The Cauchy Problem for Coupled Nonlinear Schrodinger Equations with Linear Damping:Local and Global Existence and Blowup of Solutions∗

Joo-Paulo DIASM´ario FIGUEIRAVladimir V.KONOTOP

1 Introduction

The study of blowup of solutions for a damped nonlinear Schrdinger equation has been developed in the papers by Tsutsumi[14]and by Ohta and Todorova[10].More recently,the problem was addressed by[4],for the case of inhomogeneous damping.Stimulated by the relevance for physical applications,there was also significant interest in exploring the blowup phenomenon in a system of coupled nonlinear Schrdinger equations with cubic nonlinearity,with the linear coupling[7]and without the linear coupling[12–13].A rather complete list of the available results can be found in[7].Two sufficient conditions for the finite-time blowup have also been established for the supercritical case of the coupled nonlinear Schrdinger equations,one of which has gain and another has dissipation,both balanced with each other(see[5]).

In this paper we consider the system:

with initial data u0,v0∈H1(RN),where 1≤N≤3,and≤p−1,with p−1＜if N=3,γ1,γ2∈ R describe gain(γ1,2＞ 0)or dissipation(γ1,2＜ 0),k ∈ R is the linear coupling,g1,g2＞0,g∈R,u(x,t),v(x,t)∈C,x∈RN,t≥0.The particular case p−1=2,N≥3 and γ1= −γ2was considered in[5]for the study of the possible blowup of H1-strong solutions.

The system(1.1)may appear in various physical contexts.As a few examples,we mention an optical coupler(N=1)with passive and active arms(see[1])and the self-phase modulation(described by g1and g2)stronger than the Kerr nonlinearity(p＞3).Alternatively,the model describes propagation of a pulse in an elliptically polarized medium(see[9])with dissipation where the two polarizations are linearly coupled.In two-and three-dimensional settings,the model can describe diffraction,focusing and filamentation of a transversely polarized electromagnetic wave(see[3])where the orthogonally polarized components(they are described by u and v)are linearly coupled(or alternatively,two beams are linearly coupled)and are subject to absorption or gain(described by γ1,2).In these cases,the evolutional variable t describes distance along the propagation direction of the beam.Further,at N=3,the model describes a collapse of an unstable binary mixture of Bose-Einstein condensates(see[11])subject to the removal and adding atoms.

In this paper we first study the existence and uniqueness of H1-strong solutions of the system(1.1)in the sense of Kato(see[2,8])by applying some variants of Strichartz’s inequalities(see[10])and some convenient a priori estimates(Theorems 2.1 and 3.1).In the second part of this paper,we extend the main result of[5]in the supercritical case(Theorem 4.1)and give a new result in the critical case(Theorem 4.2).

2 Local Existence in H1(RN)

In this section we will study the local existence in H1(RN)to the Cauchy problem for the system(1.1)with initial data(u0,v0)∈(H1(RN))2.Recall that we have≤p−1＜(＜+∞if N=1,2)and 1≤N ≤3.The case p−1=is called the critical case.

To prove the local existence of solutions,we apply Kato’s method(see[2,8])by adapting the proof of Theorem 4.4.6 in[2].

We start by writing system(1.1)in the form:

where

We decompose G1,2as follows:

with the functions G(·)and(·,·)defined as follows:

Now we easily derive

and the same estimate for||u1|2v1−|u2|2v2|.Moreover,

and we get a similar estimate for||v1|p−1v1− |v2|p−1v2|.

With r=(p− 1)+2=p+1,we derive,for r′such thatH1(RN)×H1(RN),

and similar estimates for G(v)and

and similar estimates for

Moreover,we have

and similar estimates for B2.

Now we fix M,T＞0 to be chosen latter as in the proof of Theorem 4.4.6 in[2]and,with r=max(2,p+1,4)=max(p+1,4),we consider the admissible pair(in Strichartz’s sense,see[2,Section 2.3])

We introduce the space

with the distance

whereand the subset

which is a complete metric space with distance d.

Now,with S(t)=eiΔtand t ∈ R,denoting the Schrdinger group in L2,we introduce,for ϕ∈E,(u0,v0)∈(H1(RN))2,

with the entries

Now,reasoning as in the proof of Theorem 4.4.6 in[2],we can prove,by the previous estimates and applying Strichartz’s inequalities,that

and for a convenient M and a sufficiently small T＞0,H(u,v)∈E and

The uniqueness in C([−T,T];H1)and the blowup alternative follow as in Theorem 4.4.6 in[2].We have the following theorem.

Theorem 2.1 Let(u0,v0)∈(H1(RN))2.Then,the Cauchy problem for the system(1.1)has a unique strong solution(u,v)∈C([0,Tmax);(H1)2(RN))with initial data(u0,v0)defined on a maximal time interval[0,Tmax).

3 Global Existence for k=0

In this section we prove the global existence of the particular case when the linear coupling is absent,and the system obeys sufficiently strong dissipation.

Given γ ∈ R,let us consider the semigroup(Sγ(t))t≥0in L2(RN)defined by

We need to apply Strichartz’s estimates(see[2]).We recall that a pair(q,r)is admissible if

with(2≤r≤∞if N=1,2≤r＜∞if N=2).

Using the same notation as in[2,10],we define

We have the following estimates(see[2,10]).

For every admissible pair(q,r)and∀ϕ∈L2,there exists a constant c＞0 such that,with Lp=Lp(RN)and T＞0,

with c independent of T;

with c independent of T,

Moreover,if 2and θ,∈]1,+∞[are such thatthen

with c independent of T and

Now,by using the Duhamel formula,we write the system(1.1),for the local solution,in the integral form.In the case k=0 for t∈[0,Tmax)we have

Next we state a global existence result of the Cauchy problem for the system(1.1)with k=0.

Theorem 3.1 Assume γ1,γ2＜ 0 and k=0.Then,for any(u0,v0) ∈ (H1(RN))2there exists γ∗(‖u0‖H1,‖v0‖H1) ＞ 0 such that,for all γ1,γ2＜ −γ∗,Tmax=+∞.

First we prove the following important result.

Lemma 3.1 Under the conditions of the Theorem 3.1,assume that there exist constants ε＞ 0 and γ ＜ 0 such that for γi≤ γ,i=1,2,we have,with

Then Tmax=+∞.

Proof Let(see(3.4)).We have

Applying(3.5)we derive,for t∈[0,T],T＜Tmax,

Now we start with the following case.

(I)p−1≥2.

In this case we have

We estimate

and similarly,

Hence,for|γi|large enough,it follows from(3.3)and(3.9)that

with c being a constant independent of T.

Now we estimate the typical termin the right-hand side of(3.12).

From(3.8)we derive

Next,we fix 0＜t≤T.By using the estimate(3.4),from the Duhamel formula(3.5),we deduce by(3.6):

Now,we remark that,for Du={x ∈ RN||u(x)|≥ 1 a.e.},with χDubeing the characteristic function of Duand for each t,

and

Sinceit follows that

with c being a constant independent of t.

The same conclusion can be obtained for the term‖|v|2v‖L?θ′(0,t;Lr′).Therefore,putting together all the terms,we obtain,for|γ1|and|γ2|big enough,

with c1being a constant independent of t.

On the other hand,using again the Duhamel formula and the Strichartz’s estimates,we derive

Next we proceed as before to estimate the last two terms on the right-hand side more precisely(with Dv={x∈RN||v(x)|≥1 a.e.}):

Then,for|γ1|and|γ2|large enough,it follows from(3.16)that

with c2being a constant independent of t.

Now,let c0=max(c1,c2)and choose ε such that 23p−2c0εp−1≤ 1.By the continuity of the functions t → ‖u‖Lθ(0,t;Lr)+ ‖v‖Lθ(0,t;Lr)and t → ‖u‖Lq(0,t;W1,r)+ ‖v‖Lθ(0,t;W1,r),it follows

from(3.15)and(3.17)that

and

The conclusion follows now from(3.12)–(3.13).

(II)p−1＜2.

Notice that,since≤p−1,the condition p−1＜2 implies N ＞2,which means in our case N=3.The proof follows the same steps used in the previous case p−1≥2.The first estimate(3.12)remains true with p−1 replaced by 2 when the admissible pair(q,r)corresponds now to r=4.The estimate(3.13)is now

and the estimates(3.15)and(3.17)are obtained by following the same scheme.For example,to estimate ‖u‖L8(0,t;L4),just like in(3.14),we use the assumption ‖Sγ1(·)u0‖L8(0,+∞,L4)≤ ε and we must only estimate using the decomposition|u|p−1u= χDu|u|p−1u+(1 − χDu)|u|p−1u.With the corresponding estimates(3.15)and(3.17)we conclude in the same way.

Now we can pass to the following proof.

Proof of Theorem 3.1 Assume first p−1≥2.We will prove that(3.6)holds.Since we have ‖S(t)u0‖Lp+1≤ c‖u0‖H1,t≥ 0,we derive

Therefore,

and the same conclusion holds for Sγ2v0.Similar estimates prove(3.7)in the case p−1＜ 2.Hence,the assumptions in Lemma 3.1 are satisfied and thus Theorem 3.1 is proved.

4 Blowup Results

In this section we will study the possible blowup of the local in-time H1-strong solutions(u,v)of the Cauchy problem for the system(1.1)with initial data(u0,v0)∈(H1(RN))2such that

In the following we perform formal calculations which can be justified by suitable regularisations that allow us to prove that

The main ideas are based on the seminal work of Glassey[6],in[7,10]and in the previous paper[5]where the case p−1=2 is studied.

We start by proving some preliminar estimates to the local solution(u,v)∈C([0,Tmax);(H1)2).

It is easy to derive,for t∈[0,Tmax)and with?·dx=?RN·dx,

with γ =max(|γ1|,|γ2|).

Then we obtain

We define the energy

From the system(1.1)we deduce

We need the following result.

Lemma 4.1 Assume p−1＞4N.Then the solution(u,v)∈C([0,Tmax);H1)of the system(1.1)with initial data(u0,v0)∈(H1)2verifies the inequality

with

and

Proof If(γ1+γ2)g ≥ 0,it follows from(4.4)that

where=max{1,g1,g2},and since Q(τ) ≤ e2γτQ(0),we obtain the result withIf(γ1+γ2)g ＜ 0,we remark that for N=1,2 with p−1≥and for N=3 with p−1≥2.The result now follows from(4.4)as before.

Now we define the variance

with

and let

We derive from(1.1)that

So

To compute the second derivative we take the derivative of Vi(t),i=1,2.First,

and

Then

Similarly,

By(4.5)we derive

So we obtain,from(4.6)–(4.7),

Now,we will assume p−1＞

Since p−1 ＞we can choose δ such that

Rearranging the terms on the right-hand side of(4.8),we derive by(4.3),

First we assume N=3.If g＞0,since p−1＞(we keep the notation with N by technical reasons),we choose δ such that

If g＜ 0,we must assume p−1≥ 2.In the case p−1＞ 2,we choose δ=2,so the term(− 4)Ng?|u|2|v|2dx in(4.9)can be canceled.If p − 1=2,we choose δ=Then we easily check that δ＜ N,p − 1=2 ＞and we have

and

Collecting all these cases and taking into account that

and

it follows from(4.9)that

with

and

By applying Lemma 4.1,from(4.10),we derive the following inequality:

with

Next,we show that the inequality(4.13),which holds for N=3,can also be verified for N=1,2,up to a few changes in the constants.

The critical point is to dominate the term(−4)Ng?|u|2|v|2dx in(4.9).

Assume now N=2.Since p−1＞=2,we have

and we choose δ= δ2＜ 2 such that

and

It follows that(4.13)holds with constants c2and c4given by

and

Finally,assume N=1.We have p−1＞4N=4 and we choose δ= δ1＜ 1 such that p−1−＞ 0.By the Gagliardo–Nirenberg inequality,we derive

with ε＞ 0 to be chosen.

Thus

and we choose ε such that

Once again we obtain(4.13)with the constants

and that the term c4e2γtis now replaced by c4(γ,k,1)e6γtwith

Now,let

From(4.13)we obtain

with

Next,we introduce the functions

We can now state a blowup theorem for the supercritical case.

Theorem 4.1 Assume p−1＞and the Cauchy problem(1.1)with initial data(u0,v0)∈(H1)2. Let(u,v)∈C([0,Tmax);(H1)2)be the corresponding local solution. Assume that(|x|u0,|x|v0)∈(L2)2and if N=3,we have

Assume also that there exists T0＞0 such that

Then the solution(u,v)blows up in finite time with Tmax≤T0.

Proof Let us define

It follows from(4.14)–(4.15)and(4.17)that,for t∈ [0,T1],

Applying Gronwall’s inequality,we obtain

Using this estimate,back to the right-hand side of(4.15),we derive

and by(4.18),Y(T1)＜M(T0).Then T1=T0.Hence,

which is absurd since Y≥0.

Remark 4.1 As can be seen by an adaptation of the proofs in Lemma 1 in[5],where the particular case p − 1=2 is considered,the blowup assumptions(4.17)–(4.18)are finished in that case,for a certain T0＞0,if the initial energy

where

with γ = γ1= − γ2＞ 0 and T0,max,T0,mindefined in Lemma 1 in[5].

Remark 4.2 The blowup result of Theorem 4.1 can be extended to higher dimensions if p−1＞4Nwith the same proof as in the case N=3.

Now,we consider a special case of the Cauchy problem for the system(1.1),which includes the critical case,although it requires the absence of the linear coupling.

Theorem 4.2 Assuming k=0,γ1=γ2=σ＞0,g＞0 and p−1≥(critical and supercritical cases).Let(u,v)∈([0,Tmax);(H1)2)be the local solution to the Cauchy problem for the system(1.1)with initial data(u0,v0)∈(H1)2,(xu0,xv0)∈(L2)2.Then,if E(0)＜0,the solution blows up in finite time,that is,Tmax＜+∞.

Proof We have in this case,by(4.4),

Since

we derive,with

so

Now,from(4.6)and(4.7),we deduce

Since g＞0 and p−1≥we derive

Finally,from(4.5)we have

So

Therefore

and the conclusion follows.

[1]Bludov,Yu.V.,Driben,R.,Konotop,V.V.and Malomed,B.A.,Instabilities,solitons and rogue waves in PT-coupled nonlinear waveguides,J.Opt.,15,2013,064010.

[2]Cazenave,T.,Semilinear Schrdinger Equations,Courant Lecture Notes,Vol.10,Amer.Math.Soc.,2003.

[3]Couairon,A.and Mysyrowicz,A.,Femtosecond Filamentation in Transparent Media,Phys.Rep.,441,2007,47–189.

[4]Dias,J.P.and Figueira,M.,On the blowup of solutions of a Schrdinger equation with an inhomogeneous damping coefficient,Comm.Contemp.Math.16,2014,1350036.

[5]Dias,J.P.,Figueira,M.,Konotop,V.V.and Zezyulin,D.A.,Supercritical blowup in coupled paritytime-symmetric nonlinear Schrdinger equations,Studies Appl.Math.,133,2014,422–440.

[6]Glassey,R.T.,On the blowing up of solutions to the Cauchy problem for nonlinear Schrdinger equations,J.Math.Phys.,18,1977,1794–1797.

[7]J¨ungel,A.and Weish¨aupl,R.M.,Blow-up in two-component nonlinear Schrdinger systems with an external driven field,Math.Models Meth.Appl.Sciences,23,2013,1699–1727.

[8]Kato,T.,On nonlinear Schrdinger equations,Ann.Inst.H.Poincar´e Phys.Th´eor.,46,1987,113–129.

[9]Menyuk,C.R.,Pulse propagation in an elliptically birefringent medium,IEEE J.Quant.Electron.,25,1989,2674.

[10]Ohta,M.,and Todorova G.,Remarks on global existence and blowup for damped nonlinear Schrdinger equations,Discrete Cont.Dyn.Syst.,23,2009,1313–1325.

[11]Pitaevskii,L.and Stringari,S.,Bose–Einstein Condensation,Clarendon Press,Oxford,2003.

[12]Prytula V.,Vekslerchik,V.and P´erez-Garc´ıa,V.M.,Collapse in coupled nonlinear Schrdinger equations:Sufficient conditions and applications,Physica D,238,2009,1462–1467.

[13]Roberts,D.C.and Newell,A.C.,Finite-time collapse of N classical fields described by coupled nonlinear Schrdinger equations,Phys.Rev.E,74,2006,047602.

[14]Tsutsumi,M.,Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrdinger equations,SIAM J.Math.Anal.,15,1984,357–366.