time-periodic solutions of a cubic wave equation - dominic...

0 29.08.2019 - Waves 2019 Dominic Scheider-

WAVES 2019, Vienna

Time-Periodic Solutions of a Cubic Wave Equation

Dominic Scheider

KIT – The Research University in the Helmholtz Association www.kit.edu

Outline


1 The main result

2 The strategy

3 Conclusion

The main result


Aim:Find solutions U = U(t, x) of the wave-type equation

∂2tU − ∆U − U = Γ(x)U3 on R×R3 (W)

which are real-valued and .... periodic in t, . localized in x , . radially symmetric in x .

Method:(i) Polychromatic ansatz

U(t, x) = ∑k∈N0

cos(kt) uk (x). (P)

(ii) Bifurcation from a stationary solution U0(t, x) = w0(x).

The main result


Aim:Find solutions U = U(t, x) of the wave-type equation

∂2tU − ∆U − U = Γ(x)U3 on R×R3 (W)

which are real-valued and .... periodic in t, . localized in x , . radially symmetric in x .


U(t, x) = ∑k∈N0

cos(kt) uk (x), (P)

uk ∈ X := {u ∈ Crad(R3,R) | sup(1+ |x |)|u(x)| < ∞}.


The main result


Aim:Find solutions U ∈ C2

per(R,X ) of the wave-type equation

∂2tU − ∆U − U = Γ(x)U3 on R×R3. (W)


U(t, x) = ∑k∈N0

cos(kt) uk (x), uk ∈ X . (P)


The main result


Let Γ ∈ L∞(R3,R) be radial and continuously differentiable.Let w0 ∈ X with −∆w0 − w0 = Γ(x)w3

0 on R3.

Theorem: (S, 2019)There exist an interval I ⊆ R, 0 ∈ I and a family (Uη)η∈I of real-valued,classical solutions Uη = Uη(t, x) ∈ C2

per(R,X ) of the wave equation

∂2tU − ∆U − U = Γ(x)U3 on R×R3 (W)

which is a continuous curve in C (R,X ) with. U0(t, x) = w0(x),. Uη non-stationary and 2π-periodic in time (η 6= 0).

The main result



0 on R3.



∂2tU − ∆U − U = Γ(x)U3 on R×R3 (W)

which is a continuous curve in C (R,X ) with

. U0(t, x) = w0(x),

. Uη non-stationary and 2π-periodic in time (η 6= 0).

The main result



0 on R3.



∂2tU − ∆U − U = Γ(x)U3 on R×R3 (W)

which is a continuous curve in C (R,X ) with. U0(t, x) = w0(x),. Uη non-stationary and 2π-periodic in time (η 6= 0).

The main result


Remarks:Family of polychromatic solutions (“breathers”)

Uη(t, x) = ∑k∈N0

cos(kt) uηk (x).

. (Excitation of s-th mode)

Given s ∈N, find Uη with ddη

∣∣η=0u

ηk 6= 0 iff k = s.

. Extension to the Klein-Gordon equation

∂2tU − ∆U +m2 U = Γ(x)U3 on R×R3.

. Open:Other space dimensions / powers (easy?); non-constant potentials (hard!).

Outline


1 The main result

2 The strategy

3 Conclusion

The strategy


Aim: Find solutions U ∈ C2per(R,X ) of the wave-type equation

∂2tU − ∆U − U = Γ(x)U3 on R×R3. (W)


U(t, x) = ∑k∈N0

cos(kt) uk (x), uk ∈ X . (P)


The strategy



∂2tU − ∆U − U = Γ(x)U3 on R×R3. (W)


U(t, x) = ∑k∈Z

eikt uk (x), uk= u−k ∈ X . (P)


The strategy



∂2tU − ∆U − U = Γ(x)U3 on R×R3. (W)


U(t, x) = ∑k∈Z

eikt uk (x), uk = u−k ∈ X . (P)


The strategy



∂2tU − ∆U − U = Γ(x)U3 on R×R3. (W)


U(t, x) = ∑k∈Z


(infinite) Helmholtz system

−∆uk − (k2 + 1)uk = ∑l+m+n=k

Γ(x) ulumun. (W ∗)


The strategy



∂2tU − ∆U − U = Γ(x)U3 on R×R3. (W)


U(t, x) = ∑k∈Z


(infinite) Helmholtz system for u = (uk )k ∈ `1(N0,X )

−∆uk − (k2 + 1)uk = Γ(x) (u ∗ u ∗ u)k . (W ∗)

(ii) Bifurcation from w = (w0, 0, 0, ...).

The strategy



∂2tU − ∆U − U = Γ(x)U3 on R×R3. (W)


U(t, x) = ∑k∈Z



−∆uk − (k2 + 1)uk = Γ(x) (u ∗ u ∗ u)k . (W ∗)


The strategy


Questions:

(1) What does “Helmholtz” mean?

(2) What is bifurcation?

(3) Why are we talking about this?

The strategy


Question 1: What does “Helmholtz” mean?

−∆u − λu = f on R3, λ > 0 (H)

. “Helmholtz” case: 0 ∈ σ(−∆− λ)

. Particular solution of (H):Limiting Absorption Principle,

u1 = <[limε→0

(−∆− λ− iε)−1f

]=

cos(| · |√

λ)

4π| · | ∗ f .

. General solution of (H):

u = u1 + u2 with any Herglotz wave − ∆u2 − λu2 = 0.

Summary: Multitude of (weakly) localized solutions.

The strategy



−∆u − λu = f (r) on R3, λ > 0 (H)

. “Helmholtz” case: 0 ∈ σ(−∆− λ)

. Particular solution of (H):

−(ru1)′′ − λ(ru1) = rf (r), u1(0) = 1, u′1(0) = 0.

. General solution of (H):

u = u1 + c · sin(| · |√

λ)

4π| · | , c ∈ R.

Radial symmetry 1-dim. solution spaces.

The strategy



−∆u − λu = g(r) · u on R3, λ > 0 (H∗)

. Asymptotically, if g is localized:

−(ru)′′ − λ(ru) ≈ 0 u(r) ≈ $∞sin(r√

λ + τ∞)

r

Lemma:(H∗) has a unique normalized solution in X . It satisfies

u(r) =sin(r√

λ + τ∞)

r+O

(1r2

)for some unique τ∞ ∈ [0,π).

The strategy



−∆u − λu = g(r) · u on R3, λ > 0 (H∗)

. Asymptotically, if g is localized:

−(ru)′′ − λ(ru) ≈ 0 u(r) ≈ $∞sin(r√

λ + τ∞)

r

Lemma:(H∗) together with an asymptotic phase condition (far field condition)

u(r) ∼ sin(r√

λ + τ)

r+O

(1r2

)(Aτ)

has a nontrivial solution in X iff τ = τ∞. (Unique up to constant multiple.)

The strategy



Lemma:(H∗) together with the asymptotic phase condition (Aτ) has a nontrivialsolution in X iff τ = τ∞. (Unique up to constant multiple.)

Remark: For τ 6= 0,

(H∗), (Aτ) ⇔ u = Rτλ[g u] =

sin(| · |√

λ + τ)

4π| · | sin(τ) ∗ [g u].

Radial symmetry ⊕ “good” phase cond. 1-dim. solution spaces.Radial symmetry ⊕ “bad” phase cond. 0-dim. solution spaces.

The strategy


Question 2: What is bifurcation?

Situation: Banach space E , u0 ∈ E and f ∈ C1(E ×R,E ) with

f (u0,λ) = 0 for all λ ∈ R.

Question: Solutions of f (u,λ) = 0 with (u,λ) ≈ (u0,λ0) but u 6= u0?

dim kerDuf (u0,λ0) = 0

Implicit Function Theorem

The strategy


Question 2: What is bifurcation?Situation: Banach space E , u0 ∈ E and f ∈ C1(E ×R,E ) with



dim kerDuf (u0,λ0) = 0

Implicit Function Theorem

The strategy


Question 2: What is bifurcation?Situation: Banach space E , u0 ∈ E and f ∈ C1(E ×R,E ) with



dim kerDuf (u0,λ0) = 1(and more)

Crandall-Rabinowitz Theorem:Bifurcationfrom a simple eigenvalue

The strategy


Question 3: Why are we talking about this? - The strategy, finally.

Aim: Solve ∂2tU − ∆U − U = Γ(x)U3, U ∈ C2

per(R3,X ).

Method:(i) Polychromatic ansatz U(t, x) = ∑k∈Z eikt uk (x)


−∆uk − (k2 + 1)uk = Γ(x) (u ∗ u ∗ u)k . (W ∗)


The strategy


Question 3: Why are we talking about this? - The strategy, finally.Aim: Solve ∂2

tU − ∆U − U = Γ(x)U3, U ∈ C2per(R

3,X ).Method:

(i) Polychromatic ansatz U(t, x) = ∑k∈Z eikt uk (x)


−∆uk − (k2 + 1)uk = Γ(x) (u ∗ u ∗ u)k . (W ∗)


The strategy



tU − ∆U − U = Γ(x)U3, U ∈ C2per(R

3,X ).Method:



with asymptotic conditions

uk = Rτkk2+1[Γ(x) (u ∗ u ∗ u)k ]. (W ∗∗)


The strategy



tU − ∆U − U = Γ(x)U3, U ∈ C2per(R

3,X ).Method:




uk = Rτkk2+1[Γ(x) (u ∗ u ∗ u)k ]. (W ∗∗)


. “Invisible” bifurcation parameter: τs τs + λ.

. Study linearization of (W ∗∗) at u = w, λ = 0:

ψk = 3Rτkk2+1[Γ(x)w

20 (x)ψk ]. (W ∗∗∗)

Lemma!

The strategy



tU − ∆U − U = Γ(x)U3, U ∈ C2per(R

3,X ).Method:




uk = Rτkk2+1[Γ(x) (u ∗ u ∗ u)k ]. (W ∗∗)

(ii) Bifurcation from w = (w0, 0, 0, ...).. “Invisible” bifurcation parameter: τs τs + λ.

. Study linearization of (W ∗∗) at u = w, λ = 0:


20 (x)ψk ]. (W ∗∗∗)

Lemma!

The strategy



tU − ∆U − U = Γ(x)U3, U ∈ C2per(R

3,X ).Method:




uk = Rτkk2+1[Γ(x) (u ∗ u ∗ u)k ]. (W ∗∗)

(ii) Bifurcation from w = (w0, 0, 0, ...).. “Invisible” bifurcation parameter: τs τs + λ.. Study linearization of (W ∗∗) at u = w, λ = 0:


20 (x)ψk ]. (W ∗∗∗)

Lemma!

The strategy


Question 3: Why are we talking about this? - The strategy, finally.


20 (x)ψk ] (W ∗∗∗)

is equivalent to

− ∆ψk − (k2 + 1)ψk = 3 Γ(x)w20 (x)ψk on R3, (H∗)

ψk (r) ∼sin(r√

λ + τk )

r+O

(1r2

). (Aτk )

Radial symmetry ⊕ “good” phase cond. 1-dim. solution spaces.Radial symmetry ⊕ “bad” phase cond. 0-dim. solution spaces.

The strategy



tU − ∆U − U = Γ(x)U3, U ∈ C2per(R

3,X ).Method:




uk = Rτkk2+1[Γ(x) (u ∗ u ∗ u)k ]. (W ∗∗)

(ii) Bifurcation from w = (w0, 0, 0, ...).. “Invisible” bifurcation parameter: τs τs + λ.. Study linearization of (W ∗∗) at u = w, λ = 0:


20 (x)ψk ]. (W ∗∗∗)

Choice of τk such that ψk ≡ 0 (k 6= s), ψs 6≡ 0.

Outline


1 The main result

2 The strategy

3 Conclusion

Conclusion




∂2tU − ∆U − U = Γ(x)U3 on R×R3 (W)

which is a continuous curve in C (R,X ) with U0(t, x) = w0(x) and (forη 6= 0) Uη nonstationary and 2π-periodic.

Key ideas:

Conclusion




∂2tU − ∆U − U = Γ(x)U3 on R×R3 (W)


Key ideas:

Conclusion


Related results:Breather solutions for the wave equation

V (x)∂2tU − ∂2

xU + c V (x)U = Γ(x)U3 on R×R.

. Blank, Chirilus-Bruckner, Lescarret, Schneider 2011,

. Hirsch, Reichel 2019.

Common feature: polychromatic ansatzDifferences:. specific, rough periodic potentials (↔ constant potentials). analysis in spectral gaps (↔ exploitation of kernel elements). strongly localized solutions (↔ power decay). large solutions (↔ close-to-stationary solutions)

Conclusion



V (x)∂2tU − ∂2




Common feature: polychromatic ansatz

Differences:. specific, rough periodic potentials (↔ constant potentials). analysis in spectral gaps (↔ exploitation of kernel elements). strongly localized solutions (↔ power decay). large solutions (↔ close-to-stationary solutions)

Conclusion



V (x)∂2tU − ∂2




Common feature: polychromatic ansatzDifferences:. specific, rough periodic potentials (↔ constant potentials). analysis in spectral gaps (↔ exploitation of kernel elements). strongly localized solutions (↔ power decay). large solutions (↔ close-to-stationary solutions)

Conclusion


Thank you for your attention!... Questions?



∂2tU − ∆U − U = Γ(x)U3 on R×R3 (W)


References


C. Blank, M. Chirilus-Bruckner, V. Lescarret and G. Schneider, BreatherSolutions in Periodic Media, Comm. Math. Phys. 3 (2011), pp. 815–841A. Hirsch and W. Reichel, Real-valued, time-periodic localized weaksolutions for a semilinear wave equation with periodic potentials,Nonlinearity (2019), to appearR. Mandel, E. Montefusco and B. Pellacci, Oscillating solutions fornonlinear Helmholtz equations, ZAMP 6 (2017), article 121R. Mandel and D. Scheider, Bifurcations of nontrivial solutions of a cubicHelmholtz system, to appear in ANONA. URL: http://www.waves.kit.edu/downloads/CRC1173_Preprint_2018-32.pdf

The main result will be part of my PhD thesis (KIT, Oct 2019).

http://www.waves.kit.edu/downloads/CRC1173_Preprint_2018-32.pdf

http://www.waves.kit.edu/downloads/CRC1173_Preprint_2018-32.pdf

time-periodic solutions of a cubic wave equation - dominic...

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