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On two coupled nonlinear Schrodinger equations

Para el cumpleanos del egregio

profesor Ireneo Peral

Eugenio MontefuscoDipartimento di MatematicaSapienza Universita di Roma

Salamanca 13.02.2007

Eugenio Montefusco on two CNLS eqs

Coauthors

Luca Fanelli (Sapienza universita di Roma),Liliane Maia (universidade de Brasılia),Benedetta Pellacci (universita di Napoli Parthenope),Marco Squassina (universita di Verona).

Pulse propagation in optical fibers

Using optical solitons (i.e. concentrated wave packet) asinformation bits in high-speed telecommunication systemsnow is a standard technique.

Obviously the interest in optical solitons transmissiongrows, motivating the study (also mathematical) of pulsepropagation in optical fibers.The main problem is to increase the bit rate of thetransmission.

Using optical solitons (i.e. concentrated wave packet) asinformation bits in high-speed telecommunication systemsnow is a standard technique.Obviously the interest in optical solitons transmissiongrows, motivating the study (also mathematical) of pulsepropagation in optical fibers.

The main problem is to increase the bit rate of thetransmission.

Using optical solitons (i.e. concentrated wave packet) asinformation bits in high-speed telecommunication systemsnow is a standard technique.Obviously the interest in optical solitons transmissiongrows, motivating the study (also mathematical) of pulsepropagation in optical fibers.The main problem is to increase the bit rate of thetransmission.

The model

Since the optical fibers are birefringent, the pulse is atraveling vector wave having two orthogonal components.The following coupled nonlinear Schrodinger (CNLS)system was derived for pulse propagation including theeffect of the interaction between the components of thevector pulse

{ihφt + h2φxx +

(φφ+ βψψ

)φ = 0

ihψt + h2ψxx +(ψψ + βφφ

)ψ = 0

where φ and ψ are complex amplitudes of wave envelopes,h� 1 is a positive constant (moreless the Planck constant)and β is the birefringence coefficient.

The model

Since the optical fibers are birefringent, the pulse is atraveling vector wave having two orthogonal components.The following coupled nonlinear Schrodinger (CNLS)system was derived for pulse propagation including theeffect of the interaction between the components of thevector pulse{

ihφt + h2φxx +(φφ+ βψψ

)φ = 0

ihψt + h2ψxx +(ψψ + βφφ

)ψ = 0

where φ and ψ are complex amplitudes of wave envelopes,h� 1 is a positive constant (moreless the Planck constant)and β is the birefringence coefficient.

The inspiring problem

First of all we want to point out that this system is weaklycoupled, so it is possible to find scalar solution, i.e. (φ, 0) or(0, ψ), and vector solution, that is (φ, ψ) (obviously we arethinking that φ, ψ 6= 0).

Question: what it is possible to say about the collisions ofsolitons? That is what we can say about the dynamics ofthe following Cauchy problem?

ihφt + h2φxx + (|φ|2 + β|ψ|2)φ = 0

ihψt + h2ψxx + (|ψ|2 + β|φ|2)ψ = 0

φ(0, x) =√

2sech ((x− x0)/h) exp(iν0x/h)

ψ(0, x) =√

2sech ((x− y0)/h) exp(iγ0x/h)

First of all we want to point out that this system is weaklycoupled, so it is possible to find scalar solution, i.e. (φ, 0) or(0, ψ), and vector solution, that is (φ, ψ) (obviously we arethinking that φ, ψ 6= 0).Question: what it is possible to say about the collisions ofsolitons? That is what we can say about the dynamics ofthe following Cauchy problem?

ihφt + h2φxx + (|φ|2 + β|ψ|2)φ = 0

ihψt + h2ψxx + (|ψ|2 + β|φ|2)ψ = 0

φ(0, x) =√

2sech ((x− x0)/h) exp(iν0x/h)

ψ(0, x) =√

2sech ((x− y0)/h) exp(iγ0x/h)

The dynamics of the problem above is amazing! Thefollowing phenomena in collisions are observed

trasmission, reflection, and trapping of the waves.It is possible to explain how these events depend on theparameters of the problem?

trasmission, reflection, and trapping of the waves.

It is possible to explain how these events depend on theparameters of the problem?

trasmission, reflection, and trapping of the waves.It is possible to explain how these events depend on theparameters of the problem?

Some references

L. Berge, Phys. Rep. 1998.

H.A. Haus, W.S. Wong, Rev. Mod. Phys. 1996.

C. Menyuk, IEEE J. Quantum Electron. 1989.

J.Q. Sun, X.Y. Gu, Z.Q. Ma, Phys. D 2004.

J. Yang, Y. Tan, Phys. Lett. A 2001.

CNLS system

Starting from the preceding considerations we want toinvestigate the mathematics of the CNLS system above inorder to clarify the dynamics of concentrated solutions.

As usual in mathematics we can scale the variables andconsider a slightly general problem: the Cauchy problemfor the following CNLS system with cubic focusingnonlinearities

i∂tφ+ ∆φ+ (|φ|2 + β|ψ|2)φ = 0

i∂tψ + ∆ψ + (|ψ|2 + β|φ|2)ψ = 0

φ(0) = φ0, ψ(0) = ψ0

In the following we assume n ≤ 3 in order to work in H1.

CNLS system

Starting from the preceding considerations we want toinvestigate the mathematics of the CNLS system above inorder to clarify the dynamics of concentrated solutions.As usual in mathematics we can scale the variables andconsider a slightly general problem: the Cauchy problemfor the following CNLS system with cubic focusingnonlinearities

i∂tφ+ ∆φ+ (|φ|2 + β|ψ|2)φ = 0

i∂tψ + ∆ψ + (|ψ|2 + β|φ|2)ψ = 0

φ(0) = φ0, ψ(0) = ψ0

In the following we assume n ≤ 3 in order to work in H1.

CNLS system

We will expose some (sometime partial) results about

local (in time) existence of solutions and conservationlaws,

existence and classification of standing waves,

blow-up or global existence of solutions,

orbital stability of standing waves.

The Cauchy problem and conservation laws

Theorem

For any (φ0, ψ0) ∈ H1 there exists a unique solution(φ, ψ) ∈ C ((−T, T ), H1) of

i∂tφ+ ∆φ+ (|φ|2 + β|ψ|2)φ = 0

i∂tψ + ∆ψ + (|ψ|2 + β|φ|2)ψ = 0

φ(0) = φ0, ψ(0) = ψ0

which depends continuously on the initial datum. Moreoverfor any t it holds ‖φ‖L2 = ‖φ0‖L2, ‖ψ‖L2 = ‖ψ0‖L2 and

E(t) =1

(‖∇φ‖2

2 + ‖∇ψ‖22

(‖φ‖4

4 + 2β‖φψ‖22 + ‖ψ‖4

)= E(0).

Consider the first equation of the system, multiplying by φand integrating we have∫

φ ·(i∂tφ+ ∆φ+

(|φ|2 + β|ψ|2

Using the divergence theorem we obtain

∫∂tφφ =

∫∇φ · ∇φ+

∫ (|φ|2 + β|ψ|2

)|φ|2

and, with a little bit of complex analysis, it follows

2i∂t‖φ‖L2 = =

∫ (∂tφφ

∫|∇φ|2 +

∫ (|φ|2 + β|ψ|2

)|φ|2.

Since the left hand side is a pure imaginary number andthe right hand side is real, it follows that

‖φ‖L2 = c.

φ ·(i∂tφ+ ∆φ+

(|φ|2 + β|ψ|2

∫∂tφφ =

∫∇φ · ∇φ+

∫ (|φ|2 + β|ψ|2

)|φ|2

2i∂t‖φ‖L2 = =

∫ (∂tφφ

∫|∇φ|2 +

∫ (|φ|2 + β|ψ|2

)|φ|2.

‖φ‖L2 = c.

φ ·(i∂tφ+ ∆φ+

(|φ|2 + β|ψ|2

∫∂tφφ =

∫∇φ · ∇φ+

∫ (|φ|2 + β|ψ|2

)|φ|2

2i∂t‖φ‖L2 = =

∫ (∂tφφ

∫|∇φ|2 +

∫ (|φ|2 + β|ψ|2

)|φ|2.

‖φ‖L2 = c.

φ ·(i∂tφ+ ∆φ+

(|φ|2 + β|ψ|2

∫∂tφφ =

∫∇φ · ∇φ+

∫ (|φ|2 + β|ψ|2

)|φ|2

2i∂t‖φ‖L2 = =

∫ (∂tφφ

∫|∇φ|2 +

∫ (|φ|2 + β|ψ|2

)|φ|2.

‖φ‖L2 = c.

Some references

T. Cazenave, 1996.

J. Ginibre, G. Velo, J. Funct. Anal. 1979.

T. Kato, Ann. I.H.P. Physique Theorique 1987.

T. Ozawa, Calc. Var. P.D.E. 2006.

C. Sulem, P.L. Sulem, 1999.

Looking for standing waves

Standing waves solution are particular solutions having theform

φ(x, t) = eitu(x), ψ(x, t) = eiω2tv(x).

Putting in the CNLS system we obtain that u and v haveto satisfy the following system{

−∆u(x) + u(x) = (|u(x)|2 + β|v(x)|2)u(x)−∆v(x) + ω2v(x) = (|v(x)|2 + β|u(x)|2)v(x)

which, in the sequel, we will call the elliptic CNLS system.

Looking for standing waves

Standing waves solution are particular solutions having theform

φ(x, t) = eitu(x), ψ(x, t) = eiω2tv(x).

Putting in the CNLS system we obtain that u and v haveto satisfy the following system{

−∆u(x) + u(x) = (|u(x)|2 + β|v(x)|2)u(x)−∆v(x) + ω2v(x) = (|v(x)|2 + β|u(x)|2)v(x)

which, in the sequel, we will call the elliptic CNLS system.

Existence of ground state solutions

Theorem

For every β > 0 and ω 6= 0 there exists a least energysolution (ground state) (uω, vω) 6= (0, 0) of the ellipticCNLS system, with uω, vω ≥ 0 and both uω and vω radial.

A ground state solution solves the minimization problem

Iβ(u, v)

where N = {(u, v) : I ′β(u, v) · (u, v) = 0} is the Neharimanifold and

Iβ(u, v) =1

(‖∇u‖2

2 + ‖∇v‖22 + ‖u‖2

2 + ω2‖v‖22

(‖u‖4

4 + 2β‖uv‖22 + ‖v‖4

Existence of ground state solutions

Theorem

For every β > 0 and ω 6= 0 there exists a least energysolution (ground state) (uω, vω) 6= (0, 0) of the ellipticCNLS system, with uω, vω ≥ 0 and both uω and vω radial.

A ground state solution solves the minimization problem

Iβ(u, v)

where N = {(u, v) : I ′β(u, v) · (u, v) = 0} is the Neharimanifold and

Iβ(u, v) =1

(‖∇u‖2

2 + ‖∇v‖22 + ‖u‖2

2 + ω2‖v‖22

(‖u‖4

4 + 2β‖uv‖22 + ‖v‖4

Scalar versus Vector ground state solutions

Theorem

If β > max

{h(ω), h

)}, h(s) =

((4 + n)s2 + (4− n))2

then there exists a vector ground state solution.

Theorem

If β > max {(4− n)ω2 + n, (4− n)ω−2 + n} /4, then thereexists a vector ground state solution.

Theorem

If there exists a vector ground state solution thenβ ≥ max

{ω(4−n)/2, ω(n−4)/2

Theorem

If β > max

{h(ω), h

)}, h(s) =

((4 + n)s2 + (4− n))2

Theorem

{ω(4−n)/2, ω(n−4)/2

Theorem

If β > max

{h(ω), h

)}, h(s) =

((4 + n)s2 + (4− n))2

Theorem

{ω(4−n)/2, ω(n−4)/2

Scalar versus Vector solutions (n = 3)

(1, 1)

ω = β2

��ω = h−1(β)

ω =√

4β − 3

ω ' 8.727

β ' 5.649β = 1

vectorgroundstate

scalargroundstate

Changing sign solutions

Theorem

Assume n = 2, 3. For every pair of nonnegative integers(h, k) it holds that

for any β > 0 there exists (uh, vk) 6= (0, 0) that achieves

minNh,k

Iβ(u, v)

that is a solution of the elliptic CNLS system with h(respectively k) nodal regions,

there exists βh,k > 0 such that if β > βh,k then uh 6≡ 0and vk 6≡ 0.

Changing sign solutions

Theorem

Assume n = 2, 3. For every pair of nonnegative integers(h, k) it holds that

for any β > 0 there exists (uh, vk) 6= (0, 0) that achieves

minNh,k

Iβ(u, v)

that is a solution of the elliptic CNLS system with h(respectively k) nodal regions,

there exists βh,k > 0 such that if β > βh,k then uh 6≡ 0and vk 6≡ 0.

Some references

B. Abdellaoui, V. Felli, I. Peral, preprint 2007.

A. Ambrosetti, E. Colorado, J. London Math. Soc.2007.

T. Bartsch, Z.Q. Wang, J. Part. Diff. Eqs 2006.

D. de Figueiredo, O. Lopes, Ann. I.H.P. 2007.

B. Sirakov, Comm. Math. Phys. 2007.

S. Terracini, G. Verzini, No.D.E.A. 2001.

Global existence and Blow–up

Theorem

A solution of the problemi∂tφ+ ∆φ+ (|φ|2 + β|ψ|2)φ = 0

i∂tψ + ∆ψ + (|ψ|2 + β|φ|2)ψ = 0

φ(0) = φ0, ψ(0) = ψ0

exists globally (in time), if n = 1,

exists globally if ‖(φ0, ψ0)‖L2 ≤ C, if n = 2,

exists globally if ‖(φ0, ψ0)‖L2 � 1, if n = 3.

An explicite example of blow–up for n = 2

1− t

), vω

1− t

(i|x|2 + 4

4(1− t)

Moreover we have proved that C = (‖uω‖L2 + ω2‖vω‖L2),where (uω, vω) is any ground state of the elliptic CNLSsystem.

An explicite example of blow–up for n = 2

1− t

), vω

1− t

(i|x|2 + 4

4(1− t)

Moreover we have proved that C = (‖uω‖L2 + ω2‖vω‖L2),where (uω, vω) is any ground state of the elliptic CNLSsystem.

A Gagliardo-Nirenberg type inequality

From the inequality

≤ (‖u‖22 + ‖v‖2

2)1−n/2

(‖∇u‖22 + ‖∇v‖2

(‖u‖44 + 2β‖uv‖2

2 + ‖v‖44)

we have an estimate on the L2 norm of the gradients

(‖∇φ‖2

2 + ‖∇ψ‖22

)− 1

(‖φ‖4

4 − 2β‖φψ‖22 + ‖ψ‖4

)≥ 1

4(‖∇φ‖2

2 + ‖∇ψ‖22) ·

[2− Cn

(‖φ‖2

2 + ‖ψ‖22

)1−n/2

·(‖∇φ‖2

2 + ‖∇ψ‖22

)n/2].

A Gagliardo-Nirenberg type inequality

From the inequality

≤ (‖u‖22 + ‖v‖2

2)1−n/2

(‖∇u‖22 + ‖∇v‖2

(‖u‖44 + 2β‖uv‖2

2 + ‖v‖44)

we have an estimate on the L2 norm of the gradients

(‖∇φ‖2

2 + ‖∇ψ‖22

)− 1

(‖φ‖4

4 − 2β‖φψ‖22 + ‖ψ‖4

)≥ 1

4(‖∇φ‖2

2 + ‖∇ψ‖22) ·

[2− Cn

(‖φ‖2

2 + ‖ψ‖22

)1−n/2

·(‖∇φ‖2

2 + ‖∇ψ‖22

)n/2].

Some references

R.T. Glassey, J. Math. Phys. 1977.

I. Peral, J.L. Vazquez, Arch. Rat. Mech. Anal. 1995,

M.I. Weinstein, Comm. Math. Phys. 1983.

Orbital stability of ground state solutions

Theorem

Assume n = 1, then any ground state solution is orbitallystable. This means that for any ε > 0 there exists δ > 0such that if (φ, ψ) is a solution of the CNLS system with

‖(φ(x, 0), ψ(x, 0))− (eiθu(x− y0), eiθv(x− y0))‖H1 ≤ δ,

then there exist functions ω1(t), ω2(t), y(t) such that

‖(φ(x, t), ψ(x, t))−(eiω1(t)u(x−y(t)), eiω2(t)v(x−y(t)))‖H1 ≤ ε.

The statement above can read in the following way: if thedynamics of the CNLS system starts near to a ground statethen the solution remains close to the orbit of the groundstate, up to translations.

Orbital stability of ground state solutions

Theorem

Assume n = 1, then any ground state solution is orbitallystable. This means that for any ε > 0 there exists δ > 0such that if (φ, ψ) is a solution of the CNLS system with

‖(φ(x, 0), ψ(x, 0))− (eiθu(x− y0), eiθv(x− y0))‖H1 ≤ δ,

then there exist functions ω1(t), ω2(t), y(t) such that

‖(φ(x, t), ψ(x, t))−(eiω1(t)u(x−y(t)), eiω2(t)v(x−y(t)))‖H1 ≤ ε.

The statement above can read in the following way: if thedynamics of the CNLS system starts near to a ground statethen the solution remains close to the orbit of the groundstate, up to translations.

Some references

T. Cazenave, P.L. Lions, Comm. Math. Phys. 1982.

M. Grillakis, J. Shatah, W. Strauss, J. Funct. Anal.1987.

O. Lopes, Nonlin. 2006.

J. Stubbe, Port. Math. 1989.

Works in progress

partial uniqueness of the ground state solution,

semiclassical limit of the evolution in presence ofpotentials,

ground state selection,

dynamics of solitons without external forces.

Thanks to Ireneo!

¡Recuerdo de un dia estupendo!

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