The aim of this section is to establish a formula that generalizes formula (1.9) to the case of non-constant capillarity and a sufficient condition under which solitary waves of large-amplitude in the Euler-Korteweg equation
Vt−Uy = 0,
Ut+p(V)y =−(κ(V)Vyy+1
2(κ(V))yVy)y
(1.18)
are stable. Let us briefly recall some notations from Section 1.4. The equations governing the profile (v, u)(ξ) connecting (v∗,0) read
−cv0=u0,
−cu0+p(v)0=−(κ(v)v00+1
2(κ(v))0v0)0. This equation is equivalent to
κ(v)v00+ 1
2(κ(v))0v0=−p(v) +p(v∗)−c2(v−v∗)
=:−∂F(v, c)
∂v
and possesses (cf. [8]) a first integral given by
Then any solitary wave is stable.
Proof. Since κ(v) >0,we still have Lemma 1.5 assuring the existence of solitary waves con-necting v∗ with speed csuch that p0(∞)< c2 <−p0(v∗).
The moment of instability is (cf. [8]) m(c) =
which, due to
∂
∂v
(κ(v)(−2F(v, c)))1/2
= κv(v)(−2F(v, c))1/2
2(κ(v))1/2 −κ(v)1/2Fv(v, c) (−2F(v, c))1/2, simplifies to
m0(c) =−4
Z κ(vm(c)−u2)1/2
Fc(vm(c)−u2, c) (−2F(vm(c)−u2, c))1/2 u du.
The second derivative can be written in the form m00(c)
2 =
Z vm(c) v∗
A(v) +B(v)
(κ(v))1/2(−2F(v, c))3/2 dv with functions
A(v) :=F(v, c)κv(v)v0m(c)Fc(v, c),
B(v) :=κ(v) (v−v∗) 2F(v, c) (v−v∗) + 2cv0m(c)
−c(v−v∗) Fv(v, c)vm0 (c) +Fc(v, c) . The following trivial proposition follows from Theorem 1.2 and finishes the proof of our theo-rem.
Proposition 1.3. Fix c > 0. The functions A(v) and B(v) are non-negative for all v ∈ (v∗, vm(c)).
Proof of the proposition. B(v) can be written in the form B(v) =κ(v)(v−v∗)T(v) with T(v)>0 defined in Proposition 1.2.
A(v) is positive since for all v ∈(v∗, vm(c)), κv(v) ≥0 by assumption and Fc(v, c) >0 while F(v, c) and v0m(c) are both negative.
Remark. In the proof we showed positivity of the integrand by showing that it consists of a sum of two positive functions, one of which – B(v) – is positive independently of the sign of κv(v). It would be interesting to find different conditions onκ (and perhaps also on p), such that the sum of A(v) and B(v) is positive when κv(v)>0.In particular in the case that
κ(v) = 1 4v4
(1.18) covers the generalized Gross-Pitaevskii equations; this will be a future project.
1.7 (In-)Stability of solitary waves in the generalized KdV equation with polynomial flux
In this part we consider the gKdV equation with polynomial flux of degree 4, i.e.,
Vt+p(V)x+Vxxx= 0 (1.19)
with
p(V) =V2+aV3+bV4. (1.20)
Without loss of generality we deal with solitary wavesv(x−ct) of speedchomoclinic tov∗ = 0.
The equations governing the profiles are
v00=cv−p(v)
=−∂F(v, c)
∂v
(1.21)
with
F(v, c) :=−1
2cv2−f(v) where
−df(v)
dv =p(v) i.e.,f(v) =−v3 3 −av4
4 −bv5 5 .
The endpoint v∗= 0 is a saddlepoint of (1.21) if and only if c >0 since the eigenvalues of the linearization at the fixed point v∗ areλ1/2=±p
c−p0(v∗) =±√ c.
We will prove the following theorems.
Theorem 1.6. Consider the generalized Korteweg-de Vries withpsatisfying (1.20)withb= 0.
Then all solitary waves are stable.
Theorem 1.7. Consider the generalized Korteweg-de Vries equation with p satisfying (1.20) with b6= 0.Then there are solitary waves that are not stable.
Remark. Note that Theorem 1.6 is not new. Kalisch and Nguyen establish the same result in [33]. It might be seen as a slight advantage that our proof does not use the explicit form of the solitary wave and therefore applies easier to other equations.
Proof of Theorem 1.6. First, letbnot necessarily vanish.
In equation (1.19) with (1.20), there occur positive solitary waves (waves of elevation) and negative solitary waves (waves of depression); we will focus on the positive case, first. Call vm(c)>0 the maximum of the wave. Then, we can expresscin terms ofF sinceF(vm(c), c) = 0 for all c.Thus, we have
c= 2
3vm(c) +a
2vm(c)2+2b
5vm(c)3. (1.22)
Similarly, we obtain an expression for v0m(c) : vm0 (c) = −Fc(vm(c), c)
Fv(vm(c), c)
= 1
2(13 +a2vm(c) +3b5vm(c)2).
As before, the convexity resp. concavity of the moment of instability decides about stability resp. instability of the corresponding solitary wave. In particular, recall Theorem (∗) in Section 1.1. With the notations of Section 1.1 we have that the moment of instability reads in our case Differentiating, we obtain (withv∗ = 0)
m00(c) =
Then the following is obvious: If for all v∈(0, vm(c)), T(v)<0,then the corresponding wave is unstable while positivity ofT(v) for all v∈(0, vm(c)) implies stability of the wave v.
Trivial manipulations show thatT(v) has the same sign as
Q(v) :=v2(vm(c)−v) 12bvm(c)2−6bvm(c)v+ 15vm(c)a−6v2b+ 20
Obviously, then v2 < 0 and v1 > vm(c), which implies positivity of the second derivative of m(c) in these cases. The case a >0, b= 0 is trivial.
Let now a, b≤0.Sincec >0 and due to (1.22), we have that 15avm(c)≥12bvm(c)2−20.
From this we can conclude that for all v∈(0, vm(c)) the last factor of Q(v) satisfies 12bvm(c)2−6bvm(c)v+ 15vm(c)a−6v2b+ 20≥6bv(−vm(c)−v)>0.
Thus,T(v)>0.Both cases together imply hence stability of all solitary waves of elevation in (1.19) for a∈R, b= 0.
Let now b= 0.To treatwaves of depression note thatv is a wave of depression for (1.19) with flux p(v) if and only if ˜v=−v is a wave of elevation of
˜
vt+ ˜p(˜v) + ˜vxxx = 0 with ˜p(˜v) =−p(−v).In our case
˜
p(˜v) =a˜v3−v˜2. Dropping the tilde, the profile equation reads
v00=cv−p(v)
=−∂F(v, c)
∂v where F is now
F(v, c) =−c 2v2+ a
4v4−1 3v3.
Orbits homoclinic to v∗ = 0 can only occur ifa >0 as otherwise the fluxp is strictly concave for all v >0.In this case, the speed csatisfies
c= avm(c)2
2 −2vm(c) 3 >0, which leads to
vm(c)> 4
3a. (1.23)
Furthermore,
vm0 (c) = 1 2(avm2(c)−13)
. As before m00(c) reads
m00(c) =
Z vm(c) v∗
−4F(v, c)v vm0 (c) +v2 Fv(v, c)v0m(c) +Fc(v, c)
(−2F(v, c))3/2 dv
while Q(v) simplifies to
Q(v) =v2(vm(c)−v)(−4 + 3avm(c)).
Q(v) can easily be seen to be positive for allv ∈(0, vm(c)) due to (1.23). Again, this implies stability of all corresponding solitary waves and finishes the proof of Theorem 1.6.
Remark. As a byproduct of our proof we obtain that all solitary waves of elevation of Vt+ (V2+aV3+bV4)x+Vxxx = 0
with either
a, b≥0 or
a, b≤0 are stable.
Proof of Theorem 1.7. Consider (1.19) with (1.20) and b <0.By scaling inx, tand V,we see that it suffices to consider the case b=−1.We will show that small-amplitude solitary waves of depression are not stable. For this purpose, again replacep(v) by−p(−v).Then, the profile equation is given by
v00 =cv−p(v)
=cv−v4−av3+v2.
Applying the same strategy as in the proof of Theorem 1.6 leads to c= 2vm(c)3
5 + avm(c)2
2 −2vm(c) 3 >0 which implies
vm(c)>−5 8a+
r25 64a2+5
3.
In particular, for a ∈ R fixed, the maximum of the wave has a lower bound β > 0 which is independent ofc,and thus, the amplitude of the wave does not tend to zero forc→0.This is in clear contrast to the small amplitude waves of Sections 1.3 and 1.4 (and might be a reason for the instability of these waves).
Furthermore, we have
v0m(c) = 1
2 3
5vm(c)2+avm2(c) −13 .
With the notation of the proof of Theorem 1.6 we have that convexity of the moment of
instabilitym(c) can be reduced to positivity of the sign ofQ(v).Again, if for allv∈(0, vm(c)), Q(v)<0,then the corresponding wave is unstable whileQ(v)>0 for allv∈(0, vm(c)) implies stability. Here, Q(v) is given by
Q(v) =v2(vm(c)−v) −6v2−6vvm(c)−20 + 12vm(c)2+ 15avm(c) .
The idea is now to investigate the limitc→0.Even if forc= 0 there is no solitary, and hence, m00(0) has no clear interpretation, all quantities introduced so far are smooth functions of c;
in particular, observe that if vm(c) =−58a+ q25
64a2+53 (this case corresponds to c= 0), then the last factor (deciding the sign of Q(v)) reduces to
−6v2+15 4 va−
r225
16 a2v2+ 60v2.
This term, however, is strictly negative for allv >0.Hence also forv∈(0, vm(c)).This implies that there is a γ >0 such thatm00(0)≤γ <0.By continuity of the integrand and integral we can conclude m00(c)<0 also for small c >0.