Construction of 2-solitons with logarithmic distance for the one-dimensional cubic Schrodinger system

We consider a system of coupled cubic Schr\"odinger equations in one space dimension \begin{equation*} \begin{cases} i \partial_t u + \partial_x^2 u +(|u|^2 + \omega |v|^2) u =0\\ i \partial_t v + \partial_x^2 v+ (|v|^2 + \omega |u|^2) v=0 \end{cases}\quad (t,x)\in {\bf R}\times{\bf R}, \end{equation*} in the non-integrable case $0<\omega<1$. First, we justify the existence of a symmetric 2-solitary wave with logarithmic distance, more precisely a solution of the system satisfying \[ \lim_{t\to +\infty}\left\| \begin{pmatrix} u(t) \\ v(t)\end{pmatrix} - \begin{pmatrix} e^{it}Q (\cdot - \frac{1}{2} \log (\Omega t) - \frac{1}{4} \log \log t) \\ e^{it}Q (\cdot + \frac{1}{2} \log (\Omega t) + \frac{1}{4} \log \log t)\end{pmatrix}\right\|_{H^1\times H^1} = 0\] where $Q = \sqrt{2}{\rm sech}$ is the explicit solution of $ Q'' - Q + Q^3 = 0$ and $\Omega>0$ is a constant. This result extends to the non-integrable case the existence of symmetric 2-solitons with logarithmic distance known in the integrable case $\omega=0$ and $\omega=1$. Such strongly interacting symmetric $2$-solitary waves were also previously constructed for the non-integrable scalar nonlinear Schr\"odinger equation in any space dimension and for any energy-subcritical power nonlinearity. Second, under the conditions $0<c<1$ and $0<\omega<\frac 12 c(c+1)$, we construct solutions of the system satisfying \[ \lim_{t\to +\infty}\left\| \begin{pmatrix}u(t) \\ v(t)\end{pmatrix} - \begin{pmatrix}e^{i c^2 t}Q_c (\cdot - \frac{1}{(c+1)c} \log (\Omega_c t) ) \\ e^{i t} Q (\cdot + \frac{1}{c+1} \log (\Omega_c t))\end{pmatrix} \right\|_{H^1\times H^1}=0\] where $Q_c(x)=cQ(cx)$ and $\Omega_c>0$ is a constant. Such logarithmic regime with non-symmetric solitons does not exist in the integrable cases $\omega=0$ and $\omega=1$ and is still unknown in the non-integrable scalar case.

First, we justify the existence of a symmetric 2-solitary wave with logarithmic distance, more precisely a solution of the system satisfying where Q = √ 2 sech is the explicit solution of Q ′′ − Q + Q 3 = 0 and Ω > 0 is a constant. This result extends to the non-integrable case the existence of symmetric 2-solitons with logarithmic distance known in the integrable case ω = 0 and ω = 1 ( [15,33]). Such strongly interacting symmetric 2-solitary waves were also previously constructed for the non-integrable scalar nonlinear Schrödinger equation in any space dimension and for any energy-subcritical power nonlinearity ( [20,22]).
1. Introduction 1.1. System of cubic Schrödinger equations. We consider the following one dimensional focusing-focusing system of coupled cubic Schrödinger equations for u(t, x), v(t, x) : R × R → C and for any parameter 0 < ω < 1. The initial data u(0, x) = u 0 (x), v(0, x) = v 0 (x) is taken in H 1 (R) × H 1 (R). The Hamiltonian system (coupled NLS) arises as a model for the propagation of the electrical field in nonlinear optics. Such systems also appear to model the interaction of two Bose-Einstein condensates in different spin states. See [1,2,32].
Let Q be the ground state, defined as unique (up to translation) H 1 solution of Q ′′ − Q + Q 3 = 0 on R.
Recall that (cubic NLS) admits solitary wave solutions, also called solitons, of the form where λ > 0, γ, σ, β ∈ R. When v = 0 (or u = 0), the system (coupled NLS) simplifies into (cubic NLS), and thus we deduce soliton solutions of (coupled NLS): for any λ j > 0, γ j , σ j , β j ∈ R (j = 1, 2). By definition, a multi-solitary wave (or multisoliton) is a solution behaving in large time as a sum of such single solitons. In this article, we focus on 2-solitons such that one solitary wave is carried by u and the other one by v.

Previous results and motivation.
Multi-solitons have been studied intensively in the integrable case, i.e. for (cubic NLS) and (MS), as well as for some nearly integrable models; see [1,7,8,13,24,32,33]. From the inverse scattering theory, there are three types of 2-solitons for (cubic NLS): (a) Two solitons with different velocities: as t → +∞, the distance between the solitons is of order t ( [33]). (b) Double pole solutions: the two solitons have the same amplitude and their distance is logarithmic in t ( [24,33]). (c) Periodic 2-solitons: the two solitons have different amplitudes and their distance is a periodic function of time ( [32,33]). More generally, the integrability theory treats the case of K-solitary waves for any K ≥ 2. Moreover, in the integrable case, multi-solitons have a pure soliton behavior for both t → +∞ and t → −∞ and describe the elastic interactions between solitons. For (MS), a trichotomy similar to (a)-(b)-(c) is studied formally and numerically in [31].
For non-integrable models, the study of multi-solitons is mostly limited to situations where solitons are decoupled, in particular, asymptotically in large time. Consider first the scalar nonlinear Schrödinger equation in any space dimension d ≥ 1 and for any energy subcritical power nonlinearity (i.e. p > 1 for d = 1, 2 and 1 < p < 1 + 4 d−2 for d ≥ 3). This equation is known to be completely integrable only for d = 1 and p = 3, i.e. (cubic NLS). Define the ground state Q as the unique radial positive H 1 solution (up to symmetries) of ∆Q − Q + Q p = 0 in R d (for more properties of the ground state, see [3,9,25,30]) and Q λ (x) = λ 2 p−1 Q(λx), for any λ > 0. The existence of K-solitary waves for (NLS) corresponding to case (a), i.e. solutions u(t) of (NLS) such that for any λ k > 0 and any two-by-two different β k ∈ R d , was established in [5,17,21].
Recently, the second author proved that the dynamics (b) is also a universal regime for (NLS), by constructing two symmetric solitary waves with logarithmic distance, [22]. The L 2 critical case (p = 1 + 4 d ), previously studied in [20], exhibits a specific blow-up behavior also related to symmetric 2-solitons with logarithmic distance in rescaled variables.
Turning back to the system (coupled NLS) in the non-integrable case, i.e. for 0 < ω < 1, the existence of multi-solitary wave solutions corresponding to case (a) for any c > 0 and any different velocities β 1 = β 2 was proved in [6] (see also [11]). A first goal of this paper is to justify the persistence of the regime (b) for the nonintegrable (coupled NLS) in presence of symmetry, following the articles [20,22] for the scalar (NLS) equation.
Second, and more importantly, we investigate the question of the (non-)persistence of the regime (c). Indeed, we exhibit a new logarithmic regime corresponding to non-symmetric 2-solitons with logarithmic distance which replaces the behavior (c). At the formal level, the system of parameters of the 2-solitons is not anymore integrable and periodic solutions disappear, see Remark 3. A logarithmic regime (see Theorem 2 and Remark 2) then takes place, which does not exist in the integrable cases ω = 0 and ω = 1. To our knowledge, such question is open for the scalar equation (NLS) in the non-integrable case (see Section 5).
Note that as t → +∞, the distance between the two solitary waves is asymptotic to Remark 1. An analogous dynamics was constructed for (cubic NLS) in [24,33] and for (NLS) in [20,22].
Second, we construct for (coupled NLS) a new logarithmic dynamics of 2-solitary waves with different amplitude. Theorem 2. For any 0 < c < 1 and 0 < ω < 1 2 c(c + 1) < 1, there exists a solution where Ω c > 0 is a constant depending on c and ω.
Note that as t → +∞, the distance between the two solitary waves is asymptotic to  We believe that there is no other logarithmic regime for (coupled NLS). In support of this conjecture, we refer to the case of the generalized Korteweg-de Vries equation, for which existence of a logarithmic regime was proved in [23] and uniqueness (in the super-critical case) was established in [12].
The case 1 2 c(c+1) ≤ ω < 1 in Theorem 2 is open (see step 1 of the proof of Proposition 1). Remark 3. The dynamics of the distance between the two solitary waves is related to nonlinear interactions. A formal study (see notably [8,13] and Chapter 4 in [32]) shows that the three behaviors (a), (b) and (c) are related to different solutions of γ = c γ e −σ sin γ σ = −c σ e −σ cos γ where γ is the phase difference, σ the relative distance and c γ , c σ are constants. For (cubic NLS), it holds c γ = c σ > 0. Denoting Y = σ +iγ, the resulting equationŸ = −c γ e Y is integrable and admits nontrivial solutions for which σ is periodic.
Remark 4. The proofs of Theorems 1 and 2 follow the overall strategy of several previous articles on multi-solitons ( [14,16,17,18,19,20,21,22,26]), particularly of [20,22] which started the study of multi-solitons with logarithmic distance in a non-integrable setting. We focus on the proof of Theorem 2, which is more original in the construction of a suitable approximate solution and the determination of the asymptotic regime (see Remark 5).
See Section 5 for a comment on the introduction of a refined energy method.

Notation and preliminaries.
For complex-valued functions f, g ∈ L 2 (R), we denote For r a positive function of time, the notation f (t, x) = O H 1 (r(t)) means that there exists a constant C > 0 such that f (t) H 1 ≤ Cr(t).
For any λ > 0 and any function f , let Note the following relation which describes the asymptotics of Q(x) as x → −∞, Throughout this paper, we consider ω and c such that The linearization of (coupled NLS) around solitons involves the following operators: Recall the special relations ( [29]) (1.5) We will use the following properties of these operators. (i) There exists µ > 0 such that, for all z ∈ H 1 , Proof. (i) The coercivity properties of L + and L − (here in the L 2 sub-critical case) are well-known facts (see e.g. [17,29,30]). Let 0 < ρ < c be such that ω = 1 2 ρ(ρ + 1). By [27] or direct computation, we see that the positive function Q ρ satisfies L c Q ρ = (c 2 − ρ 2 )Q ρ . The coercivity property follows. ( The decay properties of u then follows from standard arguments.
The following result follows directly from Lemma 1.
Approximate solution in the case 0 < c < 1 2.1. Definition of the approximate solution. Consider C 1 time-dependent real-valued functions σ 1 , σ 2 , γ 1 , γ 2 , β 1 , β 2 , to be fixed later and set Introduce the notation Define the approximate solution and Using (1.3), we obtain (2.1) for E U with F defined as in (2.2). Similarly, the equation

2.2.
Projection of the error terms. The soliton dynamics is expected to be determined by the following projections we decompose F and G as follows We compute the main order of these projections. including the refined term ϕ is necessary to determine correctly the non-symmetric logarithmic regime.
Proof. We start by proving the following estimates Proof of (2.6). By (1.3) and the condition on θ, we have Proof of (2.7). It follows from (1.3) that and (2.7) follows by integration by parts.
From the expression of F in (2.2), we have For the first term, using −( Similarly as in the proof of (2.7), using (1.3) we observe Moreover, it follows from (1.6) and the coercivity of the operator L c that Last, we check using the decay property of A in (1.7) and the condition on θ that Using also (2.6) and κ 2 = 8, we find .
From the definition of G, we have On the one hand, integrating by parts, it holds On the other hand, using (1.6) and then integration by parts , it holds Thus, also using and (2.7), we obtain b = −cα c e −2cσ + O(e −2cθσ ).

2.3.
Formal discussion. Formally, the previous computations lead us to the systeṁ Recalling σ = σ 1 − σ 2 and β = β 1 − β 2 , this gives which admits the following solution This justifies the existence of the regime (1.2) of Theorem 2. In particular, observe that the positive sign of the constant α c is responsible for the emergence of the special nonsymmetric logarithmic regime. The phase parameters γ 1 and γ 2 are not essential for the dynamics and so we do not discuss them here.
2.4. Decomposition around the approximate solution. Let T ∞ ≫ 1 to be fixed later and consider a solution u v of (coupled NLS) under the form The parameters σ 1 , σ 2 , γ 1 , γ 2 , β 1 and β 2 in the definition of U V are fixed by imposing the following orthogonality conditions where σ ∞ is to be chosen later close to 1 c log(Ω c T ∞ ) (see below (3.2)) and Indeed, by a standard argument and the initial conditions (including ε(T ∞ ) = η(T ∞ ) = 0), the orthogonality conditions are equivalent to a first order differential system in the parameters (σ 1 , σ 2 , γ 1 , γ 2 , β 1 , β 2 ), which admits a unique local solution in the regime considered in this paper. See e.g. Lemma 2.7 in [4] for a detailled argument in the case of the (gKdV) equation, and Lemma 7 in the present paper for the corresponding estimates on the time derivatives of the parameters. For technical reasons, one can fix zero initial conditions on γ 1 , γ 2 as in (2.11), but the initial conditions on σ 1 , σ 2 , β 1 and β 2 have to depend on a parameter σ ∞ to be fixed later by a topological argument. As in [20,22,26], the orthogonality conditions in (2.10) are related to (1.5). Using the conservation of masses and L 2 sub-criticality, we avoid the modulation of the scaling parameters of the solitons (see [30] and the proof of Lemma 7).
Under the bootstrap (3.1), we prove the following estimates. (3.14) Proof of Proposition 1. step 1. The coercivity property (3.13) is a consequence of the coercivity property around one solitary wave in Lemma 1, the orthogonality relations (2.10)-(3.8)) and the positivity of L c . It also involves a localization argument similar to the proof of Lemma 4.1 in [19] for the scalar case. Note that by (3.1), |J(t, ε, η)| t −1 ( ε 2 H 1 + η 2 H 1 ) and by (3.4) and (3.6), Next, we see that the following terms in the functional K are easily controlled Moreover, cubic and higher order terms in ε or η are of order t −θ 1 ε 2 H 1 + η 2 H 1 . Therefore, we are reduced to consider the following two decoupled functionals We focus on the coercivity property for W 1 , the case of W 2 is similar. Denote Φ : R → R an even function of class C 2 such that Let B > 1 and Φ B (x) = Φ(x/B). We claim that for B large enough, there exists µ 1 > 0, such that for anyε satisfying ε, Q = ε, xQ = ε, iΛQ = 0, and anyε, it holds Setting z =εΦ 1 2 B and following the proof of Claim 8 in [19], the coercivity of N 1 follows from (i) of Lemma 1 applied to the function z. A similar localization argument, using the coercivity property of L c proves the estimate for N 2 (ε) without any orthogonality condition onε. This is where our proof needs the condition (1.4).
Using these estimates withε andε such that ε = cε(c(x − σ 1 ))e iΓ 1 and ε =ε(x − σ 2 )e iΓ 2 , the orthogonality conditions (2.10) and the almost orthogonality relation (3.8), we obtain the estimate ε 2 H 1 W 1 + t −4 (log t) 2 . step 2. Time variation of the energy. Denote We prove the following estimate d dt The time derivative of t → K(t, ε(t), η(t)) splits into three parts where D t denotes the differentiation of K with respect to t, and D ε , D η the differentiation of K with respect to ε and η. In particular, Thus, using (3.9) and (3.11), we obtain (3.16) for U . The proof for V is similar. Using (3.16) and (3.1), we obtain Next, we observe We have proved (3.15). step 3. Time variation of the total mass. We claim By integration by parts, we have i∂ 2 x ε, ε = 0 so from (2.9), d dt We claim the following identity Indeed, since h(u, v)u is real, for all θ ∈ R, it holds i(U + θε), h(U + θε, V + θη) = 0.
Differentiating with respect to θ, and taking θ = 0, we obtain Moreover, with θ = 0 and θ = 1 We see that (3.18) follows from combining these identities. This yields d dt M 1 = c 2 iU, K 1 − c 2 iε, E U . Computing also d dt M 2 , we obtain (3.17). step 4. Time variation of the localized momentum. We claim By (3.1) and (3.11), we have By direct computations, and so by (3.1), (3.9) and the properties of χ, |∂ t χ j | t −1 (log t) −1 . It follows that Next, using the equation (2.9) Integrating by parts, we have Since |∂ x χ 1 | (log t) −1 and |∂ 3 For the term containing E U , we use (2.1), (3.1), (3.4) and (3.10), Then, we estimate, using |∂ x χ 1 | (log t) −1 , Collecting the above estimates, we obtain d dt We complete the proof of (3.19) by showing the following First, we prove the identity Applying this to u = U + θε and v = V + θη, we have that for all θ ∈ R Taking the derivative with respect to θ at θ = 0, we obtain Moreover, using the above identity with θ = 0 and θ = 1, we have Gathering these identities, we obtain (3.21).
(3.25) Assume for the sake of contradiction that for all ζ ♯ ∈ [−1, 1], the choice ζ(T ∞ ) = T ∞ + ζ ♯ t 2−θ 2 leads to T ⋆ ∈ (T 0 , T ∞ ]. By a continuity argument, this means that the bootstrap estimates are reached at T ⋆ . Since all estimates in (3.1) except the one on σ, have been strictly improved on [T ⋆ , T ∞ ], this yields Following the argument of [5], we remark that for any t ∈ [T ⋆ , T ∞ ] satisfying (3.26), using (3.25) and θ 2 < θ 1 , it holds (taking T 0 large enough) This transversality condition implies that T ⋆ is a continuous function of σ ∞ and thus is also a continuous function whose image is {−1, 1}, which is contradictory.

3.5.
End of the proof of Theorem 2 by compactness. We use Proposition 2 with T ∞ = n, for any n ≥ T 0 , to construct a sequence of solutions un vn ∈ C([T 0 , n], H 1 × H 1 ) of (coupled NLS) such that, for some δ > 0, on [T 0 , n], Now, we adapt from [17] (in the scalar case) and from [11] (for the vector case), the following convergence result.
From (1.9) and Lemma 5, the second and third terms in the right-hand side are bounded by σ 3 e −4σ . The last term is bounded by For the first term, using L + Q ′ = 0 and then (1.8), we compute We only have to compute Q ′ (x − σ)e x Q 2 (x)dx. First, we see Second, using (1.3) and thus σ 0 Q ′ (x − σ)e x Q 2 (x)dx = κ 3 σe −σ + O(e −σ ).

4.3.
Bootstrap estimates in the case c = 1. Fix θ 1 such that 1 < θ 1 < 2. The following bootstrap estimates are used in this case: for 1 ≪ t ≤ T ∞ , where σ ∞ is to be chosen satisfying e σ∞ Ωσ 1 2 We refer to [20,22] for similar bootstrap estimates. The rest of the proof is similar to the one of Theorem 2 and we omit it.

Discussion
For (coupled NLS), with any coupling coefficient 0 < ω < 1, we have proved the existence of symmetric 2-solitary waves (Theorem 1) and of non-symmetric 2-solitary waves (Theorem 2) with logarithmic distance. Symmetric 2-solitons with logarithmic distance were already known in the literature for the integrable cases (ω = 0 and ω = 1) and in the scalar case (NLS). In contrast, the existence of non-symmetric 2-solitary waves with logarithmic distance is new. In particular, it does not hold for the integrable case where instead a periodic regime exists.
An interesting remaining open question is whether non-symmetric logarithmic 2-solitary waves exist for the non-integrable scalar (NLS). We conjecture that it is indeed the case, as long as p = 3. Indeed, the first step of the strategy used in this paper, i.e. the computation of an approximate solution involving the main interaction terms, works equally well for (NLS) as for (coupled NLS). We expect a logarithmic regime with oscillations. However, whereas (coupled NLS) enjoys two L 2 conservation laws, the scalar equation (NLS) enjoys only one, which does not seem sufficient for the energy method to apply in a context of two solitons with logarithmic distance without symmetry.
A more technical original aspect of this article is the introduction of a refinement of the energy method. In previous articles using approximate solutions in the context of error terms of order t −k (e.g. in [20,22,23]), the energy method induces a loss of decay. Here, the additional correction term S in Section 3.3 allows an estimate of the remainder ε η directly related to the size of the error term E U E V . We believe that this general observation will be useful elsewhere.