Dynamics near a minimal-mass soliton for a Korteweg-de Vries equation

1 MB·English

by  Jeremy L. Marzuola

DYNAMICS NEAR A MINIMAL-MASS SOLITON FOR
A KORTEWEG–DE VRIES EQUATION
J.L. MARZUOLA, S. RAYNOR, AND G. SIMPSON
3 Abstract. We study soliton solutions to a generalized Korteweg
1
- de Vries (KdV) equation with a saturated nonlinearity, following
0
thelineofinquiryoftheauthorsin[5]forthenonlinearSchr¨odinger
2
equation(NLS).KdVwithsuchanonlinearityisknowntopossess
n
a minimal-mass soliton. We consider a small perturbation of a
a
J minimal-mass soliton and identify a system of ODEs, which mod-
2 elsthebehavioroftheperturbationforshorttimes. Thisconnects
2 nicely to a work of Comech, Cuccagna & Pelinovsky [1]. These
ODEs form a simple dynamical system with a single unstable hy-

]
perbolicfixedpointwithtwopossibledynamicaloutcomes. Apar-
S
P ticular feature of the dynamics are that they are non-oscillatory.
. This distinguishes the KdV problem from the analogous NLS one.
n
i
l
n
[
1. Introduction

2
v WeconsiderageneralizedKorteweg-deVriesequationequationofthe
7
form
7
6
(1.1) u +∂ (f(u))+u = 0
5 t x xxx
.
1 where f is a saturated nonlinearity; that is, f behaves subcritically at
1 high intensities and supercritically at low intensities. An example of
2
such a nonlinearity is
1
v: sp
(1.2) f(s) = .
i
X 1+δsp−q
r with p > 5 and 1 < q < 5, and with δ > 0 as an additional parameter.
a
In the computations we present, we take p = 6, q = 3 and δ = 1.
4
A traveling wave function u(x,t) = φ (x ct), is a soliton solution
c
−
to (1.1) when the profile φ satisfies the ODE
c
(1.3) cφ +f(φ )+∂ φ = 0.
c c yy c
−
Date: January 23, 2013.
ThefirstauthorwaspartiallysupportedbyanIBMJuniorFacultyDevelopment
Award through the University of North Carolina. The second author was partially
supportedbyagrantfromtheSimonsFoundation. Thethirdauthorwassupported
by NSERC, and his contribution to this work was completed under the NSF PIRE
grant OISE-0967140 and the DOE grant DE-SC0002085.
1
2.965
2.964
2.963
2.962
2
2L
(cid:2)
c2.961
φ
(cid:2)
=
2.96
c
N
2.959
2.958
2.957
2.956
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
c
Figure 1. = φ 2 possesses a local minimum for
Nc (cid:107) c(cid:107)L2
the saturated nonlinearity given in (1.2).
If we were considering a noncritical power nonlinearity, f(s) = sp, the
equation would admit solitons of arbitrarily small L2-norm. However,
saturated nonlinearities are more restrictive. For the nonlinearities we
consider, there will be a unique soliton of minimal L2-norm. See Figure
1 for a plot of the L2-norm as a function of c for one instance of (1.2).
This critical soliton is denoted φ .
c(cid:63)
It is known that there exists an interval U R so that there is
⊂
a soliton solution to (1.3) for each c U [3]; moreover, φ (y) is a
c
∈
smooth function of c on U. Indeed, via elliptic theory we see that it
holds generically that φ Cq+2 ; since φ > 0, this then implies that
φ C∞ for p,q Z. Soli∈tons can be interpreted as minimizers of the
∈ ∈
Hamiltonian energy
(cid:90) (cid:90)
1
(1.4) E(u) = ∂ u 2 F( u )dx,
x
2 | | − | |
R R
(cid:82)s
where F(s) = f(t)dt, subject to the fixed momentum condition
0
(cid:90)
1
(1.5) N(u) = u 2dx,
2 | |
R
2
with c acting as the Lagrange multiplier. Equation (1.1) also conserves
the mass:
(cid:90)
(1.6) I(u) = u dx.
R
By evaluating these conserved quantities at the soliton φ , we get func-
c
tions of c:
(1.7) = E(φ ), = N(φ ), = I(φ ).
c c c c c c
E N I
The minimal-momentum soliton is found at the value of c = c such
(cid:63)
that
d
(1.8) = φ ,∂ φ = 0.
c c c c
dcN (cid:104) (cid:105)
This first order condition would also hold at a maximal momentum
soliton.
While the stability of soliton solutions of (1.1) is well understood
away from such critical points, our understanding of the dynamics near
this critical soliton remain incomplete. Due to the saturated nature of
the nonlinearity, the equation is globally well posed in H2, so there is
no finite-time singularity. But does the perturbed soliton converge to
some nearby stable state, disperse, or engage in some other dynamic?
It is known [1], that the minimal-mass soliton itself enjoys a purely
nonlinear instability. The purpose of this work is to further examine
the dynamics of this type of solution.
To better understand these dynamics, we consider perturbations of
φ . Beginning with the ansatz
c(cid:63)
(1.9) u(x,t) = φ (x ct)+p(x ct,t)
c
− −
for a perturbed soliton, we obtain the following evolution equation for
p:
(cid:2) (cid:3)
(1.10) p = ∂ ∂2p+cp f(cid:48)(φ )p +O(p2).
t y − y − c
In order to analyze this equation, we first consider, in Section 2, the
spectrum of the linearized operator, A , where
c
(1.11) A ∂ L , L ∂ +c f(cid:48)(φ ).
c y c c yy c
≡ ≡ − −
An examination of A reveals that its generalized kernel has a dimen-
c
sion of at least two. At c , this dimension increases to least three, and
(cid:63)
could be four under special circumstances. Thus, there will be secular
growth of the perturbation at a critical point generated by the compo-
nentsofthesolutionparalleltoelementsofthegeneralizedkernel. Even
if the perturbation is initially orthogonal to these unstable directions,
the higher order terms will likely generate unstable contributions.
3
Such secular growth is eliminated by making a more general ansatz
that allows c to modulate in time. This is given in Section 3, where we
introduce a three dimensional set of scalar parameters, including the
soliton speed and wave center, and allow them to modulate. This per-
mitsforprojectionawayfromthelinearlyunstablemodes. Weseparate
the projection of p onto the discrete spectrum of A from its projection
c
onto the continuous spectrum. We then discard the continuous projec-
tion component while modulating the remaining parameters to obtain
a two dimensional system of first-order ordinary differential equations.
Finally, in Section 4, we describe the numerical methods we use to
compute key parameters and simulate (1.1). Section 5 presents the
results of these computations and discusses their implications.
2. The Linear Operator
In this section, we survey the spectral properties of A as defined in
c
(1.11), about c = c . For all values of c that admit a soliton, one can
(cid:63)
directly compute
(2.1) A ( ∂ φ ) = 0,
c y c
−
(2.2) A ∂ φ = ∂ φ .
c c c y c
−
For the remainder of this article, we will denote differentiation with
respect to c by (cid:48). The first two elements of the generalized kernel of
A , then, are:
c
(2.3) e = ∂ φ , e = φ(cid:48).
1,c − y c 2,c c
At a minimal-momentum soliton (or indeed, any soliton satisfying the
first order condition (1.8)), there is a third independent function e
3,c(cid:63)
in the generalized kernel of A , which satisfies
c
(2.4) A e = e .
c(cid:63) 3,c(cid:63) 2,c(cid:63)
To see why such a state exists, consider the adjoint operator, A∗ =
c
L ∂ . We immediately compute
c y
−
(2.5) A∗φ = 0, A∗D−1φ(cid:48) = φ ,
c c c c − c
where
(cid:90) y
(2.6) D−1f f.
≡
−∞
Hence we define:
(2.7) g = φ , g = D−1φ(cid:48).
1,c c 2,c c
4
Then, bytheFredholmalternative, weseethatA f = e hasasolution
c 2,c
provided
(cid:90) (cid:90)
d 1
(2.8) e g = φ(cid:48)φ = φ2 = (cid:48)
2,c 1,c c dc 2 c Nc
vanishes, which is condition (1.8). Thus, at a minimal-mass soliton,
there is indeed a third element of the generalized kernel of A , which
c(cid:63)
solves
A e = e .
c(cid:63) 3,c(cid:63) 2,c(cid:63)
Consequently, there is also a third element in the generalized kernel of
A∗ ,
c(cid:63)
(2.9) A∗ g = g
c(cid:63) 3,c(cid:63) 2,c(cid:63)
To see if there is a fourth element in the generalized kernel, we again
consider A f = e and recognize that we would need
c(cid:63) 3,c(cid:63)
(cid:10) (cid:11)
e ,g = e ,A∗ g = A e ,g
(cid:104) 3,c(cid:63) 1,c(cid:63)(cid:105) 3,c(cid:63) c(cid:63) 2,c(cid:63) (cid:104) c(cid:63) 3,c(cid:63) 2,c(cid:63)(cid:105)
(cid:90) (cid:18)(cid:90) y (cid:19)
= e ,g = φ(cid:48) φ(cid:48) dy
(2.10) (cid:104) 2,c(cid:63) 2,c(cid:63)(cid:105) c(cid:63) c(cid:63)
−∞
(cid:90) d 1 (cid:18)(cid:90) y (cid:19)2 1
= φ(cid:48) = ( (cid:48) )2
dy2 c(cid:63) 2 Ic(cid:63)
−∞
to vanish. Even at a critical value of c for which (cid:48) = 0, it is not
(cid:63) Nc(cid:63)
generic to observe (cid:48) = 0. Indeed, for the particular nonlinearity f
Ic(cid:63)
that we consider, our minimal-momentum soliton will not have this
fourth element. It would be of interest to find a nonlinearity that does
satisfythisadditionaldegeneracycondition, andtostudythedynamics
near the resulting doubly-critical soliton.
A particular challenge, discussed below in Section 4, is that some
of these generalized kernel elements, notably e , g and g are
3,c(cid:63) 2,c(cid:63) 3,c(cid:63)
not in L2. While e vanishes exponentially fast at + , it is only
3,c(cid:63) ∞
bounded at . g and g both vanish at , but at + the
−∞ 2,c(cid:63) 3,c(cid:63) −∞ ∞
former is only bounded and the latter grows linearly. The reader can
find a discussion the function spaces in which these kernel functions lie
in [1].
For later use, we remark that away from c , using the implicit func-
(cid:63)
tion theorem, there is a scalar λ and function e , both smooth in c,
c 3,c
such that
(cid:48)
(2.11) (A λ I)e = e , λ Nc .
c c 3,c 2,c c
− ≡ − φ ,e
c 3,c
(cid:104) (cid:105)
5
3. Modulation Equations
To overcome the secular growth due to the generalized kernel, we
now permit the equation parameters to modulate about the extremal
soliton. First, we define the moving frame
(cid:90) t
(3.1) y(x,t) x c(σ)dσ ξ(t).
≡ − −
0
Next, we consider a solution u(x,t) which is a perturbed, modulating
soliton,
(cid:18) (cid:90) t (cid:19)
u(x,t) = φ x c(σ)dσ ξ(t)
c(t)
− −
0
(3.2) (cid:18) (cid:90) t (cid:19)
+p x c(σ)dσ ξ(t),t
− −
0
= φ (y)+p(y,t).
c
Substituting this into (1.1), we get
(3.3) p +φ(cid:48)c˙ ∂ φ ξ˙ p ξ˙
t c − y c − y
1
(cid:0) (cid:1)
= ∂ L p ∂ f(cid:48)(cid:48)(φ )p2 +higher order terms.
y c(t) y c
− 2
We let
(3.4) η(t) = c(t) c ,
(cid:63)
−
and decompose
(3.5) p(x,t) = ζ(t)e +v = ζ(t)e +ζ(t)η(t)e(cid:48) +v +O(ζη2)
3,c 3,c(cid:63) 3,c(cid:63)
as in [1], equations (3.19–3.24), which results in
˙ ˙
v +A v = ξe (η˙ ζ)e (ζ λ ζ)e η˙∂ p+∂ N.
t c 1,c 2,c c 3,c x x
− − − − − −
To close the system, we introduce the constraints
g ,v = g ,v = g ,v = 0.
1,c 2,c 3,c
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
We make several observations about the resulting dynamical system.
The term ∂ N has quadratic and higher-order terms. We will preserve
x
only quadratic terms in our computations. We will also disregard all
coupling to the continuous spectrum, though some of this may also be
of quadratic order. Thus the quadratic order terms we are considering
in our approximation can be taken to be 1ζ2∂ (f(cid:48)(cid:48)(φ )e2 ). Projecting
2 x c 3,c
6
onto the canonical spectral functions, the finite dimensional system
then takes the form
 ξ˙  (cid:10)g ,e(cid:48) 1∂ (f(cid:48)(cid:48)(φ )e2 )(cid:11)
(cid:10) 1,c 3,c − 2 x c 3,c (cid:11)
(3.6) ζ˙η˙ −λζζ = −ζ2Sc(ζ)−1(cid:10)gg2,c,,ee(cid:48)3(cid:48),c − 112∂∂x((ff(cid:48)(cid:48)(cid:48)(cid:48)((φφc))ee232,c))(cid:11),
− c 3,c 3,c − 2 x c 3,c
Under this approximation, ξ is slaved to η and ζ. There is only weak
coupling between ξ and the other parameters through v.
Continuing with the above assumptions,
 (cid:10) (cid:11) 
g ,∂ e g ,e(cid:48) 0
(3.7) Sc(ζ) = Tc +ζ−−(cid:104)(cid:104)g21,,cc,∂xxe33,,cc(cid:105)(cid:105) (cid:10)(cid:10)g12,,cc,e(cid:48)33,,cc(cid:11)(cid:11) 0 = Tc +ζSˆc,
g ,∂ e g ,e(cid:48) 0
−(cid:104) 3,c x 3,c(cid:105) 3,c 3,c
where ( ) = g ,e . Assuming and are O(1) and ζ is suffi-
c jk j,c k,c c c
T (cid:104) (cid:105) T S
cientlysmall, wecanmaketheapproximation , andthusobtain,
c c
S ≈ T
for appropriate vectors R and Q :
c c
 ξ˙  (cid:10)g ,e(cid:48) 1∂ (f(cid:48)(cid:48)(φ )e2 )(cid:11)
(cid:10) 1,c 3,c − 2 x c 3,c (cid:11)
ζ˙η˙ −λζζ = −ζ2Sc(ζ)−1(cid:10)gg2,c,,ee(cid:48)3(cid:48),c − 112∂∂x((ff(cid:48)(cid:48)(cid:48)(cid:48)((φφc))ee232,c))(cid:11)
− c 3,c 3,c − 2 x c 3,c
(cid:10)g ,e(cid:48) 1∂ (f(cid:48)(cid:48)(φ )e2 )(cid:11)
(cid:10) 1,c 3,c − 2 x c 3,c (cid:11)
= −ζ2Tc−1(cid:10)g2,c,e(cid:48)3,c − 21∂x(f(cid:48)(cid:48)(φc)e23,c)(cid:11)+O(ζ3)
g ,e(cid:48) 1∂ (f(cid:48)(cid:48)(φ )e2 )
3,c 3,c − 2 x c 3,c
= ζ2 −1R +O(ζ3)
− Tc c
= ζ2Q +O(ζ3)
c
−
Therefore, the leading order equations, subject to these approxima-
tions, are
(3.8a) ξ˙ = Q ζ2,
1,c
−
(3.8b) η˙ ζ = Q ζ2,
2,c
− −
(3.8c) ζ˙ λ ζ = Q ζ2.
c 3,c
− −
Making the Taylor expansions of λ the Q ’s about c = c , and omit-
c j,c (cid:63)
ting the ξ equation, we obtain the quadratically nonlinear ODE system
(3.9a) η˙ ζ = Q ζ2
− − 2,c(cid:63)
(3.9b) ζ˙ λ(cid:48) ηζ = Q ζ2.
− c(cid:63) − 3,c(cid:63)
Critical points of the system are found at ζ = 0, η = η where η is
0 0
arbitrary, and at ζ = 1 , η = Q3 . The latter isolated critical point
Q2 λ(cid:48)c(cid:63)Q2
is a saddle point in the first quadrant.
7
0.6
0.5
0.4
0.3
0.2
ζ 0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5 0 0.5 1 1.5 2 2.5 3
η
Figure 2. A plot of a representative phase plane dia-
gram. The hyperbolic fixed point is indicated by the
◦
while the nonisolated critical points are indicated by the
solid line ζ = 0.
For the ζ = 0 critical points the linearized problem is
(cid:18) (cid:19)
0 1
0 λ(cid:48) η
c(cid:63) 0
The eigenvalues are λ(cid:48) η and 0. Thus, depending on the sign of λ(cid:48) η ,
c(cid:63) 0 c(cid:63) 0
the solution is either linearly stable or unstable. It is worth noting
in this context that λ = −Nc(cid:48) . Thus, λ(cid:48) = −Nc(cid:63)(cid:48)(cid:48) since (cid:48) = 0.
c (cid:104)φc,e3,c(cid:105) c(cid:63) (cid:104)φc,e3,c(cid:105) Nc(cid:63)
Also note that, according to [1], near c we have that φ ,e > 0.
(cid:63) c 3,c
(cid:104) (cid:105)
Thus, we expect that the sign of λ(cid:48) depends solely on whether we
c(cid:63)
are at a minimal or maximal soliton. In each case, we expect to see
that, depending on the sign of η , the critical point at (0,η ) is either
0 0
linearly stable or linearly unstable. There is semi-stability at (0,0),
depending on the sign of the initial perturbation η . See Figure 2 for
0
a representative saturated KdV phase plane diagram.
4. Computational Methods
In this section we briefly outline the computations needed to make
a comparison between (3.9) and the KdV equation, (1.1). Motivated
8
by (3.2), (3.4) and (3.5), we will take initial conditions of the form
(4.1) u = φ (x)+η e +ζ e .
0 c(cid:63) 0 2,c(cid:63) 0 3,c(cid:63)
4.1. Spectral Computations. Tocomputethecoefficientsappearing
in (3.9), the Q ’s and λ(cid:48) , we must compute
j c(cid:63)
The generalized kernels of A and A∗ from which we can get
• c(cid:63) c(cid:63)
the inner products g ,e which the matrix comprises;
(cid:104) j,c(cid:63) k,c(cid:63)(cid:105) Tc(cid:63)
∂ e and e(cid:48) to obtain the R vector, which, with , allows
• x 3,c(cid:63) 3,c(cid:63) c(cid:63) Tc(cid:63)
us to obtain the Q vector;
c(cid:63)
λ(cid:48) , which can be obtained by differentiating (2.11) and then
• c(cid:63)
computing φ(cid:48)(cid:48) .
c(cid:63)
The first few elements of the generalized kernel, e = ∂ φ , e =
1,c(cid:63) x c(cid:63) 2,c(cid:63)
φ(cid:48) and g = φ are readily obtained using the the sinc discretization
c(cid:63) 1,c(cid:63) c(cid:63)
method previously used by the authors in [5]. Briefly, this approach
solves equations like
(4.2) L f = g
c
using a sinc discretization of f, g and L , provided g is localized.
c
Derivatives of functions are easy to obtain using the discretized dif-
ferentiation matrices, and L2 inner products are just finite dimensional
inner products, multiplied by the uniform grid spacing. To obtain the
minimal-momentum soliton, we use a root-finding algorithm to solve
(cid:48) = 0. See [4,6,8,9] for additional details on the sinc discretization
Nc
method.
Computing the other elements requires a bit more care as they are
not L2-localized. First, let
(cid:90) y
(4.3) Θ(y) e .
≡ 2,c(cid:63)
+∞
Then e solves
3,c(cid:63)
(4.4) L e = Θ, lim e (y) = 0.
c(cid:63) 3,c(cid:63) y→+∞ 3,c(cid:63)
Given sinc discretization of φ(cid:48) , we can readily integrate to obtain Θ
c(cid:63)
using the techniques given in [7]. The use of such a quadrature tool
was not required in [5] as all of the NLS kernel elements belonged to
L2. This also gives us g .
2,c(cid:63)
Assuming that e grows, at most, algebraically at we can drop
3,c(cid:63) −∞
the f(cid:48)(φ )e at large negative values of y term to estimate
c(cid:63) 3,c(cid:63)
(4.5) ∂2e +c e (cid:48) = 0.
− x 3,c(cid:63) (cid:63) 3,c(cid:63) ≈ −Ic(cid:63) (cid:54)
9
This approximation implies that e is actually bounded at :
3,c(cid:63) −∞
1
(4.6) lim e = (cid:48) .
y→−∞ 3,c(cid:63) −c(cid:63)Ic(cid:63)
One method of computing e is to split it into a piece which has the
3,c(cid:63)
above asymptotics, and a spatially localized piece,
(4.7) e e(1) +e(2),
3 3 3
≡
where e(1) L2 and e(2) vanishes at + . Using the above estimate,
3 3
∈ ∞
we set
e(2) (x) e− χ (x)
3,c(cid:63) ≡ 3,c(cid:63) · −
(4.8) 1 1
= (cid:48) (1+tanh( x)).
−c Ic(cid:63) · 2 −
(cid:63)
We have some flexibility in selecting χ . The essential feature is that
−
it should not contribute anything at + , while capturing the known
∞
asymptotic behavior at . We then solve
−∞
(4.9) L e(1) = Θ L e(2) .
c(cid:63) 3,c(cid:63) − c(cid:63) 3,c(cid:63)
As the righthand side is now localized at both , we obtain e(1) .
±∞ 3,c(cid:63)
Theadjointproblemissimilar, butrequiresslightlymorecare. First,
we solve
(4.10) L h = g
c(cid:63) 3,c(cid:63) 2,c(cid:63)
and then integrate to obtain g . While e was asypmtotically con-
3,c(cid:63) 3,c(cid:63)
stant, g will have linear growth at + . The other function which
3,c(cid:63) ∞
requires such an asymptotic splitting is e(cid:48) ,
3,c(cid:63)
To compute λ(cid:48) , we compute
c(cid:63)
(cid:10) (cid:11) (cid:10) (cid:11)
(cid:48)(cid:48) φ ,φ(cid:48)(cid:48) + φ(cid:48) ,φ(cid:48)
(4.11) λ(cid:48) = Nc(cid:63) = c(cid:63) c(cid:63) c(cid:63) c(cid:63) ;
c(cid:63) − φ ,e − φ ,e
(cid:104) c(cid:63) 3,c(cid:63)(cid:105) (cid:104) c(cid:63) 3,c(cid:63)(cid:105)
φ(cid:48)(cid:48) is L2-localized and obtained by solving
c(cid:63)
L φ(cid:48)(cid:48) = 2φ(cid:48) +f(cid:48)(cid:48)(φ )(φ(cid:48) )2.
c(cid:63) c(cid:63) − c(cid:63) c(cid:63) c(cid:63)
Computing the various inner products, we obtain the matrix and
Tc(cid:63)
the vector R , from which we can solve for Q . This provides us with
c(cid:63) c(cid:63)
all coefficients in the ODE system.
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