PDF On the spectral, modulational and orbital stability of periodic ... - UNAM

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Instituto de Investigaciones en Matem´aticas Aplicadas y en Sistemas

On the spectral, modulational and

orbital stability of periodic wavetrains for the sine-Gordon equation

Ram´on G. Plaza

Institute of Applied Mathematics (IIMAS), National Autonomous University of Mexico (UNAM)

II Workshop on Nonlinear Dispersive Equations. Universidade Estadual de Campinas, Brazil. October 6 to 9, 2015.

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Collaborators:

• Christopher K. R. T. Jones (Univ. of North Carolina, Chapel Hill)

• Robert Marangell (Univ. of Sydney)

• Peter D. Miller (Univ. of Michigan)

• Jaime Angulo Pava (Univ. of Sao Paulo)

This research was partially supported by: DGAPA-UNAM,

project PAPIIT IN-104814.

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1 Introduction

2 Analysis of the monodromy map

3 Spectral (in)stability results

4 Multidimensional orbital stability

Ram´on G. Plaza — Stability of periodic nonlinear Klein-Gordon waves — II Workshop on Nonlinear Dispersive Equations. Universidade Estadual de Campinas, Brazil. October 6 to 9, 2015.

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The nonlinear Klein-Gordon equation

Nonlinear Klein-Gordon with periodic potential:

u _tt − u _xx + V ⁰ (u) = 0. (nKG) for (x, t) ∈ R × [0, +∞) , u scalar, V ∈ C ² , periodic.

Sine-Gordon equation:

u _tt − u _xx + sin u = 0. (SG)

V (u) = 1 − cos u

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The nonlinear Klein-Gordon equation

Nonlinear Klein-Gordon with periodic potential:

u _tt − u _xx + V ⁰ (u) = 0. (nKG) for (x, t) ∈ R × [0, +∞) , u scalar, V ∈ C ² , periodic.

Sine-Gordon equation:

u _tt − u _xx + sin u = 0. (SG)

V (u) = 1 − cos u

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Applications (sine-Gordon):

• Surfaces with negative Gaussian curvature (Eisenhart, 1909)

• Propagation of crystal dislocations (Frenkel and Kontorova, 1939)

• Elementary particles (Perring and Skyrme, 1962)

• Propagation of magnetic flux on a Josephson line (Scott, 1969)

• Dynamics of fermions in the Thirring model (Coleman, 1975)

• Oscillations of a rigid pendulum attached to a

stretched rubber band (Drazin, 1983)

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Assumptions on the potential:

(a) V : R → R is of class C ² in all its domain and it is periodic with fundamental period P _.

(b) V has only non-degenerate critical points.

(c) V ⁰ (u) ⁴ (V(u)/V ⁰ (u) ² ) ⁰⁰ ≥ 0 for all u _under consideration.

Assumption (c) implies monotonicity of the period map with respect to the energy.

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Traveling waves

u(x, t) = f (x − ct) , z = x − ct , solution to the nonlinear pendulum equation:

(c ² − 1)f _zz + V ⁰ (f (z)) = 0,

Sine-Gordon case:

(c ² − 1)f _zz + sin(f (z)) = 0,

c ∈ R (wave speed), c ² 6= 1 .

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Upon integration:

1 2 (c ² − 1)f _z ² = E − V(f ), E = constant (energy). Under assumptions:

0 < E ₀ = max V(u)

Sine-Gordon case: V(u) = 1 − cos u , E ₀ = 2 ,

1 2 (c ² − 1)f _z ² = E − 1 + cos f (z).

W.l.o.g.

(d) V has fundamental period P = 2π and min u∈ R

V(u) = 0, max

u∈ R

V(u) = 2.

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Classification

First dichotomy (wave speed):

• Subluminal waves: c ² < 1

• Superluminal waves: c ² > 1 Second dichotomy (energy E _):

• Librational wavetrain: f (z + T ) = f (z) . Closed trajectory inside the separatrix in the phase portrait.

• Rotational wavetrain: f (z + T) = f (z) ± 2π . Open trajectory outside the separatrix in the phase plane.

Sign f _z _{is fixed.}

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Classification

First dichotomy (wave speed):

• Subluminal waves: c ² < 1

• Superluminal waves: c ² > 1 Second dichotomy (energy E _):

• Librational wavetrain: f (z + T ) = f (z) . Closed trajectory inside the separatrix in the phase portrait.

• Rotational wavetrain: f (z + T) = f (z) ± 2π . Open trajectory outside the separatrix in the phase plane.

Sign f _z _{is fixed.}

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-2π -π 0 π 2π -2

-1 0 1 2

-2π -π 0 π 2π

-2 -1 0 1 2

-2π -π 0 π 2π

-4 -3 -2 -1 0 1 2 3 4

-2π -π 0 π 2π

-4 -3 -2 -1 0 1 2 3 4

Figure : Phase portrait sine-Gordon case: V (u) = 1 − cos u :

superluminal c ² > 1 (left); subluminal c ² < 1 (right).

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Figure : Phase portrait for V(u) = 1 − (0.861)(cos u + ¹ ₃ sin(2u)) _: superluminal c ² > 1 (left); subluminal c ² < 1 _(right).

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Superluminal librational: c ² > 1 , 0 < E < E ₀ _.

K ^{(E) =} ^{u ^∈ R : (E − V(u))/(c ² − 1) ≥ 0} = disjoint union of intervals in (0, π) . In (v ₁ , v ₂ ) , only one

non-degenerate zero of V ⁰ . Librational (closed) periodic orbit.

f _z =

√ 2

√ c ² − 1

p E − V (f ), where f ∈ (v ₁ , v ₂ ) ⊂ K ^(E) ^.

T = √ 2 p

c ² − 1 Z _v ₂

v ₁

dη p E − V(η) .

Sine-Gordon: wave oscillates around f = 0 , in

(v ₁ , v ₂ ) = (−Arc cos(−E + 1), Arc cos(−E + 1))

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Superluminal librational: c ² > 1 , 0 < E < E ₀ _.

K ^{(E) =} ^{u ^∈ R : (E − V(u))/(c ² − 1) ≥ 0} = disjoint union of intervals in (0, π) . In (v ₁ , v ₂ ) , only one

non-degenerate zero of V ⁰ . Librational (closed) periodic orbit.

f _z =

√ 2

√ c ² − 1

p E − V (f ), where f ∈ (v ₁ , v ₂ ) ⊂ K ^(E) ^.

T = √ 2 p

c ² − 1 Z _v ₂

v ₁

dη p E − V(η) .

Sine-Gordon: wave oscillates around f = 0 , in (v ₁ , v ₂ ) = (−Arc cos(−E + 1), Arc cos(−E + 1))

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Subluminal librational: c ² < 1 , 0 < E < E ₀ _.

K ^{(E) =} ^{u ^∈ R : (V(u) − E)/(1 − c ² ) ≥ 0} = disjoint union of intervals in (0, π) . In (v ₃ , v ₄ ) , only one

non-degenerate zero of V ⁰ . Librational (closed) periodic orbit.

f _z =

√ 2

√ 1 − c ²

p V(f ) − E, where f ∈ (v ₃ , v ₄ ) ⊂ K ^(E) ^.

T = √ 2 p

1 − c ² Z _v ₄

v ₃

dη p V(η) − E .

Sine-Gordon: wave oscillates around f = π , in

(v ₃ , v ₄ ) = (−Arc cos(−E + 1), 2π − Arc cos(−E + 1))

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Subluminal librational: c ² < 1 , 0 < E < E ₀ _.

K ^{(E) =} ^{u ^∈ R : (V(u) − E)/(1 − c ² ) ≥ 0} = disjoint union of intervals in (0, π) . In (v ₃ , v ₄ ) , only one

non-degenerate zero of V ⁰ . Librational (closed) periodic orbit.

f _z =

√ 2

√ 1 − c ²

p V(f ) − E, where f ∈ (v ₃ , v ₄ ) ⊂ K ^(E) ^.

T = √ 2 p

1 − c ² Z _v ₄

v ₃

dη p V(η) − E .

Sine-Gordon: wave oscillates around f = π , in

(v ₃ , v ₄ ) = (−Arc cos(−E + 1), 2π − Arc cos(−E + 1))

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Superluminal rotational: c ² > 1 , E > E ₀ _, E − V(f ) > 0 and K ^{(E) =} R . Rotation, f _z has fixed sign. Orbit outside the separatrix and f (z + T) = f (z) ± π for all z .

f _z ² = 2(E − V (f )) c ² − 1 > 0,

T =

√ c ² − 1

√ 2 Z _π

0 dη p E − V(η)

Subluminal rotational: c ² < 1 , E < 0 , V(f ) − E ≥ 0 and

K ^{(E) =} R ^{with .} f _z has fixed sign. Orbit outside the separatrix and f (z + T ) = f (z) ± π for all z _.

T =

√ 1 − c ²

√

Z _π dη

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Superluminal rotational: c ² > 1 , E > E ₀ _, E − V(f ) > 0 and K ^{(E) =} R . Rotation, f _z has fixed sign. Orbit outside the separatrix and f (z + T) = f (z) ± π for all z .

f _z ² = 2(E − V (f )) c ² − 1 > 0,

T =

√ c ² − 1

√ 2 Z _π

0 dη p E − V(η)

Subluminal rotational: c ² < 1 , E < 0 , V(f ) − E ≥ 0 and

K ^{(E) =} R ^{with .} f _z has fixed sign. Orbit outside the separatrix and f (z + T ) = f (z) ± π for all z _.

T =

√ 1 − c ²

√ 2

Z _π

0 dη p V (η) − E

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G ^lib < = {c ² < 1, 0 < E < E ₀ }, (subluminal librational) , G ^rot < = {c ² < 1, E < 0}, (subluminal rotational) , G ^lib > = {c ² > 1, 0 < E < E ₀ }, (superluminal librational) , G ^rot > = {c ² > 1, E > E ₀ }, (superluminal rotational) ,

(E, c) ∈ G := G ^lib < ∪ G ^rot < ∪ G ^lib > ∪ G ^rot >

Lemma

For each fixed z ∈ R ^, f (z; E, c) is of class C ² in (E, c) ∈ G ^.

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G ^lib < = {c ² < 1, 0 < E < E ₀ }, (subluminal librational) , G ^rot < = {c ² < 1, E < 0}, (subluminal rotational) , G ^lib > = {c ² > 1, 0 < E < E ₀ }, (superluminal librational) , G ^rot > = {c ² > 1, E > E ₀ }, (superluminal rotational) ,

(E, c) ∈ G := G ^lib < ∪ G ^rot < ∪ G ^lib > ∪ G ^rot >

Lemma

For each fixed z ∈ R ^, f (z; E, c) is of class C ² in (E, c) ∈ G ^.

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Spectral problem

Solution f (z) + u(z, t) , with u = perturbation:

u _tt − 2cu zt + (c ² − 1)u _zz + V ⁰ (u + f ) − V ⁰ (f ) = 0.

Linearized equation:

u _tt − 2cu _zt + (c ² − 1)u _zz + V ⁰⁰ (f (z))u = 0.

u = w(z)e ^λt , λ ∈ C ^, w ∈ X _Banach:

(c ² − 1)w _zz − 2cλw _z + (λ ² + V ⁰⁰ (f (z)))w = 0. (P)

Quadratic “pencil” in λ .

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Spectral problem

Solution f (z) + u(z, t) , with u = perturbation:

u _tt − 2cu zt + (c ² − 1)u _zz + V ⁰ (u + f ) − V ⁰ (f ) = 0.

Linearized equation:

u _tt − 2cu _zt + (c ² − 1)u _zz + V ⁰⁰ (f (z))u = 0.

u = w(z)e ^λt , λ ∈ C ^, w ∈ X _Banach:

(c ² − 1)w _zz − 2cλw _z + (λ ² + V ⁰⁰ (f (z)))w = 0. (P) Quadratic “pencil” in λ .

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Floquet spectrum

Formally: λ ∈ σ _F is a Floquet eigenvalue if there exists a bounded solution w to (P). (Precise definition in a

moment.)

We say the wave is spectrally stable if σ _F ⊂ { Re λ ≤ 0} .

Otherwise it is spectrally unstable.

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Floquet spectrum

Formally: λ ∈ σ _F is a Floquet eigenvalue if there exists a bounded solution w to (P). (Precise definition in a

moment.)

We say the wave is spectrally stable if σ _F ⊂ { Re λ ≤ 0} . Otherwise it is spectrally unstable.

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Previous results

• A.C. Scott , Proc. IEEE (1969). Spectral stability.

• G.B. Whitham, Linear and nonlinear waves (1974).

“Modulational” stability results. Based on Modulation theory (Whitham, 1965).

• Forest, MacLaughlin (1982); Murakami (1986);

Ercolani, Forest, McLaughlin (1990); Parkes (1991);

etc. (abridged list).

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Previous results

• A.C. Scott , Proc. IEEE (1969). Spectral stability.

• G.B. Whitham, Linear and nonlinear waves (1974).

“Modulational” stability results. Based on Modulation theory (Whitham, 1965).

• Forest, MacLaughlin (1982); Murakami (1986);

Ercolani, Forest, McLaughlin (1990); Parkes (1991);

etc. (abridged list).

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Previous results

• A.C. Scott , Proc. IEEE (1969). Spectral stability.

• G.B. Whitham, Linear and nonlinear waves (1974).

“Modulational” stability results. Based on Modulation theory (Whitham, 1965).

• Forest, MacLaughlin (1982); Murakami (1986);

Ercolani, Forest, McLaughlin (1990); Parkes (1991);

etc. (abridged list).

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Summary of stability results

Wave Whitham (1974) Scott (1969)

Subluminal rotational stable stable Superluminal rotational stable unstable Subluminal librational unstable unstable Superluminal librational unstable unstable

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Whitham (1965, 1974):

Modulation theory: well established (formal) physical method based on WKB expansions. Exact wave f = f (x − ct) = f ˜ (kx − ωt) . Allowing dependence k = k(x, t), ω = ω(x, t) , under “slow modulations”, if the PDE system on (k, ω) is well-posed then the wave is

“stable”.

References: Whitham (1974), J. Fluid Mech. (1965), Proc.

Roy. Soc. Ser. A (1965).

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Whitham (1965, 1974):

Modulation theory: well established (formal) physical method based on WKB expansions. Exact wave f = f (x − ct) = f ˜ (kx − ωt) . Allowing dependence k = k(x, t), ω = ω(x, t) , under “slow modulations”, if the PDE system on (k, ω) is well-posed then the wave is

“stable”.

References: Whitham (1974), J. Fluid Mech. (1965), Proc.

Roy. Soc. Ser. A (1965).

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Scott (1969):

y = exp

−cλz c ² − 1

,

y _zz + V ⁰⁰ (f (z)) c ² − 1 y =

λ c ² − 1

2 y =: νy. (H) Hill’s equation with period T . ν ∈ σ H (Floquet spectrum of (H)) if there is a bounded solution y _.

Scott assumed that the transformation is isospectral:

( σ H = σ F ). This is not true. Actually:

Lemma (JMMP1)

If λ ∈ σ _H ∩ σ _F then λ ∈ i R ^.

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Scott (1969):

y = exp

−cλz c ² − 1

,

y _zz + V ⁰⁰ (f (z)) c ² − 1 y =

λ c ² − 1

2 y =: νy. (H) Hill’s equation with period T . ν ∈ σ H (Floquet spectrum of (H)) if there is a bounded solution y _.

Scott assumed that the transformation is isospectral:

( σ H = σ F ). This is not true. Actually:

Lemma (JMMP1)

If λ ∈ σ _H ∩ σ _F then λ ∈ i R ^.

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References:

• C.K.R.T. Jones, R. Marangell, P.D. Miller, R.P., On the stability of periodic traveling sine-Gordon waves, Phys. D 251 (2013) (JMMP1).

• C.K.R.T. Jones, R. Marangell, P.D. Miller, R.P., Spectral and modulational stability of periodic

wavetrains for the nonlinear Klein-Gordon equation. J.

Differential Equations 257 (2014) (JMMP2).

• J. Angulo-Pava, R.P., Nonlinear orbital stability of

subluminal periodic sine-Gordon wavetrains of

rotational type. Preprint (2015) (AP) .

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Summary:

JMMP1:

• Correct proof of Scott’s results (spectral) JMMP2:

• More generic potentials

• Analysis of the monodromy map

• Modulational stability index

• Relation to Whitham’s modulation theory AP:

• Orbital (nonlinear) stability of subluminal rotational waves

• Multidimensional orbital stability (e.g. 2d sine-Gordon)

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Other references (Whitham vs. spectral):

• Serre (2005); Oh, Zumbrun (2006) (viscous conserv.

laws)

• Bronski, Johnson (2009); Johnson, Zumbrun (2010);

Bronski, Johnson, Zumbrun (2010) (gKdV)

• Johnson (2010) (BBM)

• Noble, Rodrigues (2013) (Kuramoto-Sivashinski)

• Benzoni, Noble, Rodrigues (2013) (Hamiltonian

PDEs)

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1 Introduction

2 Analysis of the monodromy map

3 Spectral (in)stability results

4 Multidimensional orbital stability

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Floquet spectrum

Problem (P) (quadratic pencil) can be written as a first order system:

w _z = A(z, λ)w,

w :=

w w _z

,

A(z, λ) :=







0 1

− (λ ² + V ⁰⁰ (f (z))) c ² − 1

2cλ c ² − 1







.

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Family of closed, densely defined operators:

T ^(λ) ^: D ^⊂ ^X ^→ ^X

T ^(λ)W ^:= ^W z − A(z, λ)W.

E.g.:

D ⁼ ^H ¹ ⁽ R ; C ² ), X = L ² ( R ; C ² ),

Spectral stability of periodic waves with respect to localized perturbations.

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Family of closed, densely defined operators:

T ^(λ) ^: D ^⊂ ^X ^→ ^X

T ^(λ)W ^:= ^W z − A(z, λ)W.

E.g.:

D ⁼ ^H ¹ ⁽ R ; C ² ), X = L ² ( R ; C ² ),

Spectral stability of periodic waves with respect to localized

perturbations.

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Definition (cf. Sandstede (2002))

The resolvent ρ , the point spectrum σ pt and the essential spectrum σ ess of problem (P):

ρ := {λ ∈ C : T ^(λ) is one-to-one and onto, and

T ^(λ) ⁻¹ ^{is bounded} ^},

σ pt := {λ ∈ C : T ^(λ) is Fredholm with zero index and has a non-trivial kernel },

σ ess := {λ ∈ C : T ^(λ) is either not Fredholm or has index different from zero }.

The spectrum is σ = σ ess ∪ σ pt . ( T ^(λ) ^closed ^⇒ ^ρ ⁼ C \σ .)

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Comments:

• The transformation v ₁ = w _, v ₂ = λw _{defines a}

cartesian product in X = L ² which allows to write as a standard eigenvale problem:

λv =

0 1

−(c ² − 1)∂ ² _z − cos f (z) 2c∂ z

v =: L ^v, ^v ⁼

v ₁ v ₂

.

• Equivalent definition to the standard one (essential

spectrum of Weyl, 1910): 1-to-1 correspondence

between Jordan chains

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Lemma (Gardner, 1997)

Since X = L ² all spectrum of problem (P) is “continuous”, that is, σ = σ ess and σ pt is empty.

Since (nKG) is a real Hamiltonian system:

Lemma

σ is symmetric with respect to reflection in real and imaginary axes: λ ∈ σ ⇒ −λ, λ ∈ σ .

Spectral stability is equivalent to σ ⊂ i R ^.

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Lemma (Gardner, 1997)

Since X = L ² all spectrum of problem (P) is “continuous”, that is, σ = σ ess and σ pt is empty.

Since (nKG) is a real Hamiltonian system:

Lemma

σ is symmetric with respect to reflection in real and imaginary axes: λ ∈ σ ⇒ −λ, λ ∈ σ .

Spectral stability is equivalent to σ ⊂ i R ^.

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Lemma (Gardner, 1997)

Since X = L ² all spectrum of problem (P) is “continuous”, that is, σ = σ ess and σ pt is empty.

Since (nKG) is a real Hamiltonian system:

Lemma

σ is symmetric with respect to reflection in real and imaginary axes: λ ∈ σ ⇒ −λ, λ ∈ σ .

Spectral stability is equivalent to σ ⊂ i R ^.

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Monodromy matrix

M(λ) := F(T , λ)

F(z, λ) = fundamental solution with F(0, λ) = I . M(λ)F(z, λ) = F(z + T, λ)

Important feature: A entire in λ , Picard iterates converge

for F in z _bounded ⇒ M is an entire (analytic) function of

λ ∈ C ^.

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Monodromy matrix

M(λ) := F(T , λ)

F(z, λ) = fundamental solution with F(0, λ) = I . M(λ)F(z, λ) = F(z + T, λ)

Important feature: A entire in λ , Picard iterates converge for F in z _bounded ⇒ M is an entire (analytic) function of λ ∈ C ^.

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Floquet multipliers:

λ ∈ σ if and only if there exists at least one µ ∈ C ^(Floquet multiplier) with |µ| = 1 such that

D(λ,µ) ˆ := det(M(λ) − µI) = 0.

µ = µ(λ) = e ^iθ(λ) are the eigenvalues of M(λ) . θ = θ(λ)

are called the Floquet exponents.

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Periodic Evans function

Definition (Gardner, 1997)

The periodic Evans function D : C × R → C îs D(λ, κ) := D(λ, ˆ e îκT ) = det(M(λ) − e îκT I), for each (λ, κ) ∈ C × R ^.

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Properties: (Gardner 1997, 1998)

• σ is the set of all λ ∈ C ^{such that} D(λ, κ) = 0 for some real κ .

• D is analytic in λ and κ .

• The order of the zero in λ is the multiplicity of the eigenvalue.

• D(λ, ˆ 1) = D(λ, 0) detects spectra corresponding to

perturbations which are T -periodic.

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Floquet spectrum

Boundary value problem of the form

(c ² − 1)w _zz − 2cλw z + (λ ² + V ⁰⁰ (f (z)))w = 0,

w(T) w _z (T)

= e ^iθ

w(0) w _z (0)

, θ ∈ R .

For a given θ ∈ R ^{we define} σ _θ ⊂ C to be the set of complex λ for which there exists a nontrivial solution. The Floquet spectrum σ _F is defined then as the union over θ of these sets:

σ _F := ^[

−π<θ≤π

σ _θ .

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Observations:

• Clearly σ = σ F

• Each set σ _θ is discrete: zero set of the entire function det(M(λ) − e ^iθ I)

• The set σ ₀ (with θ = 0 ) is the part of the spectrum corresponding to perturbations which are co-periodic (periodic partial spectrum)

• θ - local coordinate; curves of spectrum: if

D _λ (λ ₀ , µ ₀ ) 6= 0 , D _µ (λ ₀ , µ ₀ ) 6= 0 then σ is a smooth local curve

• At points where derivatives vanish: spectral analytic

arcs (e.g. at λ = 0 !)

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Bloch wave decomposition

Transformation: y = e ^−iθz/T w yields y _z = A(z, e λ, θ)y,

A(z, e λ, θ) = A(z, λ) − (iθ/T )I, Boundary conditions:

y(0) = y(T)

λ ∈ σ iff for some −π < θ ≤ π there exists a non-trivial solution y ∈ H _per ¹ ([0, T]; C ² )

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Bloch wave decomposition

Transformation: y = e ^−iθz/T w yields y _z = A(z, e λ, θ)y,

A(z, e λ, θ) = A(z, λ) − (iθ/T )I, Boundary conditions:

y(0) = y(T)

λ ∈ σ iff for some −π < θ ≤ π there exists a non-trivial

solution ¹ ²

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Equivalently, w = e ^1θz q transforms the spectral pencil (P) into

(c ² −1) ∂ _z + iθ T

2 −2cλ ∂ _z + iθ T

+(λ ² +cos f (z)) q = 0,

q(T) = q(0), q _z (T) = q _z (0)

Scalar domain base space H _per ¹ ([0, T ]; C ) ⊂ L ² _per ([0, T]; C ) . Make p ₁ = q _, p ₂ = λq , we obtain:

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Family of Bloch operators

λp =

0 1

−(c ² − 1) ∂ _z + ^iθ _T 2

− cos f (z) 2c ∂ _z + ^iθ _T

p =: L θ p,

p = p ₁

p ₂

,

L θ : D ⁼ ^H ¹ per ([0, T ]; C ² ) → L ² _per ([0, T]; C ² ),

λ ∈ σ (continuous) iff λ ∈ σ pt ( L θ ) for some θ ∈ (−π, +π]

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Solutions at λ = 0

f = f (z; E, c) , (E, c) ∈ G ^. w solution to pencil (P), with initial conditions:

w(0; E, c) = f (0;E, c)

=

( f (T; E, c), E ∈ (0, E ₀ ), (lib), f (T; E, c) − π, E ∈ (−∞, 0) ∪ (E ₀ ,+∞), (rot),

∂ _z w(0; E,c) = f _z (0; E,c) = f _z (T ; E, c)

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System at λ = 0 :

w _z = A(z, 0)w,

A(z, 0) =

0 1

−V ⁰⁰ (f (z))/(c ² − 1) 0

.

Lemma

The two-dimensional vector space of solutions is spanned by

w ₀ (z) = f _z

f _zz

, and w ₁ (z) = f _E

f _Ez

.

det(w (z), w (z)) = f f − f f = 1

6= 0

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System at λ = 0 :

w _z = A(z, 0)w,

A(z, 0) =

0 1

−V ⁰⁰ (f (z))/(c ² − 1) 0

.

Lemma

The two-dimensional vector space of solutions is spanned by

w ₀ (z) = f _z

f _zz

, and w ₁ (z) = f _E

f _Ez

.

det(w ₀ (z), w ₁ (z)) = f _z f _Ez − f _E f _zz = 1 c ² − 1 6= 0

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Solution matrix:

Q(z, 0) := (w ₀ (z), w ₁ (z))

F(z, 0) = Q(z, 0)Q(0, 0) ⁻¹ .

M(0) = F(T, 0) = Q(T, 0)Q(0, 0) ⁻¹

Q(z, 0) ⁻¹ = (c ² − 1)

f _Ez −f _E

−f _zz f _z

.

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Lemma

If T _E 6= 0 , there exists a basis in R ² such that the monodromy map M(λ) at λ = 0 has the Jordan form

M(0) ∼

1 −T _E

0 1

.

Q(T, 0) − Q(0, 0) is a rank-one matrix provided that T _E 6= 0 :

Q(T, 0) = Q(0, 0) + 0 −T _E v ₀ (E, c) 0 −T _E ^V ⁰ ^(u ⁰ ^(E,c))

c ² −1

!

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Lemma

If T _E 6= 0 , there exists a basis in R ² such that the monodromy map M(λ) at λ = 0 has the Jordan form

M(0) ∼

1 −T _E

0 1

.

Q(T, 0) − Q(0, 0) is a rank-one matrix provided that T _E 6= 0 :

Q(T, 0) = Q(0, 0) + 0 −T _E v ₀ (E, c) 0 −T _E ^V ⁰ ^(u ⁰ ^(E,c))

c ² −1

!

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Under Assumption (c), we have monotonicity of the period map (Chicone, 1987: criterion for planar Hamiltonian systems):

Lemma

Under assumptions there holds T _E 6= 0 . More precisely we have:

(i) T _E > 0 in the rotational subluminal and librational superluminal cases.

(ii) T _E < 0 in the rotational superluminal and librational subluminal cases.

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Lemma

If we define

∆ ¯ := − T _E c ² − 1 then

(a) ∆ ¯ > 0 for rotational waves.

(b) ∆ ¯ < 0 for librational waves.

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Solutions series expansions

Q = Q(z, λ) solution to dQ

dz = A(z, λ)Q.

Q(0, λ) = Q(0, 0) = (w ₀ (0), w ₁ (0)) By analyticity, seek series expansion

Q(z, λ) =

+∞

n=0 ∑

λ ⁿ Q _n (z)

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Solutions series expansions

Q = Q(z, λ) solution to dQ

dz = A(z, λ)Q.

Q(0, λ) = Q(0, 0) = (w ₀ (0), w ₁ (0)) By analyticity, seek series expansion

Q(z, λ) =

+∞

n=0 ∑

λ ⁿ Q _n (z)

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Collecting like powers of λ we obtain a hierarchy:

(c ² − 1) dQ ₁

dz = A ₀ (z)Q ₁ + A ₁ Q ₀ (c ² −1) dQ _n

dz = A ₀ (z)Q _n +A ₁ Q _n−1 + A ₂ Q _n−2 , n = 2, 3, . . . Solution by variation of parameters:

Q ₁ (z) = Q ₀ (z) c ² − 1

Z _z

0 Q ₀ (y) ⁻¹ A ₁ Q ₀ (y) dy

Q _n (z) = Q ₀ (z) c ² − 1

Z z 0

Q ₀ (y) ⁻¹ A ₁ Q _n−1 (y)+ A ₂ Q _n−2

dy, n ≥ 2

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Collecting like powers of λ we obtain a hierarchy:

(c ² − 1) dQ ₁

dz = A ₀ (z)Q ₁ + A ₁ Q ₀ (c ² −1) dQ _n

dz = A ₀ (z)Q _n +A ₁ Q _n−1 + A ₂ Q _n−2 , n = 2, 3, . . . Solution by variation of parameters:

Q ₁ (z) = Q ₀ (z) c ² − 1

Z _z

0 Q ₀ (y) ⁻¹ A ₁ Q ₀ (y) dy

Q _n (z) = Q ₀ (z) c ² − 1

Z z 0

Q ₀ (y) ⁻¹ A ₁ Q _n−1 (y)+ A ₂ Q _n−2

dy, n ≥ 2

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By Abel’s identity:

Lemma

For all z ∈ R ^, λ ∈ C , there holds

det Q(z, λ) = exp 2cλz/(c ² − 1) c ² − 1 . After (tedious) computations:

Lemma

tr Q ₀ (T)Q ₀ (0) ⁻¹ = 2.

tr Q ₁ (T)Q ₀ (0) ⁻¹ = 2cT c ² − 1 . tr Q ₂ (T)Q ₀ (0) ⁻¹ = c ² T ²

(c ² − 1) ² − T _E c ² − 1

Z _T

0 f _z (y) ² dy.

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By Abel’s identity:

Lemma

For all z ∈ R ^, λ ∈ C , there holds

det Q(z, λ) = exp 2cλz/(c ² − 1) c ² − 1 . After (tedious) computations:

Lemma

tr Q ₀ (T)Q ₀ (0) ⁻¹ = 2.

tr Q ₁ (T)Q ₀ (0) ⁻¹ = 2cT c ² − 1 .

−1 c ² T ²

− T _E ^Z ^T ₂

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Perturbation of the Jordan block

By analyticity of the monodromy map:

M(λ) =

+∞

n=0 ∑

λ ⁿ n!

d ⁿ M dλ ⁿ (0).

(Standard perturbation theory, Kato.) In general, the Floquet multipliers bifurcate from λ = 0 in Pusieux series.

Fundamental matrix:

F(z, λ) = Q(z, λ)Q ₀ (0) ⁻¹ =

+∞

∑

n=0

λ ⁿ Q _n (z)Q ⁻¹ ₀ =:

+∞

∑

n=0

λ ⁿ F _n (z)

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Perturbation of the Jordan block

By analyticity of the monodromy map:

M(λ) =

+∞

n=0 ∑

λ ⁿ n!

d ⁿ M dλ ⁿ (0).

(Standard perturbation theory, Kato.) In general, the Floquet multipliers bifurcate from λ = 0 in Pusieux series.

Fundamental matrix:

F(z, λ) = Q(z, λ)Q ₀ (0) ⁻¹ =

+∞

∑

n=0

λ ⁿ Q _n (z)Q ⁻¹ ₀ =:

+∞

∑

n=0

λ ⁿ F _n (z)

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Lemma

We have convergent series expansions

M(λ) =

+∞

∑

n=0

λ ⁿ Q _n (T )Q ₀ (0) ⁻¹ ,

tr M(λ) =

+∞

∑

n=0

λ ⁿ tr Q _n (T )Q ₀ (0) ⁻¹ ,

and det M(λ) =

+∞

∑

n=0

2cT c ² − 1

n

λ ⁿ n! ,

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Expansion of the Floquet multipliers

µ , solutions to:

D(λ, ˆ µ) = det(M(λ) −µI) = µ ² −(tr M(λ))µ+ detM(λ) = 0

µ _± (λ) = 1 2

tr M(λ) ± (tr M(λ)) ² − 4 det M(λ) 1/2

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Expanding:

tr M(λ) ² − 4 det M(λ) =

tr Q ₀ (T)Q ₀ (0) ⁻¹ + λtr Q ₁ (T)Q ₀ (0) ⁻¹ +λ ² tr Q ₂ (T)Q ₀ (0) ⁻¹ 2

+

− 4

1 + 2cT

c ² − 1 λ + 2c ² T ² (c ² − 1) ² λ ²

+ O(λ ³ )

= 4∆λ ² + O(λ ³ ), where,

∆ := − T _E c ² − 1

Z _T

0 f _z (y) ² dy

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The two Floquet multipliers are analytic functions of λ at λ = 0 . Asymptotic form:

µ _± (λ) = 1 + cT

c ² − 1 ± ∆ ^1/2

λ + O(λ ² )

Definition

We define the modulational instability index to be the quantity

γ _M := sgn∆

Clearly sgn ∆ = sgn ¯ ∆

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The two Floquet multipliers are analytic functions of λ at λ = 0 . Asymptotic form:

µ _± (λ) = 1 + cT

c ² − 1 ± ∆ ^1/2

λ + O(λ ² )

Definition

We define the modulational instability index to be the quantity

γ _M := sgn∆

Clearly sgn ∆ = sgn ¯ ∆

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The two Floquet multipliers are analytic functions of λ at λ = 0 . Asymptotic form:

µ _± (λ) = 1 + cT

c ² − 1 ± ∆ ^1/2

λ + O(λ ² )

Definition

We define the modulational instability index to be the quantity

γ _M := sgn∆

Clearly sgn ∆ = sgn ¯ ∆

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Expansion of D near the origin

Lemma

The periodic Evans function D(λ, κ) , for (λ,κ) ∈ C × R ^, has the following expansion in a neighborhood of (λ, κ) = (0, 0) ,

D(λ, κ) = −∆λ ² +

iκ − cT c ² − 1 λ

2 + O(3),

where O(3) denotes terms of order three or higher in (λ, k) .

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Lemma

If γ _M = 1 then the solutions to D(λ, κ) = 0 near

(λ, κ) = (0, 0) emerge from the origin tangentially to the imaginary axis in the complex λ -plane:

λ(κ) = −ivκ + O(κ ² ),

with v ∈ R ^{, for} |κ| 1 .

If γ _M = −1 then the solutions emerge from the origin tangentially to two lines passing through the origin and forming non-zero angles with the imaginary axis:

λ(κ) = −(α + iβ)κ + O(κ ² ),

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Im

Re λ

λ Im

Re λ λ

Figure : Qualitative sketch of σ near the origin. γ M = 1 _(left); γ M = −1 (right).

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Theorem

Under assumptions (a), (b) and (c):

• γ _M = −1 for librational waves. Spectrally unstable.

• γ _M = 1 for rotational waves. The spectrum is tangent to the imaginary axis at λ = 0 .

Theorem

Under the non-degeneracy condition T _E 6= 0 if the

modulational instability index is γ _M = −1 then the

underlying periodic traveling wave is spectrally unstable.

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Theorem

Under assumptions (a), (b) and (c):

• γ _M = −1 for librational waves. Spectrally unstable.

• γ _M = 1 for rotational waves. The spectrum is tangent to the imaginary axis at λ = 0 .

Theorem

Under the non-degeneracy condition T _E 6= 0 if the modulational instability index is γ _M = −1 then the underlying periodic traveling wave is spectrally unstable.

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Relation to Whitham’s modulation theory

Reference: Whitham, Proc. Roy. Soc. Ser. A (1965).

WKB approximations of the form:

u(x, t) = f z(x,t) ε

+ O(ε),

k, ω are no longer constant (and hence, E _and c _{). We have} c = ω/k and k = θ _x , ω = −θ _t , θ = kx − ωt . Conservation of fluxons:

k _t + ω _x = 0

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Averaged Lagrangian

I[u] = Z Z

L(u, u _x , u _t ) dx dt, L(u, u _x , u _t ) = ¹ ₂ u ² _t − ¹ ₂ u ² _x − V(u).

In the wave u = f (x − ct) = Φ(kx − ωt) :

L(u,u _x , u _t ) = ¹ ₂ (ω ² − k ² )Φ _θ (θ) ² − V (Φ(θ))

Averaged Lagrangian:

hLi = 1 kT

Z _kT

0 1

2 (ω ² −k ² )Φ _θ (θ) ² −V (Φ(θ)) dθ = L ˜ ^(ω, ^k, ^E).

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Averaged Lagrangian variational principle

δ Z Z

L ˜ ^(ω, ^k, ^E) ^{dx dt} ⁼ ^0,

L ˜ E = 0, dispersion relation

Whitham’s system:

k _t + ω _x = 0

( L ˜ ω ) _t − ( L ˜ k ) _x = 0. ^() If the last system () is hyperbolic (Cauchy problem

well-posed) then the wave is stable under slow

modulations (Whitham, 1974).

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Averaged Lagrangian variational principle

δ Z Z

L ˜ ^(ω, ^k, ^E) ^{dx dt} ⁼ ^0,

L ˜ E = 0, dispersion relation

Whitham’s system:

k _t + ω _x = 0

( L ˜ ω ) _t − ( L ˜ k ) _x = 0. ^() If the last system () is hyperbolic (Cauchy problem

well-posed) then the wave is stable under slow modulations (Whitham, 1974).

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Equivalently (Whitham, 1965) we may express (*) in terms of E _and c

hLi = 1 T

Z _T

0 1

2 (c ² − 1)f _z (z) ² − V(f (z)) dz

=

√ 2 T

I

((c ² − 1)(E − V(η))) ^1/2 dη − E =: L ^(E, ^c).

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L ^(E, ^c) ⁼ ²

√ 2 T

p c ² − 1 Z _v ₂

v 1

p E − V (η)dη − E, (sup, lib) ,

L ^(E, ^{c) =} ⁻ ²

√ 2 T

p 1 − c ² Z _v ₄

v 3

p V(η) − E dη − E, (sub, lib) ,

L ^(E, ^{c) =}

√ 2 T

p c ² − 1 Z _π

0 p E − V(η) dη − E, (sup, rot) ,

L ^(E, ^{c) =} ⁻

√ 2 T

p 1 − c ² Z _π

0 p V(η) − E dη − E, (sub, rot) .

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Classical action (mechanics):

W (E, c) = √ 2

I

((c ² − 1)(E − V(η))) ^1/2 dη,

W (E, c) := sgn (c ² − 1) q

|c ² − 1|J(E),

J(E) :=

( J _lib (E), librations , J _rot (E), rotations , J _rot (E) := √

2 Z _P

0 q

sgn (c ² − 1)(E − V (η)) dη

J _lib (E) := 2

√ 2

Z _v _j q

sgn (c ² − 1)(E − V(η)) dη

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Lemma

For each of the four cases under consideration (sub- or superluminal, libration or rotation) there hold

W _E = T , (1)

W _c = cW

c ² − 1 . (2)

Taking average of conservation of energy and momentum equations we can express the Whitham modulation system (*) as:

W _c T

t

+ cW _c

T − E

x

= 0, 1

T

t

+ c T

x = 0.

(**)

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Lemma

Whitham’s system of equations (**) is equivalent to the system:

E c

t

+ A(E, c) E

c

x

= 0, (Wh)

A(E, c) = 1 N(E, c)

c(J(E)J ⁰⁰ (E)+J ⁰ (E) ² ) −J(E)J ⁰ (E) (c ² −1) ² J ⁰ (E)J ⁰⁰ (E) c(J(E)J ⁰⁰ (E)+J ⁰ (E) ² )

,

N(E, c) = J(E)J ⁰⁰ (E) + c ² J ⁰ (E) ² .

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Lemma

Whitham system (Wh) is hyperbolic if and only if J ⁰⁰ (E) < 0.

Characteristic velocities:

c(J(E)J ⁰⁰ (E)+ J ⁰ (E) ² ) −s _± = ±|c ² −1| −J(E)J ⁰⁰ (E)J ⁰ (E) ² 1/2

.

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Proof of Whitham’s modulational instability

Lemma

sgn J ⁰⁰ (E) = −γ _M . Proof:

T _E = W _EE = sgn (c ² − 1) q

|c ² − 1|J ⁰⁰ (E).

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Corollary

The quasilinear Whitham system (Wh) is hyperbolic if and only if γ M = 1 . In this case we say that the underlying periodic traveling wave is modulationally stable (otherwise we say it is modulationally unstable).

Theorem (Proof of Whitham’s instability)

Under the non-degenerate condition T _E 6= 0 , if the periodic traveling wave is modulationaly unstable in the sense defined by Whitham then it is spectrally unstable.

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Corollary

The quasilinear Whitham system (Wh) is hyperbolic if and only if γ M = 1 . In this case we say that the underlying periodic traveling wave is modulationally stable (otherwise we say it is modulationally unstable).

Theorem (Proof of Whitham’s instability)

Under the non-degenerate condition T _E 6= 0 , if the periodic

traveling wave is modulationaly unstable in the sense

defined by Whitham then it is spectrally unstable.

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Corollary

Under the non-degenerate condition T _E 6= 0 , a necessary condition for the spectral stability of a periodic wave is that the modulational instabilty index is γ _M = 1 , or equivalently, that the Whitham modulation system is hyperbolic.

Finally we recover:

Theorem (Whitham, 1974)

• Both super- and subluminal rotational waves are modulationally stable,

• Both super- and subluminal librational waves are modulationally unstable (and whence, spectrally unstable).

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Corollary

Under the non-degenerate condition T _E 6= 0 , a necessary condition for the spectral stability of a periodic wave is that the modulational instabilty index is γ _M = 1 , or equivalently, that the Whitham modulation system is hyperbolic.

Finally we recover:

Theorem (Whitham, 1974)

• Both super- and subluminal rotational waves are modulationally stable,

• Both super- and subluminal librational waves are

modulationally unstable (and whence, spectrally

unstable).

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Interpretation: “Modulational” stability pertains to perturbations for which the wave parameters underlie small variations with respect to wavelength. (Equivalently, perturbations near the origin λ = 0 ).

Whitham’s is an instability theory .

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Figure : Numerical plots of the Floquet spectrum G(λ) = 0 _for

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1 Introduction

2 Analysis of the monodromy map

3 Spectral (in)stability results

4 Multidimensional orbital stability

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(In)stability in the rotational case

Theorem

Under assumptions we have:

(A) Superluminal rotational waves are spectrally unstable.

(B) Subluminal rotational waves are spectrally stable.

That is: if λ ∈ σ then λ is purely imaginary.

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Part (A):

Define G : C → R ^by

G(λ) = log |µ ₊ (λ)| log |µ ₋ (λ)|.

G continuous in R ² ^and λ ∈ σ if and only if G(λ) = 0 . Fact:

if µ(λ) ∈ σM(λ) (Floquet mult. for (P) then

η(λ) = exp(−λcT/(c ² − 1)) ∈ σM _H (λ) (Floquet mult. for (H)). By Abel’s identity:

G(λ) =

Re cλT c ² − 1

2 − (log |η ₊ (λ)|) ²

=

Re cλT c ² − 1

2 − (log |η ₋ (λ)|) ² .

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Thus, for λ ∈ i R ^, G ≤ 0 . Moreover, G(iβ) = 0 iff iβ ∈ σ ∩ i R = σ ^H ∩ i R ^{. Thus,}

Corollary

Suppose β ∈ R is such that iβ

c ² − 1 2

∈ / σ ^H . Then

G(iβ) < 0 .

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Moreover, we can show:

Lemma

For a superluminal rotational wave, G(λ) > 0 for λ ∈ R ^, λ 1 , and there is a iβ _∗ in the spectral gap of σ _H , that is, G(iβ) < 0 .

By continuity, there must be an eigenvalue

λ = α _∗ t + iβ _∗ (1 − t) for some t ∈ (0,1) , where G(α _∗ ) > 0 , α ∗ large and real, such that G(λ) = 0 . Clearly, Re λ > 0 .

This shows (A).

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Moreover, we can show:

Lemma

For a superluminal rotational wave, G(λ) > 0 for λ ∈ R ^, λ 1 , and there is a iβ _∗ in the spectral gap of σ _H , that is, G(iβ) < 0 .

By continuity, there must be an eigenvalue

λ = α _∗ t + iβ _∗ (1 − t) for some t ∈ (0,1) , where G(α _∗ ) > 0 , α ∗ large and real, such that G(λ) = 0 . Clearly, Re λ > 0 .

This shows (A).

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Part (B): Spectral stability of subluminal rotations.

By energy estimates: define the Hamiltonian operator H = d ² /dz ² + V ⁰⁰ (f )/(c ² − 1) so that the spectral equation (P) is:

(c ² − 1)Hw(z) − 2cλw _z (z) + λ ² w(z) = 0

Lemma

The operator H is negative semidefinite in the case of a rotational wave. For librations, H is indefinite.

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If λ ∈ σ , multiply eq. by w ^∗ and integrate by parts on a fundamental period [0, T] :

PDF On the spectral, modulational and orbital stability of periodic ... - UNAM

Instituto de Investigaciones en Matem´aticas Aplicadas y en Sistemas

On the spectral, modulational and

orbital stability of periodic wavetrains for the sine-Gordon equation

Ram´on G. Plaza

Institute of Applied Mathematics (IIMAS), National Autonomous University of Mexico (UNAM)

Collaborators:

• Christopher K. R. T. Jones (Univ. of North Carolina, Chapel Hill)

• Robert Marangell (Univ. of Sydney)

• Peter D. Miller (Univ. of Michigan)

• Jaime Angulo Pava (Univ. of Sao Paulo)

This research was partially supported by: DGAPA-UNAM,

project PAPIIT IN-104814.

1 Introduction

2 Analysis of the monodromy map

3 Spectral (in)stability results

4 Multidimensional orbital stability

The nonlinear Klein-Gordon equation

Nonlinear Klein-Gordon with periodic potential:

u tt − u xx + V 0 (u) = 0. (nKG) for (x, t) ∈ R × [0, +∞) , u scalar, V ∈ C 2 , periodic.

Sine-Gordon equation:

u tt − u xx + sin u = 0. (SG)

V (u) = 1 − cos u

The nonlinear Klein-Gordon equation

Nonlinear Klein-Gordon with periodic potential:

u tt − u xx + V 0 (u) = 0. (nKG) for (x, t) ∈ R × [0, +∞) , u scalar, V ∈ C 2 , periodic.

Sine-Gordon equation:

u tt − u xx + sin u = 0. (SG)

V (u) = 1 − cos u

Applications (sine-Gordon):

• Surfaces with negative Gaussian curvature (Eisenhart, 1909)

• Propagation of crystal dislocations (Frenkel and Kontorova, 1939)

• Elementary particles (Perring and Skyrme, 1962)

• Propagation of magnetic flux on a Josephson line (Scott, 1969)

• Dynamics of fermions in the Thirring model (Coleman, 1975)

• Oscillations of a rigid pendulum attached to a

stretched rubber band (Drazin, 1983)

Assumptions on the potential:

(a) V : R → R is of class C 2 in all its domain and it is periodic with fundamental period P .

(b) V has only non-degenerate critical points.

(c) V 0 (u) 4 (V(u)/V 0 (u) 2 ) 00 ≥ 0 for all u under consideration.

Assumption (c) implies monotonicity of the period map with respect to the energy.

Traveling waves

u(x, t) = f (x − ct) , z = x − ct , solution to the nonlinear pendulum equation:

(c 2 − 1)f zz + V 0 (f (z)) = 0,

Sine-Gordon case:

(c 2 − 1)f zz + sin(f (z)) = 0,

c ∈ R (wave speed), c 2 6= 1 .

Upon integration:

1

2 (c 2 − 1)f z 2 = E − V(f ), E = constant (energy). Under assumptions:

0 < E 0 = max V(u)

Sine-Gordon case: V(u) = 1 − cos u , E 0 = 2 ,

1

2 (c 2 − 1)f z 2 = E − 1 + cos f (z).

W.l.o.g.

(d) V has fundamental period P = 2π and min u∈ R

V(u) = 0, max

u∈ R

V(u) = 2.

Classification

First dichotomy (wave speed):

• Subluminal waves: c 2 < 1

• Superluminal waves: c 2 > 1 Second dichotomy (energy E ):

• Librational wavetrain: f (z + T ) = f (z) . Closed trajectory inside the separatrix in the phase portrait.

• Rotational wavetrain: f (z + T) = f (z) ± 2π . Open trajectory outside the separatrix in the phase plane.

Sign f z is fixed.

Classification

First dichotomy (wave speed):

• Subluminal waves: c 2 < 1

• Superluminal waves: c 2 > 1 Second dichotomy (energy E ):

• Librational wavetrain: f (z + T ) = f (z) . Closed trajectory inside the separatrix in the phase portrait.

• Rotational wavetrain: f (z + T) = f (z) ± 2π . Open trajectory outside the separatrix in the phase plane.

Sign f z is fixed.

Figure : Phase portrait sine-Gordon case: V (u) = 1 − cos u :

superluminal c 2 > 1 (left); subluminal c 2 < 1 (right).

Figure : Phase portrait for V(u) = 1 − (0.861)(cos u + 1 3 sin(2u)) : superluminal c 2 > 1 (left); subluminal c 2 < 1 (right).

Superluminal librational: c 2 > 1 , 0 < E < E 0 .

K (E) = {u ∈ R : (E − V(u))/(c 2 − 1) ≥ 0} = disjoint union of intervals in (0, π) . In (v 1 , v 2 ) , only one

non-degenerate zero of V 0 . Librational (closed) periodic orbit.

u _tt − u _xx + V ⁰ (u) = 0. (nKG) for (x, t) ∈ R × [0, +∞) , u scalar, V ∈ C ² , periodic.

u _tt − u _xx + sin u = 0. (SG)

u _tt − u _xx + V ⁰ (u) = 0. (nKG) for (x, t) ∈ R × [0, +∞) , u scalar, V ∈ C ² , periodic.

u _tt − u _xx + sin u = 0. (SG)

(a) V : R → R is of class C ² in all its domain and it is periodic with fundamental period P _.

(c) V ⁰ (u) ⁴ (V(u)/V ⁰ (u) ² ) ⁰⁰ ≥ 0 for all u _under consideration.

(c ² − 1)f _zz + V ⁰ (f (z)) = 0,

(c ² − 1)f _zz + sin(f (z)) = 0,

c ∈ R (wave speed), c ² 6= 1 .

2 (c ² − 1)f _z ² = E − V(f ), E = constant (energy). Under assumptions:

0 < E ₀ = max V(u)

Sine-Gordon case: V(u) = 1 − cos u , E ₀ = 2 ,

2 (c ² − 1)f _z ² = E − 1 + cos f (z).

• Subluminal waves: c ² < 1

• Superluminal waves: c ² > 1 Second dichotomy (energy E _):

Sign f _z _{is fixed.}

• Subluminal waves: c ² < 1

• Superluminal waves: c ² > 1 Second dichotomy (energy E _):

Sign f _z _{is fixed.}

superluminal c ² > 1 (left); subluminal c ² < 1 (right).

Figure : Phase portrait for V(u) = 1 − (0.861)(cos u + ¹ ₃ sin(2u)) _: superluminal c ² > 1 (left); subluminal c ² < 1 _(right).

Superluminal librational: c ² > 1 , 0 < E < E ₀ _.

K ^{(E) =} ^{u ^∈ R : (E − V(u))/(c ² − 1) ≥ 0} = disjoint union of intervals in (0, π) . In (v ₁ , v ₂ ) , only one

non-degenerate zero of V ⁰ . Librational (closed) periodic orbit.

f _z =

√ c ² − 1

p E − V (f ), where f ∈ (v ₁ , v ₂ ) ⊂ K ^(E) ^.

c ² − 1 Z _v ₂

v ₁

(v ₁ , v ₂ ) = (−Arc cos(−E + 1), Arc cos(−E + 1))

Superluminal librational: c ² > 1 , 0 < E < E ₀ _.

K ^{(E) =} ^{u ^∈ R : (E − V(u))/(c ² − 1) ≥ 0} = disjoint union of intervals in (0, π) . In (v ₁ , v ₂ ) , only one

non-degenerate zero of V ⁰ . Librational (closed) periodic orbit.

f _z =

√ c ² − 1

p E − V (f ), where f ∈ (v ₁ , v ₂ ) ⊂ K ^(E) ^.

c ² − 1 Z _v ₂

v ₁

Sine-Gordon: wave oscillates around f = 0 , in (v ₁ , v ₂ ) = (−Arc cos(−E + 1), Arc cos(−E + 1))

Subluminal librational: c ² < 1 , 0 < E < E ₀ _.

K ^{(E) =} ^{u ^∈ R : (V(u) − E)/(1 − c ² ) ≥ 0} = disjoint union of intervals in (0, π) . In (v ₃ , v ₄ ) , only one

non-degenerate zero of V ⁰ . Librational (closed) periodic orbit.

f _z =

√ 1 − c ²

p V(f ) − E, where f ∈ (v ₃ , v ₄ ) ⊂ K ^(E) ^.

1 − c ² Z _v ₄

v ₃

(v ₃ , v ₄ ) = (−Arc cos(−E + 1), 2π − Arc cos(−E + 1))

Subluminal librational: c ² < 1 , 0 < E < E ₀ _.

K ^{(E) =} ^{u ^∈ R : (V(u) − E)/(1 − c ² ) ≥ 0} = disjoint union of intervals in (0, π) . In (v ₃ , v ₄ ) , only one

non-degenerate zero of V ⁰ . Librational (closed) periodic orbit.

f _z =

√ 1 − c ²

p V(f ) − E, where f ∈ (v ₃ , v ₄ ) ⊂ K ^(E) ^.

1 − c ² Z _v ₄

v ₃

(v ₃ , v ₄ ) = (−Arc cos(−E + 1), 2π − Arc cos(−E + 1))

Superluminal rotational: c ² > 1 , E > E ₀ _, E − V(f ) > 0 and K ^{(E) =} R . Rotation, f _z has fixed sign. Orbit outside the separatrix and f (z + T) = f (z) ± π for all z .

f _z ² = 2(E − V (f )) c ² − 1 > 0,

√ c ² − 1

√ 2 Z _π

Subluminal rotational: c ² < 1 , E < 0 , V(f ) − E ≥ 0 and

K ^{(E) =} R ^{with .} f _z has fixed sign. Orbit outside the separatrix and f (z + T ) = f (z) ± π for all z _.

√ 1 − c ²

Z _π dη

Superluminal rotational: c ² > 1 , E > E ₀ _, E − V(f ) > 0 and K ^{(E) =} R . Rotation, f _z has fixed sign. Orbit outside the separatrix and f (z + T) = f (z) ± π for all z .

f _z ² = 2(E − V (f )) c ² − 1 > 0,

√ c ² − 1

√ 2 Z _π

Subluminal rotational: c ² < 1 , E < 0 , V(f ) − E ≥ 0 and

K ^{(E) =} R ^{with .} f _z has fixed sign. Orbit outside the separatrix and f (z + T ) = f (z) ± π for all z _.