons and integrable equations
thus
q
t
− 4
−
1
4
∂
2
x
− q +
1
2
q
x
I
+
q
x
= 0,
which is the KdV equation!
Associated with the operator L there is a hierarchy of equations (sometimes
referred to as the Lenard hierarchy) given by
u
t
+ L
k
u
x
= 0, k = 1, 2,...
The first two subsequent higher-order equations in the KdV hierarchy are given
by
u
t
−
1
4
u
5x
+ 10uu
3x
+ 20u
x
u
xx
+ 30u
2
u
x
= 0,
u
t
+
1
16
(u
7x
+ 14uu
5x
+ 42u
x
u
4x
+ 70u
xx
u
3x
+ 70u
2
u
3x
+280uu
x
u
xx
+ 70u
3
x
+ 140u
3
n
x
= 0
where u
nx
= ∂
n
u/∂x
n
.
In the above we studied two different scattering problems, namely the
classical time-independent Scr
¨
odinger equation
v
xx
+ (u + k
2
)v = 0, (8.53)
which can be used to linearize the KdV equation, and the 2 × 2 scattering
problem
v
x
= k
−i 0
0 i
v +
0 q
r 0
v, (8.54)
which can linearize the nonlinear Schr
¨
odinger, mKdV and Sine–Gordon equa-
tions. While (8.53) can be interpreted as a special case of (8.54), from the
point of view of possible generalizations, however, we regard them as different
scattering problems.
There have been numerous applications and generalizations of this method
only a few of which we will discuss below. One generalization includes that
of Wadati et al. (1979a,b) (see also Shimizu and Wadati, 1980; Ishimori, 1981,
1982; Wadati and Sogo, 1983; Konno and Jeffrey, 1983). For example, instead
of (8.33), suppose we consider the scattering problem
v
1,x
= −f (k)v
1
+ g(k)q(x, t)v
2
,
v
2,x
= f (k)v
2
+ g(k)r(x, t)v
1
,
(8.55)
8.7 More general classes of nonlinear evolution equations 209
where f (k) and g(k) are polynomial functions of the eigenvalue k, and the time
dependence is given (as previously) by
v
1,t
= Av
1
+ Bv
2
,
v
2,t
= Cv
1
+ Dv
2
.
(8.56)
The compatibility of (8.55) and (8.56) requires that A, B, C, D satisfy linear
equations that generalize equations (8.36)–(8.38).
As earlier, expanding A, B, C, f and g in finite power series expansions
in k (where the expansions for f and g have constant coefficients), then one
obtains a variety of physically interesting evolution equations (cf. Ablowitz
and Clarkson, 1991, for further details).
Extensions to higher-order scattering problems have been considered by sev-
eral authors. There are two important classes of one-dimensional linear scat-
tering problems, associated with the solvable nonlinear evolution equations:
scalar and systems of equations. Some generalizations are now mentioned.
Ablowitz and Haberman (1975) studied an N × N matrix generalization of
the scattering problem (8.54):
∂v
∂x
= ikJv + Qv, (8.57a)
where the matrix of potentials Q(x) ∈ M
N
(C) (the space of N × N matrices
over C) with Q
ii
= 0, J = diag(J
2
, J
2
,...,J
N
), where J
i
J
j
for i j,
i, j = 1, 2,...,N, and v(x, t)isanN-dimensional vector eigenfunction. In this
case, to obtain evolution equations, the associated time dependence is chosen
to be
∂v
∂t
= Tv, (8.57b)
where T is also an N × N matrix. The compatibility of (8.57) yields
T
x
= Q
t
+ ik[J, T] + [Q, T].
In the same way for the 2 × 2 case discussed earlier in the chapter, associated
nonlinear evolution equations may be found by assuming an expansion for T
in powers or inverse powers of the eigenvalue k
T =
n
j=0
k
j
T
j
. (8.58)
Ablowitz and Haberman showed that this scattering problem can be used to
solve several physically interesting equations such as the three-wave inter-
action equations in one spatial dimension (with N = 3 and n = 1) and the
210 Solitons and integrable equations
Boussinesq equation (with N = 3 and n = 2). The associated recursion oper-
ators have been obtained by various authors, cf. Newell (1978) and Fokas and
Anderson (1982). The inverse problem associated with the scattering prob-
lem (8.57) in the general N × N case has been rigorously studied by Beals
and Coifman (1984, 1985), Caudrey (1982), Mikhailov (1979, 1981) and Zhou
(1989).
The following Nth order generalization of the scattering problem (8.53)was
considered by Gel’fand and Dickii (1977)
d
N
v
dx
N
+
N
j=2
u
j
(x)
d
N−j
v
dx
N−j
= λv, (8.59)
where u
j
(x) are considered as potentials; they investigated many of the alge-
braic properties of the nonlinear evolution equations solvable through this
scattering problem. The general inverse problem associated with this scatter-
ing problem is treated in detail in the monograph of Beals, Deift, and Tomei
(1988).
The scattering problems (8.53) and (8.54), and their generalizations (8.59)
and (8.57), are only some of the examples of scattering problems used in
connection with the solution of nonlinear partial differential equations in
(1+1)-dimensions; i.e., one spatial and one temporal dimension. The reader can
find many more interesting scattering problems related to integrable systems in
the literature (Camassa and Holm, 1993a; Konopelchenko, 1993; Rogers and
Schief, 2002; Matveev and Salle, 1991; Dickey, 2003; Newell, 1985).
Exercises
8.1 Use (8.17) to find two conserved quantities additional to (8.19); i.e., w
5
and w
6
. Then also obtain the conserved fluxes.
8.2 Use the method explained in Section 8.6 associated with (8.33)–(8.34)
to find the nonlinear evolution equation corresponding to
A = a
4
k
4
+ a
3
k
3
+ a
2
k
2
+ a
1
k + a
0
.
Show that this agrees with the nonlinear equations obtained via the
operator approach in Section 8.7, i.e., equations (8.46a)–(8.48).
8.3 Use the pair of equations
v
xx
+ (λ + q)v = 0
v
t
= Av + Bv
x
Exercises 211
and the method of Section 8.6 to find an “integrable” nonlinear evolution
equation of order 5 in space, i.e., of the form u
t
+ u
xxxxx
+ ···= 0. Show
that this agrees with the the nonlinear equations obtained via the operator
approach in Section 8.7, i.e., equation (8.52).
8.4 Suppose we wish to find a real integrable equation associated with (8.51)
having a dispersion relation
ω =
1
1 + k
2
.
Find this equation via (8.52).
8.5 Consider the scattering problem (8.57a) of order 3 × 3 with time depen-
dence (8.57) with T givenby(8.58) with n = 1. Find the nonlinear
evolution equation. Hint: see Ablowitz and Haberman (1975). Extend
this to scattering problems of order N × N.
8.6 Consider the KdV equation
u
t
+ 6uu
x
+ u
xxx
= 0.
(a) Show that u(x, t) = −2k
2
csech
2
[k(x − 4k
2
t)] is a singular solution.
(b) Obtain the rational solution u(x, t) = −2/x
2
by letting k → 0.
(c) Show that the singular solution can also be obtained from the clas-
sical soliton solution u(x, t) = 2k
2
sech
2
[k(x − 4k
2
t) − x
0
] by setting
exp(2x
0
) = −1.
8.7 Show that the concentric KdV equation
u
t
+
1
2t
u + 6uu
x
+ u
xxx
= 0
has the following conservation laws:
(i)
∞
−∞
√
tu dx = constant,
(ii)
∞
−∞
tu
2
dx = constant,
(iii)
∞
−∞
√
txu − 6t
3/2
u
2
dx = constant.
8.8 Consider the KP equation
(u
t
+ 6uu
x
+ u
xxx
) + 3σu
yy
= 0
where σ = ±1. Show that the KP equation has the conserved quantities:
(i) I
1
=
∞
−∞
∞
−∞
udxdy= constant;
(ii) I
2
=
∞
−∞
∞
−∞
u
2
dxdy = constant;
212 Solitons and integrable equations
(iii)
∞
−∞
u
yy
dx = constant
and the integral relation
(iv)
d
dt
∞
−∞
∞
−∞
xu dxdy = αI
2
where α is constant.
8.9 Show that the NLS equation
iu
t
+ u
xx
+ |u|
2
u = 0
has the combined rational and oscillatory solution
u(x, t) =
1 −
4(1 + 2it)
1 + 2x
2
+ 4t
2
exp(it).
8.10 (a) Use the pair of equations
v
xx
+ (λq)v = 0
v
t
= Av + Bv
x
and the method of Section 8.6, with A, B taken to be linear in λ with
the “isospectral” condition λ
t
= 0, to find the nonlinear evolution
q
t
= (q
−1/2
)
xxx
.
This equation is usually referred to as the Harry–Dym equation,
which can be shown to be integrable via “hodograph transforma-
tions” cf. Ablowitz and Clarkson (1991).
(b) Show that the method of Section 6.8 with
v
xx
+ λ(m + α)v = 0,α= κ + 1/4
v
t
= Av + Bv
x
, A = −
u
x
2
, B = u −
1
2λ
and m = u
xx
− u results in the “isospectral” equation
u
t
+ 2κu
x
− u
xxt
+ 3uu
x
− 2u
x
u
xx
− uu
xxx
= 0.
The above equation was shown to have an infinite number of sym-
metries by Fokas and Fuchssteiner (1981); the inverse scattering
problem, solutions and relation to shallow-water waves was con-
sidered by Camassa and Holm (1993b). When κ = 0, the equation
has traveling wave “peakon” solutions, which have cusps; they have
a shape of the form e
−|x−ct|
. The above equation, which is usu-
ally referred to as the Camassa–Holm equation, has been studied
extensively due to its many interesting properties.
Exercises 213
8.11 Consider the Harry–Dym equation
u
t
+ 2(u
−1/2
)
xxx
.
(a) Let u = v
2
x
and show that v satisfies
v
t
=
v
xxx
v
3
x
−
3v
2
xx
2v
4
x
.
(b) Show that the “hodograph” transformation τ = t,ξ = v,η= x, yields
the relations
η
ξ
=
1
v
x
,η
τ
=
−v
t
v
x
.
(c) Using the above relations find the equation
w
τ
= w
ξξξ
−
⎛
⎜
⎜
⎜
⎜
⎜
⎝
3w
2
ξ
2w
⎞
⎟
⎟
⎟
⎟
⎟
⎠
ξ
where w = η
ξ
, which is known to be “integrable” (linearizable).
Hint: see Clarkson et al. (1989).
9
The inverse scattering transform for the Korteweg–de
Vries (KdV) equation
We have seen in the previous chapter that the Korteweg–de Vries (KdV)
equation is the result of compatibility between
v
xx
+ (λ + u(x, t))v = 0 (9.1)
and
v
t
= (γ + u
x
)v + (4λ + 2u)v
x
, (9.2)
where γ is constant. More precisely, the equality of the mixed derivatives v
xxt
=
v
txx
with λ
t
= 0 (“isospectrality”) leads to the KdV equation
u
t
+ 6uu
x
+ u
xxx
= 0. (9.3)
In the main part of this chapter we will use the above compatibility relations
in order to carry out the inverse scattering transform (IST) for the KdV equa-
tion on the infinite interval. This includes a basic description of the direct and
inverse problems of the time-independent Schr
¨
odinger equation and the time
evolution of the scattering data. Special soliton solutions as well as the scatter-
ing data for a delta function and box potential are given. Also included are the
derivation of the conserved quantities/conservation laws of the KdV equation
from IST. Finally, as an extension, the IST associated with the second-order
scattering system and associated time (8.33)–(8.34), discussed in the previous
chapter, is carried out.
For (9.1), the eigenvalues and the behavior of the eigenfunctions as |x|→∞
determine what we call the scattering data S (λ, t) at any time t, which depends
upon the potential u(x, t). The direct scattering problem maps the potential into
the scattering data. The inverse scattering problem reconstructs the potential
u(x, t) from the scattering data S (λ, t). The initial value problem for the KdV
equation is analyzed as follows. At t = 0 we give initial data u(x, 0) which
we assumes decays sufficiently rapidly at infinity. The initial data is mapped to
214