Solitary Waves and
Undercompressive Shocks
Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften
vorgelegt von
Johannes H¨owing
an der
Universit¨at Konstanz
Mathematisch-Naturwissenschaftliche Sektion Fachbereich Mathematik und Statistik
Tag der m¨undlichen Pr¨ufung: 30.01.2013 Referenten:
Prof. Dr. Heinrich Freist¨uhler (Universit¨at Konstanz)
Prof. Dr. Sylvie Benzoni-Gavage (Universit´e Claude Bernard Lyon I)
Abstract 1
Introduction 3
1 Stability of solitary waves 11
1.1 Stability of large-amplitude solitary waves in the generalized KdV equation . . 11
1.2 Stability of large-amplitude solitary waves in the generalized Boussinesq equation 17 1.3 (In-)Stability of small-amplitude solitary waves in the generalized KdV equation 22 1.4 (In-)Stability of small-amplitude solitary waves in the Euler-Korteweg equation 23 1.5 Remarks . . . 28
1.6 Stability of large-amplitude solitary waves in the Euler-Korteweg equation . . . 29
1.7 (In-)Stability of solitary waves in the generalized KdV equation with polynomial flux . . . 32
1.8 Stability of large- and small-amplitude solitary waves in the generalized BBM equation . . . 36
2 Stability of undercompressive shocks 41 2.1 Situation and results . . . 41
2.2 Links between the full system and the associated system . . . 44
2.3 Proof of Part (i) of Proposition 2.1 . . . 47
2.4 Proof of part (ii) – Outmost regime . . . 50
2.5 Proof of part (ii) – Outer regime I . . . 53
2.6 Proof of part (ii) – Outer regime II . . . 56
2.7 Proof of part (ii) – Outer regime III . . . 59
2.8 Proof of Theorem 2.2 . . . 61
2.9 Appendix – Notes from Geometric Singular Perturbation Theory . . . 65
Bibliography 71
Zusammenfassung in deutscher Sprache 77
This dissertation establishes spectral stability of traveling waves in two different settings. In the first part, we prove stability of solitary waves in Hamiltonian partial differential equations, notably in the generalized Korteweg-de Vries equation, the generalized Boussinesq equation, and equations which are closely related with both. Under natural and physically meaningful assumptions on the nonlinearity, we establish stability of large- and small-amplitude solitary waves in this context. In the second part, we prove spectral stability of small undercompressive shocks in viscous systems of non-strictly hyperbolic conservation laws. Here, at a pointv∗∈Rn, the nonlinearity f(v) has the property that two of the eigenvalues of Df(v∗) coincide. We show that in this situation, which frequently arises in applications, the essential dynamics are governed by the dynamics of an associated two dimensional system. We finish this thesis with a careful investigation of the latter.
This thesis establishes spectral stability of traveling waves in partial differential equa- tions. We focus on two different settings, namely, solitary waves in Hamiltonian differential equations and undercompressive shocks in systems of viscous conservation laws.1
In the first chapter, our motivation is the following. While the stability of solitary waves for the generalized Boussinesq equation
Vt−Ux= 0,
Ut+p(V)x=−κVxxx with p0 <0 and the generalized Korteweg-de Vries (gKdV) equation
Vt+p(V)x=−κVxxx has been studied in great completeness for the case that
p00(V) =Vq−2
with some integer q−2 ≥0, the case of more general p seems to have escaped attention. It is, however, interesting from the point of view of applications. In particular, the generalized Boussinesq equation with
p(V) =kV−γ with γ ≥1, k >0 (0.1)
describes the flow of an inviscid isothermal ideal (barotropic) fluid with capillarity.
Moreover, when
p(V) =V3/2+αV2 withα∈R, (0.2)
the gKdV equation is called (generalized) Schamel equation and describes ion acoustic waves [48].
1Parts of this introduction, namely the parts that present the contents of Sections 1.1 to 1.5, were published in [25].
We mainly establish the following new stability results:
Theorem 1.1. Consider the generalized Korteweg-de Vries equation with a smooth function p satisfyingp00>0 and p000 ≤0. Then any solitary wave is stable.
Theorem 1.2. Consider the generalized Boussinesq equation withp:R→Rorp: (0,∞)→R satisfying p0 <0, p00>0 and p000≤0.Then any solitary wave is stable.
These results complement the findings of [10] and [9], respectively; the only overlap of Theorems 1.1 and 1.2 with those in [10, 9] consists exactly in the quadratic nonlinearity p000 ≡0.Note, however, that Theorems 1.1 and 1.2 are not restricted to pure power laws. Notably, our result implies stability of all corresponding solitary waves in the Boussinesq equation withpsatisfying (0.1) and in the gKdV equation withp satisfying (0.2).
Motivated by [18], we prove without any assumption on the third derivative of p stability of small-amplitude solitary waves. Assuming that p00(v∗) > 0 for some fixed state v∗ ∈ R, we establish stability of small-amplitude solitary waves homoclinic to v∗ for the gKdV equation (Theorem 1.3), and a generalization of the Boussinesq equation with variable capillarityκ(V)>
0, the so-called Euler-Korteweg equation (Theorem 1.4) which read in Lagrangian coordinates (cf. [8])
Vt−Ux= 0,
Ut+p(V)x=−(κ(V)Vxx+1
2(κ(V))xVx)x withp0 <0.
Theorems 1.1 to 1.4 will be proved in Sections 1.1 to 1.4, respectively. Sections 1.3 and 1.4 also contain Theorems 1.3 a and 1.4 a which provide some stability / instability findings in situations which are similar to, but different from those considered in Theorems 1.3 and 1.4.
Finally, Section 1.5 presents obvious examples which show that strict convexity of p alone cannot preclude instability for general amplitudes.
In the remaining sections of the first chapter, we prove several theorems that can be seen as applications and generalizations of the preceding results. In Section 1.6 we extend Theorem 1.2 to the Euler-Korteweg equation (Theorem 1.5) to prove stability of large-amplitude solitary waves, withpsatisfying the assumptions of Theorem 1.2 andκwithκ0 ≥0.It should be noted that a physically more meaningful condition seems to be that κis a decreasing function of the specific volume.
In Section 1.7 we prove (in-)stability of solitary waves in the gKdV equation with polynomial
flux of at most fourth order, i.e., p has the form
p(V) =V2+aV3+bV4 witha, b∈R.
Solitary waves in these equations occur in several physical contexts (cf., for example, [13, 31, 34]); of particular interest is the case b = 0, which is sometimes called the extended KdV or Gardner equation. In the case b = 0, we prove stability of all solitary waves (Theorem 1.6) while for b <0 we prove occurrence of unstable waves (Theorem 1.7). Note that Theorem 1.6 is not new (cf. [33]); our proof, however, does not use the explicit form of the solution which might be seen as a slight advantage. The unstable waves we identify in the proof of Theorem 1.7 are small-amplitude waves of depression. As a byproduct we show that, in the caseab >0, all solitary waves of elevation are stable.
Finally, in Section 1.8 we prove stability of small- and large-amplitude solitary waves for the generalized Benjamin-Bona-Mahony (gBBM) or regularized Long Wave equation (Theorems 1.8 and 1.9)
Ut+p(U)x−Uxxt= 0
with psatisfying similar assumptions as in the preceding theorems.
The proofs of all theorems are based on a tool which is called moment of instability.
The idea is to use the energy conserving nature of the equation to show stability of waves and goes back to Boussinesq [12] and was made rigorous by Benjamin [2], Bona [11] and Grillakis et al. [23]. The rough argument is the following. Consider a Hamiltonian partial differential equation
ut=JδH[u(t)] (0.3)
whereJ is a skew-symmetric linear operator andHis a functional, the so-called Hamiltonian or energy of the system, which is a conserved quantity for the flow of (0.3). With a further conserved quantityI[u] the profile equation for a solitary wave Φcis given by
Jδ(H[Φc] +cI[Φc]) = 0.
Let now ∆H[u,Φc] := H[u]− H[Φc]≥0 and assume that d1(u(0),Φc) is small at t = 0 for a metric d1; further let α, β >0 exist such that
α(d1(u(0),Φc))2 ≥∆H[u,Φc] att= 0 and β(d2(u(t),Φc))2 ≤∆H[u,Φc] fort≥0
with another (pseudo-)metric d2. Then, the fact that also ∆H is a conserved quantity is the key idea to show stability. The lower and upper bounds of ∆Hare related to the convexity of
the moment of instability m(c) defined as
m(c) := (H+cI)[Φc].
We give a precise analysis for each equation in the respective sections of Chapter 1.
A primary motivation for the topic of the second chapter comes from the following observation. In systems of viscous conservation laws
vt+f(v) =vxx
with v ∈ Rn, undercompressive shocks are shocks for which the classical Lax condition does not hold; the number of outgoing characteristics being n instead ofn−1 (if nis the number of unknowns). For such shocks the existence of a traveling wave profile is not implied by the Rankine-Hugoniot relations, but by an extra condition often called a kinetic rule (cf. [17]).
Undercompressive shocks arise in multiphase fluid dynamics [1, 3, 5, 6, 51, 52], magnetohydro- dynamics [1, 19, 35, 36, 40, 41], and other contexts. We focus on the case of small-amplitude undercompressive shocks that occur in the neighborhood of umbilic points; these are points where two of the eigenvalues of the Jacobian of f coincide. Marchesin, Plohr, Temple et al. and Schaeffer and Shearer have studied the prototypical situation of quadratic systems [27, 28, 29, 30, 45, 46, 47, 49], which will play a key role in our investigation.
Theorem 2.1. Let
vt+f(v)x =vxx (0.4)
be a non-strictly hyperbolic system of conservation laws with an umbilic pointv∗.Consider also the associated quadratic system
w1 w2
!
t
+ α1w21+β1w1w2+γ1w22 α2w21+β2w1w2+γ2w22
!
x
= w1 w2
!
xx
(0.5) with
αi = 1 2
∂2fi
∂u2k(v∗), βi = ∂2fi
∂ukuk+1(v∗), γi= 1 2
∂2fi
∂u2k+1(v∗), for i= 1,2.
Now, if (ˆv−,ˆv+, φ0) is an undercompressive shock in (0.5), then (0.4) admits a family of undercompressive shocks
(v−ε, vε+, φε) with
ε→0lim 1 εφεx
ε
= (0, . . . ,0, φ0(x),0, . . . ,0).
If 0 < ε ≤ ε0 with ε0 > 0 sufficiently small these undercompressive shocks (v−ε, vε+, φε) are spectrally stable.
Note that spectral stability implies linear and nonlinear stability (cf. [24, 44]).
Theorem 2.1 is motivated by Theorem 1 in [18]. Freist¨uhler and Szmolyan establish stability of Lax-shocks that are associated with a genuinely nonlinear mode in strictly hyperbolic systems of conservation laws. Briefly, φis spectrally stable if the ODE system
κp=p00−((df(φ)−sI)p)0 (0.6)
has only one solution p(ξ) with p(±∞) = 0 in H:={κ∈C: Re κ≥0},namely at κ= 0,and 0 is a simple eigenvalue. Since the amplitude of the wave is small, after appropriate scaling, one can apply geometric singular perturbation theory [15, 53] to reveal slow and fast dynamics in the eigenvalue problem. Especially for |κ| 1,careful investigations of the dynamics are necessary. In the proof we follow ideas presented in [18, 20, 54] where spectral stability of noncharacteristic Lax-shocks in strictly hyperbolic systems of viscous conservation laws close to an isolated point, where genuine nonlinearity is violated, is proven.
In contrast to the situation in [18, 20, 43, 54] where the associated system is the (modified) Burgers equation, in which all shocks are known to be spectrally stable, the situation here is more delicate. Stability of small-amplitude waves in (0.4) can only be established if the limiting objects in (0.5) are stable. This is the statement of Theorem 2.2, which is also of independent interest.
Theorem 2.2. All undercompressive shocks in (0.5) are spectrally stable.
To my knowledge there is only one stable undercompressive shock known in quadratic systems with an umbilic point, namely in the complex Burgers equation [38]. Theorem 2.2 was part of my diploma thesis [26].
In the proofs of both theorems we use Evans function techniques. The Evans function is a scalar analytic functionE(κ) depending on the spectral parameterκ; zeros ofE correspond to solutions of (0.6), and the order of the root is equal to the algebraic multiplicity of the eigen- value (cf. [18] and references therein). Moreover, the proof of Theorem 2.1 uses geometric singular perturbation theory in many places; an appendix with some notes from this theory completes Chapter 2.
Even though solitary waves in Hamiltonian system and undercompressive shocks in non-strictly hyperbolic systems are quite different in nature, there are close links between Parts I and II of this thesis. Apart from the obvious fact that in both parts we deal with stability of traveling waves, there is also a nice connection between the tools we use, namely
the Evans function and the moment of instability. This connection was pointed out in [42] for the gKdV equation and in [55, 4] for the generalized Boussinesq / Euler-Korteweg equation.
The idea is the following: In both cases, we have that the eigenvalue problem associated with a solitary wave Φc can have at most one solution in{κ∈C: Reκ >0}; thisκ must be real.
Furthermore, κ = 0 is an eigenvalue of order two, i.e., E(0) = E0(0) = 0. Finally, note that E(∞)>0.It is obvious, then, that
E00(0)>0 ⇒ there is no solutionκ of the eigenvalue problem withκ >0, while
E00(0)<0 ⇒ there is exactly one solution κ of the eigenvalue problem withκ >0.
The following relation then connects both methods:
sgnm00(c) = sgnE00(0).
Another common theme is the use of scaling adapted to the small-amplitude limit. In both parts, the scaling reveals convergence, in the small-amplitude limit, to limiting reference ob- jects. These reference objects determine the essential dynamics in the small-amplitude limit.
Acknowledgments
I would like to thank my supervisor Heinrich Freist¨uhler for his continuous support during the last years. I hope that I was able adopt some of his great mathematical intuition and clarity.
I am very grateful to Sylvie Benzoni-Gavage for agreeing to write a report.
Furthermore, I would like to thank my friends and colleagues Andreas Klaiber, Matthias Kotschote and Johannes W¨achtler for numerous discussions and helpful comments. I am also very grateful to Linda Sass and Thomas H¨owing for many helpful comments and careful read- ing of the manuscript.
I gratefully acknowledge the support of the University of Konstanz and, in particular, of the members of the Department of Mathematics and Statistics.
The work for this project was funded by Deutsche Forschungsgemeinschaft through a grant to the University of Konstanz and by the Cusanuswerk; to both organizations I am indebted.
In this chapter we prove stability of solitary waves in Hamiltonian differential equations. Sec- tions 1.1 to 1.5 were published in [25], while Sections 1.6 to 1.8 present hitherto unpublished results.
1.1 Stability of large-amplitude solitary waves in the generalized KdV equation
Consider the generalized Korteweg-de Vries (gKdV) equation
Vt+p(V)x=−κVxxx (1.1)
with
p00 >0 and p000 ≤0, (1.2)
and a positive constantκ >0 which we without loss of generality from now on assume to equal one. We are interested in solitary wave solutions, i.e., (non-constant) traveling wave solutions
V(x, t) =v(x−ct) with lim
ξ→∞v(ξ) = lim
ξ→−∞v(ξ).
Obviously,vis a solitary wave if and only if it is a homoclinic orbit of the ordinary differential equation
−cv0+p(v)0 =−v000. With v∗ := lim
|ξ|→∞v(ξ),this is equivalent to
v00=cv−cv∗−p(v) +p(v∗). (1.3) Define f (up to a constant) by the relation
−df(v)
dv =p(v)
and let
F(v, c) :=−1
2c(v−v∗)2−f(v) +f(v∗)−p(v∗)(v−v∗).
Then (1.3) reads
v00=−∂F(v, c)
∂v ; a first integral is given by
I(v, v0) = 1
2v02+F(v, c). (1.4)
Lemma 1.1. Consider (1.1) with p satisfying (1.2)and fix an arbitrary state v∗ ∈R.There is a smooth bijection vm : (p0(v∗), p0(∞))→ (v∗,∞) such that the following holds. A solitary wave homoclinic to v∗ and of speed c exists if and only if p0(v∗)< c < p0(∞) and vm(c)> v∗. Proof. As
∂2F
∂v2(v∗, c) =−c+p0(v∗), (v∗,0) is a saddle point for
v0 =w,
w0 =−∂F(v, c)
∂v
if and only if c > p0(v∗).Consider now only such c.In view of
∂F
∂v(v, c) =−c(v−v∗) +p(v)−p(v∗), the following is immediate. In any case,
F(v, c)<0 for all v < v∗;
ifc≥p0(∞),then alsoF(v, c)<0 for all v > v∗; if, however, c < p0(∞),then F(vm(c), c) = 0
for a unique value vm(c).
Definition 1.1. A traveling wave v of (1.1) is called orbitally stable if for each ε >0, there exists a δ >0such that for any solutionV ∈v+C([0,∞);Hs(R)), s >3/2,of (1.1), closeness at initial time,
kV(·,0)−v(·)kH1(R)< δ, implies closeness at any time
σ∈infR
kV(·, t)−v(·+σ)kH1(R) < ε for all t >0.
Recall that the existence time of any solution V ∈C([0, T), Hs(R)) withs > 3/2 that has a uniform H1 bound is ∞ (Theorem 2.1 of [10]).
The main result of this section is
Theorem 1.1. Consider (1.1) with a smooth function p satisfying (1.2). Then any solitary wave v for (1.1)is stable.
Proof. In order to show nonlinear orbital stability, we apply a technique that was developed in [23] and first used for the gKdV equation in [10]; cf. also [42]. The stability of solitary waves is decided by the convexity behavior of so-called moment of instability. To recapitulate this very briefly, recall first that equation (1.1) has the Hamiltonian form
Vt=JδH[V] with H[V] = Z
H(V) where
H(V) = 1
2(Vx)2+f(V)−f(v∗) +p(v∗)(V −v∗), and J =∂x. Introduce the functional
I[V] = Z
I(V) with I(V) = 1
2(V −v∗)2
and denote for the moment the traveling wave by Φc.Then the profile equation (1.3) reads Jδ(H[Φc] +cI[Φc]) = 0
The following theorem is due to [10, 42]:
Theorem (∗). Let X=H1(R). Assume
(A1) (Local Well-posedness) For any µ >0, there exists T > 0 such that for all U0 ∈X with kU0kX ≤µ, the system Ut=JδH[U]has a solution defined at least on [0, T) such that U(0) =U0. U satisfies H[U(t)] =H[U0] andI[U(t)] =I[U0]for all t∈[0, T).
(A2) (Existence of solitary waves) There are an interval (c−, c+) and a mapping (c−, c+)→ X
c 7→ Φc
such that for each c∈(c−, c+), (i) δ(H+cI)[Φc] = 0,
(ii) Φ00c ∈X, and
(iii) Φ0c6= 0.
(A3) (Spectrum of the Hessian) For each c∈(c−, c+), the operator Lc=δ2(H+cI)[Φc] has precisely one negative eigenvalue which is simple and its kernel is spanned by (Φ0c).The rest of its spectrum is positive and bounded away from zero.
Then, the solitary wave with profile Φc is stable if and only if its moment of instability m(c) := (H+cI)(Φc)
satisfies
d2m
dc2 (c)>0. (1.5)
For gKdV, local well-posedness (A1) is due to Kato [32]. Assumption (A3) has been proved by Pego and Weinstein [42]. As (A2) is obvious in our case, we will have proved Theorem 1.1 once we have shown the convexity of m(c).For our case,
m(c) = Z ∞
−∞
1
2v02+f(v)−f(v∗) +p(v∗)(v−v∗) + c
2(v−v∗)2dx
= Z ∞
−∞
v02 dx= 2
Z vm(c) v∗
v0dv = 2
Z vm(c) v∗
(−2F(v, c))1/2 dv
= 4
Z (vm(c)−v∗)1/2 0
−2F(vm(c)−u2, c)1/2
u du, where u:= (vm(c)−v)1/2.Differentiating, we obtain
m0(c) = 2vm0 (c) (−2F(v∗, c))1/2+ 4
Z (vm(c)−v∗)1/2 0
d dc
−2F(vm(c)−u2, c)1/2
u du
= 4
Z (vm(c)−v∗)1/2 0
−Fv(vm(c)−u2, c)v0m(c) +Fc(vm(c)−u2, c) (−2F(vm(c)−u2, c))1/2 u du
= 2vm0 (c)
Z vm(c) v∗
∂
∂v
(−2F(v, c))1/2
dv −
−4
Z (vm(c)−v∗)1/2 0
Fc(vm(c)−u2, c)
(−2F(vm(c)−u2, c))1/2u du and thus
m00(c) =
Z vm(c) v∗
−4F(v, c)(v−v∗)vm0 (c) + (v−v∗)2 Fv(v, c)vm0 (c) +Fc(v, c)
(−2F(v, c))3/2 dv.
Obviously, the following proposition will finish the proof of the theorem.
Proposition 1.1. For fixed c,define T(v) as
T(v) :=−4F(v, c)vm0 (c) + (v−v∗) Fv(v, c)vm0 (c) +Fc(v, c) . Then T(v)>0 for allv∈(v∗, vm(c)).
Proof of the proposition. We introduce two smooth functionsα1, α2 via f(v) =f(v∗) +f0(v∗)(v−v∗) +α1(v)
2 (v−v∗)2 and p(v) =p(v∗)−α2(v)(v−v∗)
and numbers α3, α4 via
α3 : = −p(vm(c)) +p(v∗) vm(c)−v∗
=α2(vm(c)), α4 : = −p(¯v∗(c)) +p(v∗)
¯
v∗(c)−v∗
=α2(¯v∗(c));
here, ¯v∗(c) denotes the point in (v∗, vm(c)) where Fv(v, c) vanishes. By construction c=−α4 and
F(v, c) = (v−v∗)2
2 (α4−α1(v)), Fv(v, c) = (v−v∗)(α4−α2(v)), Fc(v, c) =−1
2(v−v∗)2. Finally, as for all c
F(vm(c), c)≡0, it follows that
vm0 (c) = (vm−v∗)
−2(α3+c). It remains to show that
Q(v) := (v−v∗)(α3−α4) + (vm−v∗)(2α1(v)−α4−α2(v))≥0.
The following lemma establishes properties of the αi :
Lemma 1.2. Let αi defined as above. Then we have for all v ∈(v∗, vm(c)) (i) α3≤α4,
(ii) α3≤α2(v), (iii) α4≤α1(v),
(iv) α2(v)≤α1(v).
Proof of the lemma. The proof of the lemma uses the convexity ofpin many places. Relations (i) and (ii) can be seen easily comparing the slope of the secant line connecting v∗ and vm (which corresponds to α3) with those connecting interior points (corresponding to α2(v) and α4 resp.).
To show relation (iii), recall that F(v, c)≤0 between v∗ and vm(c) and so we see from F(v, c) = 1
2(v−v∗)2(α4−α1(v)) that α4 ≤α1(v).
The last relation (iv) follows from the following observation: We have α1(v)−α2(v) = (v−v1
∗)2
n
2f(v)−2f(v∗) +p(v∗)(v−v∗) +p(v)(v−v∗) o
= (v−v2
∗)
n−(v−v1
∗)
Rv
v∗p(s)ds+p(v∗)+p(v)2 o
>0.
Lemma 1.3. Q(v) satisfiesQ(vm(c))≥0.
Proof of the lemma. By definition, we have α2(vm(c)) =α3 which yields
Q(vm(c)) = (vm(c)−v∗)(α3−2α4+ 2α1(vm(c))−α2(vm(c)))≥0 by Lemma 1.2 (iii).
Lemma 1.4. Q0(v)<0 for all v∈(v∗, vm(c)).
Proof of the lemma. As
α01(v) = 1 v−v∗
(2α2(v)−2α1(v)) and α02(v) = 1
v−v∗
(−p0(v)−α2(v)), we obtain
Q0(v) =α3−α4+vm(c)−v∗
v−v∗
(5α2(v)−4α1(v) +p0(v)).
Since α3−α4 ≤0,it is sufficient to show that for v > v∗
5α2(v)−4α1(v) +p0(v)<0
or, equivalently,
A(v) :=−8f(v) + 8f(v∗)−5p(v)(v−v∗)−3p(v∗)(v−v∗) +p0(v)(v−v∗)2 <0.
This, however, is true as A(v∗) = 0 and A0(v)≤0 for v > v∗.To see the latter expand p as a function of v∗ :
p(v∗) =p(v)−p0(v)(v−v∗) +1
2p00(τ)(v−v∗)2 where τ ∈(v∗, v).
Then
A0(v) = 3p(v)−3p(v∗)−3p0(v)(v−v∗) +p00(v)(v−v∗)2
= (p00(v)− 3
2p00(τ))(v−v∗)2
<(p00(v)−p00(τ))(v−v∗)2 ≤0 sincep000≤0.
Lemmata 1.3 and 1.4 imply that Q, and thus T are positive.
Remark. Replacing V with −V shows that Theorem 1.1 remains true if (1.2) is replaced by p00 <0 and p000 ≤0.
1.2 Stability of large-amplitude solitary waves in the generalized Boussinesq equation
Consider the generalized Boussinesq equation
Vt−Uy = 0,
Ut+p(V)y =−Vyyy (1.6)
with a smooth p:R→Rorp: (0,∞)→R satisfyingp0<0 and
p00 >0 and p000 ≤0. (1.7)
In contrast to the gKdV equation where the sign of the first derivative is irrelevant, we now need p to be strictly decreasing. The profile equation for a solitary wave (v, u)(ξ) connecting
(v∗,0) reads
v00 =−c2v+c2v∗−p(v) +p(v∗)
=:−∂F(v, c)
∂v
(1.8)
and admits a first integral given by I(v, v0) = 1
2v02+F(v, c)
= 1
2v02+1
2c2(v−v∗)2−f(v) +f(v∗)−p(v∗)(v−v∗) with
−df(v)
dv =p(v).
Lemma 1.5. Consider (1.6) with a smooth function p:R→ R or p : (0,∞)→ R satisfying p0<0and (1.7). Fix an arbitrary statev∗ in the interval of definition ofp. There is a smooth bijection
vm(c) : (p
−p0(∞),p
−p0(v∗))∪(−p
−p0(v∗),−p
−p0(∞))→(v∗,∞)
such that the following holds: A solitary wave homoclinic to v∗ and of speed c exists if and only if p0(∞)< c2 <−p0(v∗) and vm(c)> v∗.
Proof. Lemma 1.5 follows by Lemma 1.1 and the fact that the profile equation of the gKdV equation with speedcis the profile equation of the generalized Boussinesq equation with speed
−c2.In the sequel, we restrict attention to the cases c >0 without loss of generality.
Let us recall the definition of orbital stability for the generalized Boussinesq equation.
Definition 1.2. A traveling wave (v, u) of (1.6) is called orbitally stable if for each ε > 0, there exists a δ >0 such that for any solution (V, U) ∈(v, u) +C([0, T);H3(R)×H2(R)) of (1.6), closeness at initial time,
k(V, U)(·,0)−(v, u)(·)kH1×L2 < δ implies closeness at any time
σ∈infR
k(V, U)(·, t)−(v, u)(·+σ)kH1×L2 < ε for all t >0.
In this case global well-posedness follows from [9]. Our main result in this section is
Theorem 1.2. Consider (1.6) with a smooth function p : R → R or p : (0,∞) → R with p0<0 satisfying (1.7). Then any solitary wavev for (1.6) is stable.
Proof. The moment of instability is closely related to the moment of instability of the gKdV equation (cf. also Section 1.5, Remark 3). Briefly, equation (1.6) can be written as
Wt=JδH[W] with H[W] = Z
H(W), where
W = V
U
!
, H(W) = 1
2(Vy)2+1
2U2+f(V)−f(v∗) +p(v∗)(V −v∗) and J = 0 ∂y
∂y 0
! .
Introducing the functional I[W] =
Z
I(W) with I(W) =U(V −v∗) and denoting the profile by
Φc= v u
! , the profile equation (1.8) equivalently reads
Jδ(H[Φc] +cI[Φc]) = 0.
In this context, Theorem (∗) has been established in [9, 55](see also [37]) with X =H3(R)× H2(R).
Assumptions (A1) and (A3) having been proved in [9], we are again left with showing the convexity of m(c).We obtain
m(c) = Z ∞
−∞
(v0)2dx= 2
Z vm(c) v∗
(−2F(v, c))1/2 dv
= 4
Z (vm(c)−v∗)1/2 0
(−2F(vm(c)−u2, c))1/2 u du, with u:= (vm(c)−v)1/2. Differentiating twice yields
m00(c)
2 =
Z vm(c)
v∗
(v−v∗) 2F(v, c)((v−v∗) + 2cvm0 (c))−c(v−v∗)(Fv(v, c)v0m(c) +Fc(v, c))
(−2F(v, c))3/2 dv.
(1.9) Again, we will prove the positivity of the integrand, which will be similar to but slightly more difficult than in the gKdV-case.
Proposition 1.2. For fixed c, define T(v) as
T(v) := 2(v−v∗)F(v, c) + 4cvm0 (c)F(v, c)−c(v−v∗) Fv(v, c)vm0 (c) +Fc(v, c) . Then T(v)>0 for all v∈(v∗, vm(c)).
Proof of the proposition. Define functions α1(v), α2(v),and numbers α3 and α4 as in Propo- sition 1.1 with vm(c) and ¯v∗(c) now referring to F(v, c) satisfying (1.8).Recall Lemma 1.2. It follows that c2=α4 and
F(v, c) = (v−v∗)2
2 (α4−α1(v)), Fv(v, c) = (v−v∗)(α4−α2(v)), Fc(v, c) =c(v−v∗)2.
vm0 (c) can now be expressed as
vm0 (c) =−c(vm−v∗) (α4−α3) . After trivial manipulations, we see that it suffices to show that
Q(v) := (v−v∗)α1(v)α3+ (vm−v)α1(v)α4+ (vm−v∗)α4(α1(v)−α2(v)−α4)≥0.
Lemma 1.6. Q(v) satisfiesQ(vm(c))≥0.
Proof of the lemma.
Q(vm(c)) = (α3+α4)(α1(vm(c))−α4)≥0
by Lemma 1.2 (iii) and the fact that for the generalized Boussinesq equation allαi are positive since p0 <0.
Lemma 1.7. Q0(v)<0 for all v∈ (v∗, vm(c))
Proof of the lemma. Here, a little more than in the gKdV-case has to be done. We have Q0(v) =α1(v)(α3−α4) +α3 2α2(v)−2α1(v)
+ + vm−v
v−v∗
α4(2α2(v)−2α1(v)) + + vm−v∗
v−v∗
α4 3α2(v)−2α1(v) +p0(v) . Obviously by Lemma 1.2,
α1(v)(α3−α4)<(α1(v)−α2(v))(α3−α4),
and so
Q0(v)<(α1(v)−α2(v))(α3−α4) +α3 2α2(v)−2α1(v) + + vm−v
v−v∗
α4(2α2(v)−2α1(v)) + + vm−v∗
v−v∗
α4 3α2(v)−2α1(v) +p0(v) . The proof of the lemma will be finished once we prove that
2(α2(v)−α1(v))n(v−v∗)
2 (α4−α3) + (v−v∗)α3+ (vm(c)−v) + (vm(c)−v∗) α4o
+ + (vm(c)−v∗)α4(α2(v) +p0(v)) is less or equal to zero. Equivalently,
−1≥2α2(v)−α1(v) α2(v) +p0(v)
v−v∗
vm(c)−v∗α3+ (vvm(c)−v
m(c)−v∗ + 1)α4+2(vv−v∗
m(c)−v∗)(α4−α3)
α4 .
As the third factor can be estimated from below by 32,it remains to show that α1(v)−α2(v)
α2(v) +p0(v) ≥ 1 3 or, equivalently,
B(v) := 6f(v)−6f(v∗) + 6p(v∗)(v−v∗) + 4(p(v)−p(v∗))(v−v∗)−p0(v)(v−v∗)2 ≥0.
We have B(v∗) = 0 and with
p(v∗) =p(v)−p0(v)(v−v∗) +1
2p00(τ)(v−v∗)2 with τ ∈(v∗, v), we obtain
B0(v) =−2p(v) + 2p(v∗) + 2p0(v)(v−v∗)−p00(v)(v−v∗)2
= (p00(τ2)−p00(v))(v−v∗)2 ≥0.
Again, Lemmata 1.6 and 1.7 imply thatQ and henceT are positive.
1.3 (In-)Stability of small-amplitude solitary waves in the generalized KdV equation
Theorem 1.3. Consider the gKdV equation and assume that p00(v∗)>0 for some fixed state v∗ ∈R. Then there exists anε1>0 such that for all ε∈(0, ε1) there exists a small-amplitude solitary wave of speed p0(v∗) +εand this wave is orbitally stable.
Proof. Shifting and scaling v andp, we assume without loss of generality that v∗= 0 and p(v) =v2+O(v3) (and thusp0(v∗) = 0).
Let z:=ε1/2ξ. The scaling
v(ξ) =:εζ(z)
transforms the equation governing the profile (1.3) (with speed c=ε) into ζ¨=ζ−ζ2+εζ3g(ζ, ε)
with a smooth function g. The corresponding first integral now reads Iε(ζ,ζ) =˙ 1
2
ζ˙2−ζ2 2 +ζ3
3 +εζ4G(ζ, ε)
with a smooth functionG(ζ,ζ).˙ In order to prove the nonlinear orbital stability we apply again the moment of instability. We obtain
m(ε) = Z ∞
−∞
v02 dξ
=ε5/2 Z ∞
−∞
−ζ(ζ−ζ2+εζ3g(ζ(z), ε))dz
=:ε5/2A(ε).
We differentiate twice to get
m00(ε) = 15
4 ε1/2A(ε) + 5ε3/2A0(ε) +ε5/2A00(ε)
SinceAis continuous on (0, ε1),it remains to show thatA0(ε) andA00(ε) are bounded indepen- dently of ε. Smooth dependence of ε and exponential decay of ζ(z), and its derivatives with respect to ε follows from the ODE of the profile with ε as parameter; this implies that the leading term for ε→0 is 15/4ε1/2A(ε).The limitA(0) though is the moment of instability of
a soliton of the KdV equation with p(v) =v2,and we know [10] that A(0)>0.
Theorem 1.3 a. (i) The conclusion of Theorem 1.3 holds also if p00(v∗) = 0 and either p000(v∗)>0
or
p000(v∗) = 0 and p0000(v∗)>0.
(ii) If, however,
k:= min
j≥2{p(j)(v∗)6= 0}>5,
then there exists an ε1 >0 such that for all ε∈(0, ε1) there exists a small-amplitude solitary wave and this wave is not stable.
Proof. This fact is related to the transition from stability to instability for power lawsp(v) = kvq fromq ≤4 toq >5.For this dichotomy see [10]. (i) Without loss of generality we assume again that p(v) =vq+O(vq+1) with q∈ {3,4}.Adopting the notation from the above proof, the scaling
v(ξ) :=ε1/(q−1)ζ(ε1/2ξ) transforms the corresponding profile equations into
ζ¨=ζ−ζq+ε1/(q−1)ζq+1g(ζ, ε).
Again, the ε→0 limit object is a stable KdV soliton which shows the stability for smallε.
(ii) The finite-size solitary wave corresponding toε= 0 is an unstable KdV soliton. Its moment of instability is concave, and the linearization has an eigenvalue in the right half-plane; cf. [42].
This property is robust.
1.4 (In-)Stability of small-amplitude solitary waves in the Euler-Korteweg equation
The Euler-Korteweg equations describe the motion of a compressible, inviscid fluid with a variable internal capillarity. They have been studied intensively in the last years, particularly by Benzoni-Gavage and co-authors; see [8, 7] and references therein. In one space dimension
and a Lagrangian coordinate y, the one-dimensional isothermal model has the form Vt−Uy = 0,
Ut+p(V)y =−(κ(V)Vyy+1
2(κ(V))yVy)y,
(1.10)
whereU, V, andp(V) denote velocity, specific volume, and pressure of the fluid, andκ(V)>0 its volume-dependent capillarity. The special caseκ≡1 is the generalized Boussinesq equation.
In [8] the authors were mainly interested in nonlinear waves in so-called van der Waals fluids, i. e. media exhibiting phase changes. We will here concentrate on the easier situation of a classical monotone, convex pressure function, i. e., we only assume that at a reference value v∗ of the specific volume,
p0(v∗)<0 and p00(v∗)>0. (1.11) 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.
(1.12)
This equation is equivalent to κ(v)v00+ 1
2(κ(v))0v0=−p(v) +p(v∗)−c2(v−v∗) (1.13) and possesses (cf. [7]) a first integral given by
I(v, v0) = 1
2κ(v)v02−f(v) +f(v∗)−p(v∗)(v−v∗) +1
2c2(v−v∗)2, (1.14) with f the classical part of the free energy,
−df(v)
dv =p(v).
We can adapt Definition 1.2 for the Euler-Korteweg equation; note, however, that it is natural to define stability independently of a requirement for temporally global existence; cf. [7]. Our main result is
Theorem 1.4. Consider (1.10) with smooth functions p satisfying (1.11). Then:
(i) There is ac1 < c∗ :=p
−p0(v∗)such that for eachc∈(c1, c∗)there exists a small-amplitude solitary wave solution with profile (vc, uc).The map c7→(vc, uc) isC∞((c1, c∗)).
(ii) There is a c2 ∈(c1, c∗) such that for eachc∈(c2, c∗) the solitary wave (vc, uc) is orbitally stable.
Proof. Shifting and scalingv and p, we assume without loss of generality that v∗= 0 and p(v) =−v+v2+O(v3) (and thusc∗ = 1).
(i) Letε:= 1−c2 and z:=ε1/2ξ. The scaling
v(ξ) =:εζ(z) (1.15)
transforms (1.13) into
κ(εζ) ¨ζ =−1 2
d(κ(εζ)) dz
ζ˙+ζ−ζ2+εζ3g(ζ, ε) (1.16) with a smooth function g. The first integral (1.14) now reads
Iε(ζ,ζ) =˙ 1
2κ(εζ) ˙ζ2−1 2ζ2+1
3ζ3+εζ4G(ζ, ε)
with a smooth function G. A way to prove the existence of homoclinic orbits for smallεis to show that
F(ζ, ε) :=Iε(ζ,0) vanishes for ζ >0. As
F3 2,0
= 0 and ∂F
∂ζ 3
2,0 6= 0,
the implicit function theorem shows that there is a smooth curve ζ =ζ0(ε),0 ≤ε < ε1 along which F(ζ0(ε), ε) = 0. Recalling thatε= 1−c2, we see that the first sentence of (i) is proved.
As the homoclinic orbit connects a hyperbolic fixed point to itself, the solution (v, u) together with its derivatives tends exponentially fast to zero as |z| → ∞. Smooth dependence of c follows by a standard ODE argument.
(ii) In order to show nonlinear orbital stability, we apply again the moment of instability that was first used for the Euler-Korteweg equations in [8]. The moment of instability is defined in the same way as for the generalized Boussinesq equation. The Hamiltonian though now reads
H(W) = 1
2U2+f(V) + 1
2κ(V)Vy2, and the momentum simplifies to
I(W) =U V.
Theorem (∗) for the Euler-Korteweg equation is due to [8]. As assumptions (A1) and (A3) are fulfilled for the Euler-Korteweg equation [7, 8] and as statement (i) of our theorem just means
(A2) with (c−, c+) = (c1,1),statement (ii) will be proved once we have verified the convexity of m(c).
Using the identities
0 =I(v, v0)−I(0,0) = 1
2κ(v)(v0)2−f(v) +1 2c2v2 and u=−cv, we obtain
m(c) = Z ∞
−∞
(H+cI)(v, u)dy
= Z ∞
−∞
1
2u2+f(v) +1
2κ(v)(vy)2+cvu dy
= Z ∞
−∞
κ(v)(vy)2dy.
Integration by parts, the scaling (1.15), and equation (1.16) yield m(c) =ε3
Z ∞
−∞
κ(εζ(ε1/2ξ))( ˙ζ(ε1/2ξ))2 dξ
=ε52 Z ∞
−∞
−ζ(z)dκ(εζ(z))
dz ζ˙(z) +κ(εζ(z)) ¨ζ(z) dz
=ε52 Z ∞
−∞
− 1
2
dκ(εζ(z)) dz
ζ(z)ζ˙ (z) +ζ2(z)−ζ3(z) +εζ4(z)g(ζ(z), ε)
dz.
Recall that ε(c) = 1−c2. Let A(c) denote the integral term in the last line, i .e., m(c) =
−ε5/2A(c).To see thatm(c) is convex forc→1, we differentiate twice to get m00(c) =
+15 (1−c2)1/2c2−5 1−c23/2
A(c)−10(1−c2)3/2A0(c) + (1−c2)5/2A00(c) whose leading term for c→ 1 is 15(1−c2)1/2c2A(c). As A(1) is the moment of instability of a finite-amplitude soliton for the Boussinesq equation, we know [9] that
A(1)>0. (1.17)
Since A is continuous on (c1,1], it remains to show that A0(c) and A00(c) are bounded inde- pendently of c. This follows, however, directly from the following observation.
Lemma 1.8. The scaled profileζ(z) and its derivatives∂cζ(z), ∂c2ζ(z) with respect to c decay to zero exponentially for z→ ∞.
Proof of the lemma. Letη:= dζdε and consider κ(εζ) ¨ζ =−1
2κ0(εζ)( ˙ζ)2+ζ+ζ2+εζ3g(ζ, ε), κ(εζ)¨η =−κ0(εζ)(ζ+εη)
κ(εζ) −1
2κ0(εζ)( ˙ζ)2+ζ+ζ2+εζ3g(ζ, ε)
−
− 1
2κ00(εζ)(ζ+εη)( ˙ζ)2−κ0(εζ) ˙ζη˙+η−2ζη+O(ζ2),
˙ ε= 0.
Its linearization at the fixed point (ζ,ζ, η,˙ η) = (0,˙ 0,0,0) has eigenvalues λ=±κ(0)−1/2 and λ= 0 which shows exponential decay ofζ andη for|z| → ∞ as well as smooth dependence of ε, and hencec.The same argument applies to ∂ε2ζ.
Theorem 1.4 a. (i) The conclusion of Theorem 1.4 holds also if p0(v∗) < 0 and p00(v∗) = 0 and either
p000(v∗)>0 or
p000(v∗) = 0 and p0000(v∗)>0.
(ii) If, however, p0(v∗)<0 and
k:= min
j≥2{p(j)(v∗)6= 0}>5,
then assertion (i) of Theorem 1.4 again holds, but there now is a c2 ∈ (c1, c∗) such that for each c∈(c2, c∗) the solitary waves (φc, ψc) are not stable.
Proof. (i) Without loss of generality we assume again that p(v) = −v+vq+O(vq+1) with q ∈ {3,4}.Adopting the notation from the above proof, the scaling
φ(ξ) :=ε1/(q−1)ζ(ε1/2z) transforms the corresponding profile equations into
κ(ε1/(q−1)ζ) ¨ζ =−1 2
d(κ(ε1/(q−1)ζ)) dz
ζ˙+ζ−ζq+ε1/(q−1)ζq+1g(ζ, ε).
Again, theε→0 limit object is a stable Boussinesq soliton which shows the stability for small ε.
(ii) The finite-size solitary wave corresponding to c= 1 is an unstable Boussinesq soliton. Its
moment of instability is partly concave, and the linearization has an eigenvalue in the right half-plane; cf. [55] (see also [37]). This property is robust.
1.5 Remarks
1. As an obvious consequence of our small-amplitude results, we note that no constant state solution
(v(y, t), u(y, t)) = (v∗, u∗) with (1.11)
can be time-asymptotically stable towards perturbations which have the form of a small- amplitude soliton.
2. Obviously, there are well-known examples for flux functions p(v) for the gKdV equation as well as for the generalized Boussinesq equation which are convex but whose first derivative is not concave and whose solitary waves are not stable, notably
p(v) =vq with q >5 in the gKdV-case (cf. [10, 42]) and
p(v) =−v+vq withq >5 for the generalized Boussinesq equation (cf. [55]).
3. Theorem 1.3 follows actually from Theorem 1.4. Let us briefly sketch this idea: We connect the stability of waves in the gKdV equation with that of waves in the generalized Boussinesq equation. To make this nice little connection, we start by observing that the moment of instability only depends on the profile equation. Let without loss of generality p(v) =−v+vq+O(vq+1) andv∗ = 0.Denote bymB(c) resp. mK(ε) the moment of instability for solitary waves in the generalized Boussinesq resp. gKdV equations. They satisfy
mB(c) = Z ∞
−∞
((φBc)0)2 dξ and
mK(ε) = Z ∞
−∞
((φKε )0)2 dξ
where φBc resp. φKε are respective profiles of the generalized Boussinesq and gKdV equations, solving
(φBc)00= (1−c2)φBc −(φBc)q−h(φBc) resp.
(φKε )00= (1 +ε)φKε −(φKε )q−h(φKε )
with h(v) = p(v) +v−vq. The profiles differ only in their dependence on the speed; more precisely, we have
mB(c) =mK(g(c)) with g(c) =−c2 and thus
(mK)00(g(c)) = (mB)00(c)
4c2 −(mB)0(c) 4c3 .
We investigate the behavior of (mK)00(g(c)) for c→ p0(v∗) = 1. Using the notation from the proof of Theorem 1.4, with κ≡1, we find
(mK)00(−c2) = 15√ 1−c2
4 A(c) +O((1−c2)3/2).
Due to (1.17) and Theorem 1.4, this implies in the caseq≤4 thatm(ε) is convex for sufficiently small ε. In the caseq >5, A(1) is negative and we can conclude instability.
1.6 Stability of large-amplitude solitary waves in the Euler-Korteweg equation
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 I(v, v0) = 1
2κ(v)v02−f(v) +f(v∗)−p(v∗)(v−v∗) +1
2c2(v−v∗)2
= 1
2κ(v)v02+F(v, c) with
−df(v)
dv =p(v).
Our Theorem is
Theorem 1.5. Consider (1.18) with a smooth positive function κ:R→R or κ: (0,∞)→R with κ0 ≥0 and a smooth function p:R→Ror p: (0,∞)→R with p0 <0 satisfying
p00>0 and p000 ≤0.
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) =
Z ∞
−∞
κ(v)v02dξ
= 2
Z vm(c) v∗
κ(v)v0 dv.
Due to I(v, v0)≡0 along solutions, we have m(c) = 2
Z vm(c) v∗
(κ(v))1/2 (−2F(v, c))1/2 dv
= 4
Z (vm(c)−v∗)1/2 0
κ(vm(c)−u2)1/2
−2F(vm(c)−u2, c)1/2
u du,
where u:= (vm(c)−v)1/2.Its first derivative is m0(c) = 4
Z d dc
n
κ(vm(c)−u2)1/2
−2F(vm(c)−u2, c)1/2o u du
= 4
Z κv(vm(c)−u2)vm0 (c) 2 (κ(vm(c)−u2))1/2
−2F(vm(c)−u2, c)1/2
u du + + 4
Z κ(vm(c)−u2)1/2
−Fv(vm(c)−u2, c)vm0 (c)−Fc(vm(c)−u2, c) (−2F(vm(c)−u2, c))1/2 u du
