Viscous balance laws

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Existence and spectral instability of bounded periodic waves for viscous balance laws

Ram´on G. Plaza

Institute of Applied Mathematics (IIMAS)

Universidad Nacional Aut´onoma de M´exico (UNAM) March 29, 2021

III Summer School, Universidade Estadual de Maring´a, Brasil.

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Joint work with: Enrique ´Alvarez(Ph.D. student, UNAM) Sponsors:

• DGAPA-UNAM, program PAPIIT, grant no. IN-100318.

• CONACyT, M´exico, grant no. 307909

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Viscous balance laws

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Viscous balance laws

General family of viscous balance laws:

u_t+f(u)_x =u_xx+g(u), (VBL) whereu=u(x,t)∈Randx∈R,t>0.

• f =f(u) - nonlinearfluxfunction,

• g =g(u) -balanceor production term

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Interpretation

• Viscous balance laws appear as regularizationsof scalarconservation laws:

u_t+f(u)_x = 0.

• Balance laws introduce a production term u_t+f(u)_x =g(u).

• Viscosity or diffusionappears naturally as a regularization (parabolic) term, e.g., Burgers’ equation

u_t+ ¹₂u²)_x=u_xx.

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• Viscous balance laws describe the evolution of a densityuof point particles which:

• diffuse

• get advected with speedf⁰(u)

• react a a per c´apita production rateg(u)/u

• Applications to:

• roll waves

• nozzle flow

• combustion theory

• Scalar viscous balance laws are simplified models that combine these effects into one single equation.

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Hypotheses (i)

Assumptions:

f ∈C⁴(R). (H₁)

g ∈C³(R) is oflogistic or Fisher-KPPtype:

g(0) =g(1) = 0, g⁰(0)>0, g⁰(1)<0, g(u)>0, ∀u∈(0,1), g(u)<0, ∀u∈(−∞,0).

(H2)

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Hypotheses (ii)

There existsu_∗∈(−∞,0) such that Z 0

u∗

g(s)ds+ Z 1

0

g(s)ds= 0.

(H3)

a₀:=f⁰⁰⁰(0)−f⁰⁰(0)g⁰⁰(0)

pg⁰(0) 6= 0, (genericity condition) (H₄)

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Hypotheses (iii)

Under (H₂) and (H₃),u∗∈(−∞,0) is the uniquevalue such that (H₃) holds and

Z 1 u

g(s)ds>0, ∀u∈(u∗,1).

Therefore we can define γ(u) :=

r 2

Z 1 u

g(s)ds, u∈(u_∗,1), as well as

I0:=

Z 1 u∗

γ(s)ds>0, I1:=

Z 1 u∗

f⁰(s)γ(s)ds, J:= 2

Z 1 u∗

f⁰(s)p

1 +γ⁰(s)²ds, L:= 2 Z 1

u∗

p1 +γ⁰(s)²ds.

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Hypotheses (iv)

I0J6=LI1, (non-degeneracy condition) (H5)

f⁰(1)6=I₁ I0

, (saddle condition) (H₆)

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Example

Burgers-Fisherequation:

ut+uux=uxx+u(1−u), x∈R,t>0, Burgers’ flux:

f(u) =1 2u², Logistic or Fisher-KPPreaction:

g(u) =u(1−u).

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Figure: Logistic reaction functiong(u) =u(1−u).

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Periodic traveling waves

Aspatially periodic traveling wave is a solution of the form u(x,t) =ϕ(x−ct),

wherec∈R-wave speed, andϕ∈C¹(R) -profile function.

The wave isperiodicwith fundamental periodT >0 if ϕ(z+T) =ϕ(z), ∀z∈R, z:=x−ct, The wave isboundedif

|ϕ(z)|,|ϕ⁰(z)| ≤C, ∀z∈R, some C>0.

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Main results

Theorem (existence of small amplitude periodic waves)

Under (H1)thru (H4), there exist acritical speed c0:=f⁰(0),andε0>0 such that, for each0<ε<ε0 there exists a unique (up to translations) periodic traveling waveto (VBL)of the form u(x,t) =ϕ^ε(x−c(ε)t), traveling with speed c(ε) =c₀+ε if a₀>0, or c(ε) =c₀−ε if a₀<0, with fundamental period,

T_ε= 2π

pg⁰(0)+O(ε), as ε→0⁺.

The profileϕ^ε=ϕ^ε(·)is of class C³(R), satisfiesϕ^ε(z+T_ε) =ϕ^ε(z)for all z∈Rand is ofsmall amplitude,

|ϕ^ε(z)|,|(ϕ^ε)⁰(z)| ≤C√ ε, for all z∈Rand some uniform C>0.

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Theorem (existence of large period waves)

Under (H1)-(H3),(H5)and (H6), there is acritical speed c1:=I1/I0

such that the equation(VBL)has atraveling pulsesolution (homoclinic orbit), u(x,t) =ϕ⁰(x−c1t), with speed c1,ϕ⁰∈C³(R)and

|ϕ⁰(z)−1|,|(ϕ⁰)⁰(z)| ≤Ce^−κ|z|,

for all z∈R, someκ>0. Moreover, there existsε1>0 such that, for each0<ε<ε1there exists a uniqueperiodic traveling wavesolution to (VBL)of the form u(x,t) =ϕ^ε(x−c(ε)t), with speed c(ε) =c₁+εif f⁰(1)<c1or c(ε) =c1−ε if f⁰(1)>c1, withlarge fundamental period and bounded amplitude,

T_ε=O(|logε|)→∞, |ϕ^ε(z)|,|(ϕ^ε)⁰(z)|=O(1), asε→0⁺.

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Theorem (existence of large period waves (continuation)) Moreover, the periodic orbitsconverge to the homoclinic(or traveling pulse) asε→0⁺ and satisfy (after a suitable reparametrization of z),

sup

z∈[−^T₂^ε,^T₂^ε]

|ϕ⁰(z)−ϕ^ε(z)|+|(ϕ⁰)⁰(z)−(ϕ^ε)⁰(z)|

≤Cexp

−κT_ε 2

,

|c1−c(ε)|=ε≤Cexp −κT_ε , for some uniform C>0, sameκ>0and for all0<ε<ε1.

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Theorem (spectral instability of small-amplitude waves)

Under conditions (H1)thru (H4), there exists 0<ε¯0<ε0such that every small-amplitudeperiodic waveϕ^ε with0<ε<¯ε0isspectrally unstable:

the Floquet spectrum of the linearized operator around the wave intersects the unstable half planeC+={λ ∈C:Reλ >0}.

Theorem (spectral instability of large period waves)

Under assumptions (H1)-(H3),(H5)and (H6), there exists0<ε¯1<ε1

such that everylarge periodwaveϕ^ε with0<ε<¯ε1is spectrally unstable: the Floquet spectrum of the linearized operator around the wave intersects the unstable half planeC+={λ∈C:Reλ>0}.

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Existence of small amplitude

periodic waves

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Associated ODE system

Substituteu(x,t) =ϕ(x−ct) into (VBL):

−cϕ⁰+f⁰(ϕ)ϕ⁰=ϕ⁰⁰+g(ϕ), ⁰ =d/dz.

DenoteU:=ϕ(z),V :=ϕ⁰(z) to obtain the first order ODE planar system:

U⁰=F(U,V,c) :=V,

V⁰=G(U,V,c) :=−cV+f⁰(U)V−g(U). (ODE) (H₁) and (H₂) imply: F,G∈C³(R³) and there existtwo equilibrium points:

P₀= (0,0), P₁= (1,0).

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Linearization at equilibria (i)

Jacobian with respect to (U,V) of the right hand side of (ODE):

A(U,V) := F_U F_V G_U G_V

!

= 0 1

f⁰⁰(U)V−g⁰(U) −c+f⁰(U)

! .

A0=A(0,0) andA1=A(1,0) are thelinearizations of (ODE) evaluated atP0andP1:

A₀= 0 1

−g⁰(0) −c+f⁰(0)

!

, A₁= 0 1

−g⁰(1) −c+f⁰(1)

! .

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Linearization at equilibria (ii)

•Eigenvalues ofA1: λ₁^±(c) =1

2 f⁰(1)−c

±1 2

p(f⁰(1)−c)²−4g⁰(1),

From (H2),g⁰(1)<0 andP1= (1,0) is ahyperbolic saddlefor each value ofc∈R.

•Eigenvalues ofA0: λ₀^±(c) =1

2 f⁰(0)−c

±1 2

(f⁰(0)−c)²−4g⁰(0)1/2

. HenceP₀= (0,0) isa node, a focus or a center, depending onc∈R.

c∈Ris thebifurcation parameter.

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Andronov-Hopf bifurcation (i)

Planar system of the form:

U⁰=F(U,V,µ) V⁰=G(U,V,µ)

F,G of classC³,µ∈R-bifurcationparameter. AssumeP0= (U0,V0) is an equilibrium point; eigenvalues of the linearization atP0:

λ^±(µ) =α(µ)±iβ(µ). Assume that forµ=µ0 the following conditions hold:

(a) non-hyperbolicity condition: α(µ0) = 0,β(µ0) =ω06= 0, and sgn(ω0) =sgn((∂G/∂U)(U0,V0,µ0)).

(b) transversality condition:

dα

dµ(µ0) =d06= 0.

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Andronov-Hopf bifurcation (ii)

(c) genericity condition: a06= 0, wherea0is thefirst Lyapunov exponent, a₀:= 1

16 F_UUU+F_UVV+G_UUV+G_VVV + + 1

16ω0

F_UV(F_UU+F_VV)−G_UV(G_UU+G_VV)−F_UUG_UU+F_VVG_VV ,

(derivatives of F andG are evaluated at (U0,V0,µ0)).

Then there existsε>0 such that aunique family of closed periodic orbit solutionsbifurcates from the equilibrium point into the region:

((µ₀,µ0+ε), ifa₀d₀<0, (µ0−ε,µ0), ifa0d0>0.

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Andronov-Hopf bifurcation (iii)

Theiramplitude and fundamental periodbehave like

|ϕ|,|ϕ_z|=O(p

|µ−µ0|), T(µ) = 2π

|ω0|+O(|µ−µ0|), asµ→µ0. Stabilityas solutions to the ODE.Orbits are:

stable (supercritical Hopf bifurcation)ifa0<0 unstable (subcritical Hopf bifurcation)ifa0>0.

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Existence of small-amplitude periodic waves (i)

Proof of Theorem.

Write eigenvalues ofA0asλ₀^±=α(c)∓iβ(c), where α(c) :=1

2(f⁰(0)−c), β(c) :=−1 2

p4g⁰(0)−(f⁰(0)−c)², c≈f⁰(0).

Thus,α(c0) = 0 for the onlybifurcation valueof the speed: c0:=f⁰(0).

Atc=c0 the origin is a center with eigenvalues λ₀⁺(c0) =−ip

g⁰(0), λ₀⁻(c0) =ip g⁰(0).

Notice thatω0:=β(c0) =−p

g⁰(0)6= 0. SinceG_U=f⁰⁰(U)V−g⁰(U), (G_U)|₍₀_,₀_,c₀₎=−g⁰(0)<0,

yieldingsgn(ω0) =sgn((G_U)|₍₀_,₀_,c₀₎) =−1, that is, the non-hyperbolicity condition (a).

Thetransversality condition (b)also holds:

dα

dc (c0) =−1

2 =:d0<0.

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Existence of small-amplitude periodic waves (ii)

The Lyapunov exponent reduces to a0= 1

16 G_UUV+G_VVV

|₍₀_,₀_,c₀₎− 1

16ω₀ G_UV(G_UU+G_VV)

|₍₀_,₀_,c₀₎.

Upon calculations,

a0= 1 16

f⁰⁰⁰(0)−f⁰⁰(0)g⁰⁰(0) pg⁰(0)

=a0

166= 0,

in view of (H4). This verifies thegenericity condition (c). Sinced0<0 andsgn(a₀) =sgn(a₀) we obtain the result.

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Examples

(I) Burgers-Fisher equation The Burgers-Fisher equation:

u_t+uu_x=u_xx+u(1−u), x∈R,t>0, Burgers’ flux:

f(u) =1 2u², Logistic or Fisher-KPP reaction:

g(u) =u(1−u).

f andg satisfy (H1) and (H2). Hereu∗=−1/2 is such that Z 1

u∗

g(s)ds= Z 1

−¹₂

s(1−s)ds= 0.

Thus, (H₃) is also satisfied.

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Sinceg⁰(u) = 1−2u,g⁰⁰(u) =−2,f⁰(u) =u,f⁰⁰(u) = 1, a₀=−f⁰⁰(0)g⁰⁰(0)

pg⁰(0) = 2>0, Thus, the genericity condition (H4) holds.

Bifurcation speed: c0=f⁰(0) = 0. Sincea0>0 then for each speed value c∈(0,ε0) with 0<ε01 there is a small amplitude periodic wave. This corresponds to asubcritical Hopf bifurcation. Their fundamental period isT= 2π+O(c), forc≈0⁺.

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Figure: Phase portrait for the speed valuec=−0.05<0. The origin is a repulsive node and all nearby solutions move away from it.

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Figure: Phase portrait whenc=c0= 0; a subcritical Hopf bifurcation occurs.

The origin is a center and solutions move away if they start far enough from the origin and locally rotate around a linearized center otherwise.

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Figure: Phase portrait forc= 0.005. The orbit in red is a numerical

approximation of the unique small amplitude periodic wave for this speed value, the origin is an attractive node and nearby solutions inside the periodic orbit approach zero, whereas solutions outside the periodic orbit move away since the orbit is unstable.

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Figure: Graph (in red) of the periodic profileϕ forc= 0.005 as a function of z=x−ct

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Examples

(II) Logistic Buckley-Leverett model

Consider the following viscous balance law u_t+∂_x u²

u²+¹₂(1−u)²

!

=u_xx+u(1−u), x∈R, t>0, underlies the nonlinearBuckley-Leverett flux function,

f(u) = u² u²+¹₂(1−u)².

It captures the main features of two phase fluid flow in a porous medium.

Again,logisticreaction: g(u) =u(1−u).

f andg satisfy (H₁), (H₂) and (H₃). After computing the derivatives, a0=f⁰⁰⁰(0)−f⁰⁰(0)g⁰⁰(0)

pg⁰(0) = 32, and the genericity condition (H₄) holds.

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Bifurcation speed: c0=f⁰(0) = 0. Sincea0>0 then for each speed value c∈(0,ε0) with 0<ε01 there is a small amplitude periodic wave. This corresponds to asubcritical Hopf bifurcation. Their fundamental period isT= 2π+O(c), forc≈0⁺.

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Figure: Phase portrait for the speed valuec=−0.05<0. The origin is a repulsive node and all nearby solutions move away from it.

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Figure: Phase portrait whenc=c0= 0; a subcritical Hopf bifurcation occurs.

The origin is a center and solutions move away if they start far enough from the origin and locally rotate around a linearized center otherwise.

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Figure: Phase portrait forc= 0.0025. The orbit in red is a numerical

approximation of the unique small amplitude periodic wave for this speed value, the origin is an attractive node and nearby solutions inside the periodic orbit approach zero, whereas solutions outside the periodic orbit move away since the orbit is unstable.

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Figure: Graph (in red) of the periodic profileϕ forc= 0.0025 as a function of z=x−ct

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Examples

(III) Modified generalized Burgers-Fisher equation Here

f(u) =1 4u⁴−1

3u³, g(u) =u−u⁴.

Clearly, assumptions (H1) - (H3) hold, where the unique value u_∗≈ −0.72212 is approximated numerically. Upon calculation of the derivatives,

a0=−2<0.

Thus, (H₄) holds.

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Bifurcation speed: c0=f⁰(0) = 0. Sincea0<0 then the family occurs for negativespeed valuesc(ε) =−ε<0 =c₀=f⁰(0), sufficiently small. This is asupercritical Hopf bifurcationand the small amplitude periodic orbits arestable as solutions to the ODE, with speed valuec(ε) =−ε∈(−ε0,0) with 0<ε01. Their fundamental period isT= 2π+O(c), forc≈0⁺.

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Figure: Phase portrait for the speed valuec= 0.05>c0= 0. The origin is an attractive node and all nearby solutions converge to it.

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Figure: Phase portrait whenc=c0= 0; a supercritical Hopf bifurcation occurs.

The origin is a center and solutions move away if they start sufficiently far from the origin and rotate locally around a linearized center otherwise.

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Figure: Phase portrait forc=−0.005. The orbit in red is a numerical

approximation of the unique small amplitude periodic wave for this speed value, the origin is a repulsive node and nearby solutions both inside and outside the periodic orbit approach the periodic wave because it is stable as a solution to the ODE.

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Figure: Graph (in red) of the periodic profileϕ forc=−0.005 as a function of z=x−ct

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Existence of large period waves

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Augmented system

First we establish the existence of anhomoclinic loop. We apply Melnikov’s method. Consider theaugmented system:

U⁰=V,

V⁰=−cV+af⁰(U)V−g(U), (A) wherea∈Ris a new auxiliary parameter. Write it innear-Hamiltonian form:

U⁰=∂VH+εR(U,V,µ),

V⁰=−∂_UH+εQ(U,V,µ), (PS)

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where:

µ= (µ1,µ2)∈R², a=:ε µ1,c=:ε µ2, H(U,V) := 1

2V²+ Z U

0

g(s)ds, is theHamiltonian, and

R(U,V,µ)≡0,

Q(U,V,µ) :=µ1f⁰(U)V−µ2V, areautonomous perturbations. Here 0<ε1 is small.

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Hamiltionan system

The associatedHamiltonian (unperturbed)system is U⁰=∂VH=V,

V⁰=−∂_UH=−g(U). (HS)

Observations:

• P0= (0,0) andP1= (1,0) are equilibrium points forboththe Hamiltonian system (HS) and the perturbed system (PS).

• It is easy to check that: P0 is acenterandP1 is ahyperbolic saddle for the Hamiltonian system (HS).

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• Important: P1= (1,0) is also ahyperbolic saddle for the perturbed system (PS) for any parameter valuesaandc(equivalently, for any ε, µ1andµ2). Indeed, the linearization of (PS) aroundP1= (1,0) is

Ae^ε(1,0) = 0 −g⁰(1) 1 af⁰(1)−c

! ,

having eigenvalues λ_±^ε=1

2

af⁰(1)−c±p

(af⁰(1)−c)²−4g⁰(1) ,

and in view of (H2), we haveλ−^ε <0<λ₊^ε for all values ofaandc, yielding a hyperbolic saddle.

• P₀= (0,0) is acenterfor the perturbed system only ifc=af⁰(0) (which happens whenε= 0).

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Energy levels for the Hamiltonian system (i)

• Theenergy levelsatP₀= (0,0) andP₁= (1,0) as equilibria of the Hamiltonian system (HS) are

β:=H(1,0) = Z 1

0

g(s)ds>0, andH(0,0) = 0.

• The set

Γ^β :={(U,V)∈R²:H(U,V) =β},

is a homoclinic loopfor the Hamiltonian system joiningP1= (1,0) with itself. It is given explicitly by

V(U) =±V^β(U) :=± s

2 β−

Z U 0

g(s)ds

=±γ(U), U∈(u∗,1).

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Energy levels for the Hamiltonian system (ii)

• There exists a family ofperiodic orbitsfor the Hamiltonian system (HS),

Γ^h:={(U,V)∈R²:H(U,V) =h}, h∈(0,β), such that

(i) Γ^h→P0= (0,0) ash→0⁺, and (ii) Γ^h→Γ^β ash→β⁻.

IfGe(u) =^R₀^ug(s)ds then under (H₂) and (H₃) it is easy to check that for eachh∈(0,β) there exist unique valuesu1(h)∈(u∗,0) and u₂(h)∈(0,1) such thatGe(u₁(h)) =G(ue ₂(h)) =hand the periodic orbits are given by

V(U) =±V^h(U) :=± s

2 h−

Z U 0

g(s)ds , forU∈(u₁(h),u₂(h)),h∈(0,β).

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Figure: Homoclinic loop Γ^β in the (U,V)-plane and periodic orbits Γ^h, h∈(0,β), for the Hamiltonian system. Hereg(u) =u(1−u) (logistic).

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Energy levels for the Hamiltonian system (iii)

• IfT =T(h) is thefundamental period of the periodic orbit Γ^h, h∈(0,β), then T(h)→∞ash→β⁻. The period can be computed explicitly:

T(h) =√ 2

Z u2(h) u1(h)

dy

ph−^R₀^yg(s)ds, h∈(0,β).

From standard properties of Hamiltonian systems, 0<T(h)<∞for each h∈(0,β) and T(h)→∞ash→β⁻, which is theinfinite period of the homoclinic loopΓ^β.

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Melnikov integrals for the perturbed system

Define the associatedMelnikov integralsfor the perturbed system (cf.

Han and Yu (2012), Chicone (2006)):

M(h,e µ) :=

Z

int(Γ_h)

(∂_UR+∂_VQ)dU dV. They satisfy (see Han and Yu (2012)):

• Me∈C^∞for|ε|+|h−h0| 1 small for anyh0∈(0,β), all µ∈R².

• Thederivative ofMe with respect tohis determined by:

∂hMe(h,µ) = I

Γ^h

(∂_UR+∂VQ)dσ_h, h∈(0,β), µ∈R², where dσ_h denotes the arc length measure on Γ^h.

The Melnikov integrals precisely ath=β are M(µ) :=M(βe ,µ) =

Z

int(Γ^β)

(∂_UR+∂VQ)dU dV, M1(µ) :=∂hM(βe ,µ) =

I

Γ^β

(∂_UR+∂VQ)dσ_β.

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Melnikov’s theorem

Theorem (Melnikov’s method for perturbed homoclinic orbits) Suppose that P1 is a hyperbolic saddle for the unperturbed Hamiltonian system, with a homoclinic loopΓ^β. Ifε>0 is small then the perturbed system(PS)has aunique hyperbolic saddle P₁(ε) =P₁+O(ε).

Moreover, if M(µ0) = 0and M1(µ0)6= 0then, for eachε>0sufficiently small, the perturbed system withµ=µ0 has aunique hyperbolic homoclinic loopΓ^β_ε relative to the stable and unstable manifolds of the hyperbolic saddle P1(ε). If M(µ)has no zeroes and|ε| 6= 0is small, then the stable and unstable manifolds of P₁(ε)do not intersect.

– Classical theorem by Melnikov (1963). See also Wiggins (2003), Chicone (2006).

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Existence of a homoclinic loop

Proposition

Under (H1) - (H3) and (H5), system (A) has aunique homoclinic orbit joining the hyperbolic saddle pointP₁= (1,0) with itself for the parameter valuesa= 1 andc=c1=I1/I0.

Proof. Follows upon application ofMelnikov’s method. Since R(U,V,µ) = 0 andQ(U,V,µ) =µ1f⁰(U)V−µ2V, we evaluate the Melnikov integrals:

M(µ) = Z

int(Γ^β)

(µ1f⁰(U)−µ2)dV dU= Z1

u∗

Z _γ(U)

−γ(U)

(µ1f⁰(U)−µ2)dV dU

= 2 µ₁

Z 1

u_∗

f⁰(U)γ(U)dU−µ₂ Z 1

u_∗

γ(U)dU

= 2 µ1I1−µ2I0 . Hence,M(µ) = 0 only when

µ2=I1

I0

µ1. (*)

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EvaluateM1at any µ∈R² satisfying (*):

M₁(µ) = I

Γ^β

(µ₁f⁰(U)−µ2)dσ_β=µ1

I

Γ^β

f⁰(U)dσ_β−µ2

I

Γ^β

dσ_β

= 2µ₁ Z1

u∗

f⁰(U)p

1 +γ⁰(U)²dU−µ2|∂Ω_β|

=µ₁J−µ₂L

=µ1

J−I₁

I₀

L

6= 0

ifµ16= 0 and in view of (H5). This implies that there is a wholeline of simple zeroesof the Melnikov function determined by

C :=n

(µ₁,µ2)∈R² :µ2=I₁ I0

µ1

o\{(0,0)}.

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Now, fixε>0 sufficiently small. By Melnikov’s theorem, the perturbed system (PS) has auniquehyperbolic pointP1(ε) =P1+O(ε). But since P₁= (1,0) is a hyperbolic saddle for system (PS) for any parameter values, we obtain thatP1(ε)≡P1= (1,0). Moreover, for any fixed µ0∈C,µ06= (0,0), the perturbed system (PS) with µ=µ0has a unique hyperbolic homoclinic loop, Γ^β_ε, relative to the stable and unstable manifolds of the saddleP1. For thisfixed value of ε>0, let us define

µ1:=1

ε >0, µ2=I1

I₀

µ1,

so thatc=ε µ2=I₁/I₀anda=ε µ1= 1. Therefore,

µ0:= (µ1,µ2) = (1/ε,(I0/εI1))∈C ⊂R²is a bifurcation value for which the Melnikov integral has a simple zero. In this case the critical value for the speed isc=c1:=I1/I0. Now, sinceM(µ0) = 0 andM1(µ0)6= 0 we conclude that the perturbed system has aunique homoclinic loop Γ^β_ε relative to the stable and unstable manifolds atP₁, for parameter values a= 1 andc=c1.

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Traveling pulse

Corollary (existence of a traveling pulse)

The system(ODE)has a homoclinic loop for the speed value c=c₁=I₁/I₀, which we denote asΓ₀:={(ψ,ψ⁰)(z) :z∈R}, with ψ∈C³(R)and such that(ψ,ψ⁰)(z)→(1,0)as z→ ±∞. The convergence is exponential: there exist constants C,κ>0such that

|ψ(z)−1|,|ψ⁰(z)| ≤Ce^−κ|z|, as |z| →∞.

This homoclinic orbit is associated to a unique (up to translations) traveling pulsesolution to (VBL)of the form u(x,t) =ψ(x−c₁t)and traveling with speed c=c1.

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Andronov-Leontovich’s theorem

Theorem (Andronov-Leontovich)

Consider a planar system of the form U⁰=F(U,V,µ) V⁰=G(U,V,µ)

where F and G are smooth andµ∈R. Assume that (U0,V0)is a hyperbolic saddlefor allµ near µ0; that atµ=µ0 the eigenvalues are λ1(µ0)<0<λ2(µ0); and that the system has ahomoclinic orbitΓ0at the saddle. Let us define thesaddle quantityas

Σ₀:=λ1(µ₀) +λ2(µ₀), and suppose thatΣ06= 0.

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Theorem (Andronov-Leontovich (continuation))

Then:

(a) IfΣ0<0 then for sufficiently smallµ−µ0>0there exists aunique stable periodic orbit Γ(µ)bifurcating fromΓ0which as µ→µ₀⁺gets closer to the homoclinic loop atµ=µ0. Whenµ<µ0there are no periodic orbits.

(b) IfΣ0>0 then for sufficiently smallµ−µ0<0there exists aunique unstable periodic orbitΓ(µ)bifurcating fromΓ0which asµ→µ₀⁻ becomes the homoclinic loop atµ=µ0. Whenµ>µ0 there are no periodic orbit.

– Classical theorem by Andronov and Leontovich (1937). See, e.g., Shilnokov et al. (2001).

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Periodic wavetrains with large fundamental period (i)

Proof of theorem.

Assume (H1) - (H3), (H5) and (H6). We know that whenc=c1, P₁= (1,0) is a hyperbolic saddle for system (ODE) and there is a homoclinic orbit joiningP1with itself. The eigenvalues of the linearization atP1,

λ1(c1) =1

2(f⁰(1)−c1)−1 2

p(f⁰(1)−c1)²−4g⁰(1)<0, λ2(c₁) =1

2(f⁰(1)−c₁) +1 2

p(f⁰(1)−c₁)²−4g⁰(1)>0,

satisfyλ1(c₁)<0<λ2(c₁). Hence thesaddle quantity is non-zero, Σ0=f⁰(1)−c16= 0, in view of (H6). Andronov-Leontovich’s theorem implies there exists ˜ε1>0 small such that, iff⁰(1)>c1 (respectively, f⁰(1)<c1) then for each c∈(c1−˜ε1,c1) (respectively, c∈(c1,c1+ ˜ε1)) there exists aunique closed periodic orbitfor system (ODE) with fundamental periodT(c).

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Periodic wavetrains with large fundamental period (ii)

We obtain afamily of periodic orbits parametrized by ε:=|c−c₁| ∈(0,˜ε1),

denoted as (U^ε,V^ε)(z) =: (ϕ^ε,(ϕ^ε)⁰)(z),z∈R, with speed value c(ε) =c₁+ε iff⁰(1)<c₁ orc(ε) =c₁−ε iff⁰(1)>c₁, fundamental periodT_ε, and amplitude|ϕ^ε(z)|,|(ϕ^ε)⁰(z)|=O(1) asε→0⁺.

Moreover, the family of orbitsconverge to the homoclinic looprelative to the saddle pointP₁= (1,0) asε→0⁺, which we denote as

(ϕ⁰,(ϕ⁰)⁰)(z) := (ψ,ψ⁰)(z), z∈R,

with (ϕ⁰,(ϕ⁰)⁰)(z)→(1,0) exponentially fast as z→ ±∞. Hence, there existsδe(ε)>0 such thatδe(ε)→0 asε→0⁺and

|ϕ⁰(z)−ϕ^ε(z)| ≤δe(ε), for all |z| ≤T_ε 2 .

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Periodic wavetrains with large fundamental period (iii)

It can be shown that the homoclinic loop (ϕ⁰,(ϕ⁰)⁰) = (ψ,ψ⁰) is non-degenerate(see definition by Beyn (1990)). Apply Corollary 3.2 in Beyn (1990) to conclude that there exists 0<ε1<˜ε1sufficiently small and an appropriate reparametrization of the phasez such that

sup_z∈[−Tε

2,^Tε₂] |ϕ⁰(z)−ϕ^ε(z)|+|(ϕ⁰)⁰(z)−(ϕ^ε)⁰(z)|

≤Cexp

− min{λ2(c1),|λ1(c1)|}_T_ε

2

,

ε≤Cexp − min{λ2(c1),|λ1(c1)|}

T_ε ,

for each 0<ε<ε1. Setκ= min{λ2(c₁),|λ1(c₁)|}>0. This shows the desired bounds. Notice the second bound impliesT_ε=O(|logε|)→∞.

Finally, the family of orbitsϕ^ε is of classC³inz∈Rand in the bifurcation parameterc, by the regularity off andg, and to standard ODE results.

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Examples

(I) Burgers-Fisher equation

Wheng(u) =u(1−u) the functionγ=γ(u) is γ(u) = 1

√3

p1−3u²+ 2u³, u∈(−¹₂,1).

Hence,I0andI1can be computed:

I0= Z1

−¹ 2

γ(s)ds= 1

√ 3

Z 1

−¹ 2

p1−3s²+ 2s³ds=3 5, I₁=

Z1

−¹₂

f⁰(s)γ(s)ds= 1

√3 Z1

−¹₂

sp

1−3s²+ 2s³ds= 3 35. LandJ are approximated numerically,

L= 2 Z 1

−¹₂

r1−4s³+ 3s⁴

1−3s²+ 2s³ ds≈4.0734, J= 2

Z 1

−¹₂

s

r1−4s³+ 3s⁴

1−3s²+ 2s³ ds≈0.6906.

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Thenon-degeneracy condition(H5) holds: I0J≈0.41526=LI1≈0.3492.

Thehomoclinic loop speedis

c1=I1

I0

=1 7. Thesaddle condition (H6)holds: f⁰(1) = 16=c1.

Sincef⁰(1) = 1>c1= 1/7, then the family of periodic waves with large period emerge forc∈(¹₇−ε1,¹₇) withε1>0 small.

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-�� -��

-��

��

Figure: Numerical approximation of the homoclinic loop for the Burgers-Fisher equation with speed valuec1= 1/7 (in blue, dashed line) and the periodic wave nearby with speed valuec1−ε,ε≈0.05 (solid, orange line).

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Examples

(II) Logistic Buckley-Leverett model

Same logistic function, soI0= 3/5. We also have, I1=

Z 1

−1/2

f⁰(s)γ(s)ds= Z 1

−1/2

s(1−s)√

1−3s²+ 2s³

s²+¹₂(1−s)²2 ds= 0.353458.

LandJ are approximated numerically, L= 2

Z 1

−1/2

r1−4s³+ 3s⁴

1−3s²+ 2s³ ds≈4.0734, J= 2

Z 1

−1/2

s(1−s) s²+¹₂(1−s)²2

r1−4s³+ 3s⁴

1−3s²+ 2s³ ds≈1.6272.

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Thenon-degeneracy condition(H5) holds: I0J≈0.97636=LI1≈1.4398.

Thehomoclinic loop speedis c₁=I₁

I0

= 0.589097.

Thesaddle condition (H₆)holds: f⁰(1) = 06=c₁. Sincef⁰(1) = 0<c₁, then the periodic waves emerge for c∈(0.589097,0.589097 +ε1) withε1>0 small.

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-�� -��

-��

��

Figure: Numerical approximation of the homoclinic loop for the logistic Buckley-Leverett equation with speed valuec1≈0.5891 (in blue, dashed line) and the periodic wave nearby with speed valuec1+ε,ε≈0.025 (solid, orange line).

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Examples

(III) Modified generalized Burgers-Fisher equation

Whenf(u) =¹₄u⁴−¹₃u³andg(u) =u−u⁴ we have the approximations I0= 1

√5 Z1

u∗

p3−5s²+ 2s⁵ds≈0.979027,

I1= 1

√5 Z1

u∗

(s³−s²)p

3−5s²+ 2u⁵ds≈ −0.129571, and,

L= 2 Z 1

u∗

r3−8s⁵+ 5s⁸

3−5s²+ 2s⁵ds≈5.02904, J= 2

Z 1

u∗

(s³−s²)

r3−8s⁵+ 5s⁸

3−5s²+ 2s⁵ds≈ −1.27529,

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Thenon-degeneracy condition(H₅) holds:

I0J≈ −1.248546=LI1≈ −0.65162.

Thehomoclinic loop speedis c1=I1

I0

≈ −0.13235,

Thesaddle condition (H6)holds: f⁰(1) = 06=c1.

Sincef⁰(1) = 0>c1=−0.13235, then the periodic waves emerge for c∈(−0.13235−ε1,−0.13235) withε1>0 small.

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-�� -��

-��

��

Figure: Numerical approximation of the homoclinic loop for the modified Burgers-Fisher equation with speed valuec1≈ −0.13235 (in blue, dashed line) and the periodic wave nearby with speed valuec1−ε,ε≈0.05 (solid, orange line).

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Spectral instability

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The spectral stability problem

Spectral stabilityrefers to an important property of the traveling waveas a solution to the PDE.

Consider aperturbationv of a bounded periodic traveling wave ϕ=ϕ(z),z=x−ct, with speedc and fundamental periodT>0.

Substitutingv+ϕ into the viscous balance law (VBL) written in the variables (z,t) = (x−ct,t), thenv=v(z,t) is a solution to

v_t−cvz+f(v+ϕ)_z−f(ϕ)_z=v_zz+g(v+ϕ)−g(ϕ).

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For nearby perturbations the leading approximation is given by the linearizationaroundϕ:

v_t=v_zz+ (c−f⁰(ϕ))v_z+ (g⁰(ϕ)−f⁰(ϕ)_z)v.

Takev(z,t) =e^λ^tw(z), whereλ ∈C(growth rate) andw∈X Banach, we obtain aneigenvalue problem:

λw=w_zz+ (c−f⁰(ϕ))w_z+ (g⁰(ϕ)−f⁰(ϕ)_z)w. Our choice: X=L²(R;C), stability with respect to smalllocalized perturbations.

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Thelinearized operatoraround the wave is

(L :L²(R;C)−→L²(R;C),

L :=∂_z²+a1(z)∂_z+a0(z)I, (LO) with dense domainD(L) =H²(R;C), and where the coefficients,

a1(z) :=c−f⁰(ϕ), a₀(z) :=g⁰(ϕ)−f⁰(ϕ)_z,

are bounded and periodic,a_j(z+T) =a_j(z),∀z ∈R,j= 0,1.

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Resolvent and spectra

Definition (resolvent and spectra)

LetL :X →Y be a closed linear operator, with X,Y Banach and dense domainD(L)⊂X. TheresolventofL,ρ(L), is the set of all complex numbersλ ∈Csuch thatL−λ is injective and onto, and (L−λ)⁻¹is bounded. Thepoint spectrumofL, σpt(L), is the set ofλ ∈Csuch thatL−λ is a Fredholm operator with index zero and non-trivial kernel.

Theessential spectrumofL,σess(L), is the set of allλ ∈Csuch that eitherL−λ is not Fredholm, or it is Fredholm with non-zero index. The spectrumofL is defined asσ(L) =σess(L)∪σpt(L).

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Spectral stability

Definition (spectral stability)

We say that a bounded periodic waveϕ isspectrally stableas a solution to the viscous balance law (VBL) if theL²-spectrum of the linearized operator around the wave defined in (LO) satisfies

σ(L)_|L2∩ {λ∈C: Reλ>0}=∅. Otherwise we say that it isspectrally unstable.

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Floquet characterization of the spectrum

Well-known fact: since the coefficientsL are periodic inz, Floquet theory implies that theL²-spectrum ispurely essential or “continuous”, σ(L)_|L2=σess(L)_|L2, and there are no isolated eigenvalues.

We can parametrize the spectrum in terms ofFloquet multipliersof the forme^iθ∈S¹ (or byθ∈R (mod 2π)). Define the setσθ as the set of complex numbersλ for which there exists a bounded, non-trivial solution w∈L^∞(R;C) to the boundary value problem











λw=w_zz+ c−f⁰(ϕ)

w_z+ g⁰(ϕ)−f⁰(ϕ)_z w, w(T) =e^iθw(0),

w_z(T) =e^iθw_z(0), for someθ∈(−π,π].

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Floquet characterization of the spectrum (ii)

Define theFloquet spectrumσF as:

σ_F:= ^[

−π<θ≤π

σ_θ.

Lemma (Floquet characterization of the spectrum)

σ(L)_|L2=σF.

See Jones et al. (2014); Gardner (1993).

Observation: The continuous spectrumσ(L)_|L2 can be written as the union of partial spectraσθ. Each setσθ isdiscreteas it is the zero set of an analytic function. Ifθ= 0 then the boundary conditions become periodic andσ0 detects perturbations which areco-periodic. The setσπ

detectsanti-periodic perturbations.

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Bloch-wave decomposition

Defineu(z) :=e^−iθz/Tw(z). Then the non-separated boundary conditions transform into periodic ones,∂z^ju(T) =∂z^ju(0), j= 0,1, and the spectral problem is recast asLθu=λufor aone-parameter family of Bloch operators:

(Lθ:= (∂_z+iθ/T)²+a₁(z)(∂_z+iθ/T) +a₀(z)I, Lθ:L²_per([0,T];C)→L²_per([0,T];C),

with domainD(Lθ) =H_per² ([0,T];C), parametrized byθ∈(−π,π].

Their spectrum consists entirely ofisolated eigenvalues:

σ(Lθ)_|L2

per=σpt(Lθ)_|L2

per. Moreover, they depend continuously on the Bloch parameterθ, which is alocal coordinatefor the spectrumσ(L)_|L2

Conclusion: λ ∈σ(L)_|L2 if and only ifλ ∈σpt(Lθ)_L2

per for some θ∈(−π,π]:

σ(L)_|L2= ^[

−π<θ≤π

σpt(Lθ)_|L2 per.

Viscous balance laws

Existence and spectral instability of bounded periodic waves for viscous balance laws

Table of contents

Viscous balance laws

Viscous balance laws

Interpretation

Hypotheses (i)

Hypotheses (ii)

Hypotheses (iii)

Hypotheses (iv)

Example

Periodic traveling waves

Main results

Existence of small amplitude

periodic waves

Associated ODE system

Linearization at equilibria (i)

Linearization at equilibria (ii)

Andronov-Hopf bifurcation (i)

Andronov-Hopf bifurcation (ii)

Andronov-Hopf bifurcation (iii)

Existence of small-amplitude periodic waves (i)

Existence of small-amplitude periodic waves (ii)

Examples

Examples

Examples

Existence of large period waves

Augmented system

Hamiltionan system

Energy levels for the Hamiltonian system (i)

Energy levels for the Hamiltonian system (ii)

Energy levels for the Hamiltonian system (iii)

Melnikov integrals for the perturbed system

Melnikov’s theorem

Existence of a homoclinic loop

Traveling pulse

Andronov-Leontovich’s theorem

Periodic wavetrains with large fundamental period (i)

Periodic wavetrains with large fundamental period (ii)

Periodic wavetrains with large fundamental period (iii)

Examples

Examples

Examples

Spectral instability

The spectral stability problem

Resolvent and spectra

Spectral stability

Floquet characterization of the spectrum

Floquet characterization of the spectrum (ii)

Bloch-wave decomposition