Multi-existence of multi-solitons for the supercritical nonlinear Schr\"odinger equation in one dimension

For the L2 supercritical generalized Korteweg-de Vries equation, we proved in a previous article the existence and uniqueness of an N-parameter family of N-solitons. Recall that, for any N given solitons, we call N-soliton a solution of the equation which behaves as the sum of these N solitons asymptotically as time goes to infinity. In the present paper, we also construct an N-parameter family of N-solitons for the supercritical nonlinear Schr\"odinger equation, in dimension 1 for the sake of simplicity. Nevertheless, we do not obtain any classification result; but recall that, even in subcritical and critical cases, no general uniqueness result has been proved yet.

Recall also that (NLS) admits the following symmetries.
• Galilean invariance: if u(t, x) satisfies (NLS), then for any v 0 ∈ R, w(t, 4 t) also satisfies (NLS). We now consider solitary waves of (NLS), in other words solutions of the form u(t, x) = e ic0t Q c0 (x), where c 0 > 0 and Q c0 is solution of Recall that such positive solution of (1.1) exists and is unique up to translations, and is moreover the solution of a variational problem: we call Q c0 the solution of (1.1) which is even, and we denote Q := Q 1 . By the symmetries of (NLS), for any γ 0 , v 0 , x 0 ∈ R, is also a solitary wave of (NLS), moving on the line x = v 0 t + x 0 , that we also call soliton.
Finally recall that, in the supercritical case p > 5, solitons are unstable (see [8]). A striking illustration of this fact is the following result of Duyckaerts and Roudenko [5] (adapted from a previous work of Duyckaerts and Merle [4]), obtained for the 3d focusing cubic nonlinear Schrödinger equation (NLS-3d), which is also L 2 supercritical and H 1 subcritical as in our case. Proposition 1.1 ([5]). Let A ∈ R. If t 0 = t 0 (A) > 0 is large enough, then there exists a radial solution U A ∈ C ∞ ([t 0 , +∞), H ∞ ) of (NLS-3d) such that where e 0 > 0 and Y + = 0 is in the Schwartz space S.
In particular, U A (t) = e it Q if A = 0, whereas lim t→+∞ U A (t) − e it Q H 1 = 0. Note that, in the subcritical and critical cases p 5, no such special solutions U A (t) can exist, due to a variational characterization of Q. Indeed, if lim t→+∞ u(t) − e it Q H 1 = 0, then u(t) = e it Q in this case. The purpose of this paper is to extend Proposition 1.1 to multi-solitons.
• In the L 2 subcritical and critical cases, i.e. for (NLS) with p 5, there exists a large literature on the problem of existence of multi-solitons and on their properties. Merle [12] first established an existence result in the critical case, as a consequence of a blow up result and the conformal invariance. This result was extended by Martel and Merle [10] to the subcritical case, using arguments developed by Martel, Merle and Tsai [11] for the stability in H 1 of solitons. Nevertheless, we recall that no general uniqueness result has been proved, contrarily to the generalized Korteweg-de Vries (gKdV) equation (see [9]).
For other stability and asymptotic stability results on multi-solitons of some nonlinear Schrödinger equations, see [13,14,15].
• In the L 2 supercritical case, i.e. in a situation where solitons are known to be unstable, Côte, Martel and Merle [3] have recently proved the existence of at least one multi-soliton solution for (NLS): Recall that, with respect to [10,11], the proof of Theorem 1.2 relies on an additional topological argument to control the unstable nature of the solitons. Finally, recall that Theorem 1.2 was also obtained for the L 2 supercritical gKdV equation, and has been a crucial starting point in [2] to obtain the multi-existence and the classification of multi-solitons. It is a similar multi-existence result that we propose to prove in this paper.

Main result and outline of the paper
The whole paper is devoted to prove the following theorem of existence of a family of multi-solitons for the supercritical (NLS) equation.
Then there exist γ > 0 and an N -parameter family (ϕ A1,...,AN ) (A1,...,AN )∈R N of solutions of (NLS) such that, for all (A 1 , . . . , A N ) ∈ R N , there exist C > 0 and t 0 > 0 such that Finally, to prove Proposition 3.1, we follow the strategy of the proof of the similar proposition in [2], except for the monotonicity property of the energy which does not hold for the (NLS) equation. If this property of monotonicity was necessary to obtain the classification, we prove that a slightly different functional estimated regardless its sign is sufficient to reach our purpose. We also rely on refinements of arguments developed in [3], in particular the topological argument to control the unstable directions.

Preliminary results
Notation 2.1. They are available in the whole paper.

Linearized operator around a stationary soliton
The linearized equation appears if one considers a solution of (NLS) close to the soliton e it Q.
and the self-adjoint operators L + and L − are defined by The spectral properties of L are well-known (see [7,16] for instance), and summed up in the following proposition. 7,16]). Let σ(L) be the spectrum of the operator L defined on L 2 (R) × L 2 (R) and let σ ess (L) be its essential spectrum. Then Furthermore, e 0 and −e 0 are simple eigenvalues of L with eigenfunctions Y + and Y − = Y + which have an exponential decay at infinity. Finally, the null space of L is spanned by ∂ x Q and iQ, and as a consequence, the null space of L + is spanned by ∂ x Q and the null space of L − is spanned by Q.

Remark 2.3.
By standard ODE techniques, we can quantify the exponential decay of Y ± and ∂ x Y ± at infinity. In fact, there exist η 0 > 0 and C > 0 such that, for all x ∈ R, Moreover, L, L + and L − satisfy some properties of positivity or coercivity. The following proposition sums up the two properties useful for our purpose. Note that the first one is proved in [16], while the second one is proved in [4,5]. (ii) There exists κ 0 > 0 such that, for all v = v 1 + iv 2 ∈ H 1 , Finally, we extend Proposition 2.2 to the operator L c linearized around a soliton e ict Q c (x), by a simple scaling argument. In fact, we recall that if u is a solution of (NLS), then w(t, x) = λ 2 p−1 u(λ 2 t, λx) is also a solution, and moreover, we have Q c (x) = c Finally, e c and −e c are simple eigenvalues of and the null space of L c is spanned by ∂ x Q c and iQ c .
Now, suppose that there exists λ ∈ R such that Y 2 = λQ. Then, we would have L − Y 2 = −e 0 Y 1 = λL − Q = 0, and so Y 1 = 0. But it would imply L + Y 1 = 0 = e 0 Y 2 , and so Y 2 = 0, which would be a contradiction. Therefore, by (i) of Proposition 2.4, we have (

Multi-solitons results
A set of parameters (1.2) being given, we adopt the following notation.
(iv) e j = e cj , where e c = c 3/2 e 0 . Now, to estimate interactions between solitons, we denote c min = min{c k ; k ∈ [[1, N ]]}, and the small parameters From [10], it appears that γ is a suitable parameter to quantify interactions between solitons in large time. For instance, we have, for j = k and all t 0, From the definition of σ 0 and Remark 2.3, such an inequality is also true for Y ± j . Moreover, since σ 0 has the same definition as in [3], Theorem 1.2 can be rewritten as follows. There exist T 0 ∈ R, C > 0 and ϕ ∈ C([T 0 , +∞), H 1 ) such that, for all t T 0 , (2.5)

Construction of a family of multi-solitons
In this section, we prove Theorem 1.3 as a consequence of the following crucial Proposition 3.1.
..,AN , and let us show that it implies For j = 1, first note that, from the construction of ϕ A1,...,AN , the hypothesis means , and moreover and so, by difference, we have Now, if we multiply this equality by Y + σ(1) (t), integrate, and take the imaginary part of it, we obtain, by Claim 2.6 and (2.4), For the inductive step from j − 1 to j, we write similarly and we finally obtain A σ(j) = A ′ σ(j) as expected, by taking the difference of these two expressions, multiplying by Y + σ(j) (t), integrating and taking the imaginary part of it. Now, the only purpose of the rest of the paper is to prove Proposition 3.
. We want to construct a solution u of (NLS) such that

Equation of z
Since u is a solution of (NLS) and also ϕ is (and this fact is crucial for the whole proof), we get But from Corollary 2.5, we have where Y + cj ,1 = Re Y + cj and Y + cj ,2 = Im Y + cj , and so Therefore, we get the following equation for z: (3.4) By developing the nonlinearity, we find where ω(z) satisfies |ω(z)| C|z| 2 for |z| 1. Hence, we can rewrite (3.4) as where Finally, the equation of z can be written in the shorter form where ω 1 satisfies ω 1 (t) L 2 Ce −ej t for all t T 0 . We finally estimate the source term Ω in the following lemma, that we prove in Appendix A.

Compactness argument assuming uniform estimates
To prove Proposition 3.1, we follow the strategy of [10,3]. We first need some notation for our purpose.
(iii) S R k 0 (r) denotes the sphere of radius r in R k0 .
(iv) B B (r) is the closed ball of the Banach space B, centered at the origin and of radius r 0.
Let S n → +∞ be an increasing sequence of time, b n = (b n,k ) k∈K ∈ R k0 be a sequence of parameters to be determined, and let u n be the solution of   Proposition 3.4. There exist n 0 0 and t 0 > 0 (independent of n) such that the following holds. For each n n 0 , there exists b n ∈ R k0 with b n 2e −(ej+2γ)Sn , and such that the solution u n of (3.7) is defined on the interval [t 0 , S n ], and satisfies Assuming this key proposition of uniform estimates, we can sketch the proof of Proposition 3.1, relying on compactness arguments developed in [10,3]. The proof of Proposition 3.4 is postponed to the next section.
Sketch of the proof of Proposition 3.1 assuming Proposition 3.4. From Proposition 3.4, there exists a sequence u n (t) of solutions to (NLS), defined on [t 0 , S n ], such that the following uniform estimates hold: In particular, there exists C 0 > 0 such that u n (t 0 ) H 1 C 0 for all n n 0 . Thus, there exists u 0 ∈ H 1 (R) such that u n (t 0 ) ⇀ u 0 in H 1 weak (after passing to a subsequence). Moreover, using the compactness result [10, Lemma 2], we can suppose that u n (t 0 ) → u 0 in L 2 strong, and so in H sp strong by interpolation, where 0 s p < 1 is an exponent for which local well-posedness and continuous dependence hold, according to a result of Cazenave and Weissler [1]. Now, consider u solution of Fix t t 0 . For n large enough, we have S n > t, so u n (t) is defined and by continuous dependence of the solution of (NLS) upon the initial data, we have u n (t) → u(t) in H sp strong. By the uniform we finally obtain, by weak convergence, Thus, u is a solution of (NLS) which satisfies (3.1).

Proof of Proposition 3.4
The proof proceeds in several steps. For the sake of simplicity, we will drop the index n for the rest of this section (except for S n ). As Proposition 3.4 is proved for given n, this should not be a source of confusion. Hence, we will write u for u n , z for z n , b for b n , etc. We possibly drop the first terms of the sequence S n , so that, for all n, S n is large enough for our purposes.
From (3.6), the equation satisfied by z is In particular, we have

Modulated final data
Lemma 3.5. For n n 0 large enough, the following holds. For all a − ∈ R k0 , there exists a Proof. Consider the linear application Φ : Moreover, from (2.4), there exists C 0 > 0 independent of n such that, for l = k, Thus, by taking n 0 large enough, to conclude the proof of Lemma 3.5.
Claim 3.6. The following estimates at S n hold:

Equations on α ±
k Let t 0 > 0 independent of n to be determined later in the proof, a − ∈ B R k 0 (e −(ej +2γ)Sn ) to be chosen, b be given by Lemma 3.5 and u be the corresponding solution of (3.7). We now define the maximal time interval [T (a − ), S n ] on which suitable exponential estimates hold.
Proof. Following Notation 2.7, we compute Moreover, using the equation of z (3.8) and an integration by parts, we find for the second term Using the estimate ω 1 (t) L 2 Ce −ej t and Lemma 3.2, we find for the last term From the definition of γ (2.3), we deduce that and, as in the proof of Lemma 3.2, we also find Hence, we have Finally, if we denote z 1 = Re(ze −iθ k ) and z 2 = Im(ze −iθ k ), we find

Control of the stable directions
We estimate here α + k (t) for all k ∈ [ [1, N ]] and t ∈ [T (a − ), S n ]. From (3.10) and (3.9), we have Thus, |(e −e k s α + k (s)) ′ | K 2 e −(ej +e k +4γ)s , and so, by integration on [t, S n ], we get |e −e k Sn α + k (S n )− e −e k t α + k (t)| K 2 e −(ej +e k +4γ)t , which gives But from Claim 3.6 and Lemma 3.5, we have and so finally

Localized Weinstein's functional
We follow here the same strategy as in [11,10,3] to estimate the energy backwards. For this, we define the function ψ by Moreover, we set Observe that the functions h 1 and h 2 take values close to c k + Hence, we have φ k 0 and N k=1 φ k ≡ 1, and by an Abel's transform, we also have Proof. See Appendix A. Now, we define a quantity related to the energy for z, by 14) The following estimate of the variation of H is the main new point of this paper, and as its proof is long and technical, it is postponed to Appendix B.
Thus, by integration on [t, S n ], we obtain |H(t) − H(S n )| K1 √ t e −2(ej +γ)t , and so But from Claim 3.6 and Lemma 3.5, we have and so Finally, expanding |ϕ + r j + z| and so, from the definition of H (3.14) , Using (2.5), we easily obtain (3.15) by similar techniques used in the proof of Lemma 3.2 in Appendix A to replace (ϕ + r j ) by R plus an exponentially small error term.

Control of the directions of null energy
Define First, note that there exist C 1 , C 2 > 0 such that Indeed, by (2.4), we have and similarly,

Now, we compare the functionals H[ z] and H[z]
in the following lemma, that we prove in Appendix A.
Completely similarly, we find, for all k ∈ [[1, N ]], using (3.13) for k ∈ J, and (3.9) for k ∈ K. Finally, gathering all estimates from (3.18), we have proved that there exists K 0 > 0 such that, for all t ∈ [T (a − ), S n ], We want now to prove the same estimate for z, and so we have to control the parameters β k (t) and γ k (t) introduced above.

Improvement of the decay of z
Proof. By (3.16), it is enough to prove this estimate for |β k (t)| + |γ k (t)| with k ∈ [[1, N ]] fixed. To do this, write first the equation of z, from the equation of z (3.6), Then, multiply this equation by R k , integrate, and take the real part of it, so that we obtain, by (2.4), (2.5) and Lemma 3.2, In other words, we have, by (3.16) and (3.9),

Moreover, from
= Im ∂ t zR k + Im z∂ t R k , and so, as Gathering previous estimates, we find Completely similarly, if we multiply the equation on z by ∂ x Q c k (λ k )e −iθ k , integrate and take the imaginary part of it, we find Hence, we have proved that there exist C 3 , C 4 > 0 such that, for all t ∈ [T (a − ), S n ], Finally, if we choose t 0 large enough so that By integration on [t, S n ], we get |β k (t)| + |γ k (t)| |β k (S n )| + |γ k (S n )| + C t 1/4 e −(ej +γ)t . But from Claim 3.6, Lemma 3.5 and (3.16), we have and so finally, ∀t ∈ [T (a − ), S n ], |β k (t)| + |γ k (t)| C t 1/4 e −(ej +γ)t .

Control of the unstable directions for k ∈ K by a topological argument
Lemma 3.12 being proved, we choose t 0 large enough so that K0 t 1/4 0 1 2 . Therefore, we have We can now prove the following final lemma, which concludes the proof of Proposition 3.4. Note that its proof is very similar to the one in [2], by the common choice of notation, but it is reproduced here for the reader's convenience. We can now consider, for t ∈ [T, S n ], To calculate N ′ , we start from estimate (3.12): Multiplying by |α − k (t)|, we obtain where e min = min{e k ; k ∈ K}. By summing on k ∈ K, we get Therefore, we can estimate Hence, we have, for all t ∈ [T, S n ], where θ = 2(e min − e j − 2γ) > 0 by the definitions of γ (2.3) and of the set K. In particular, for all τ ∈ [T, S n ] satisfying N (τ ) = 1, we have Now, we definitely fix t 0 large enough so that K 3 e −2γt0 We finally deduce that T (a − ) − ε T ( a − ) T (a − ) + ε, as expected.
Second consequence: We can define the map Note that M is continuous by the previous point. Moreover, let a − ∈ S R k 0 (e −(ej +2γ)Sn ). As N ′ (S n ) − θ 2 by (3.21), we deduce by definition of T (a − ) that T (a − ) = S n , and so M(a − ) = a − . In other words, M restricted to S R k 0 (e −(ej +2γ)Sn ) is the identity. But the existence of such a map M contradicts Brouwer's fixed point theorem.

A Appendix
Proof of Lemma 3.2. First, we calculate Hence, from the expression of Ω (3.5), it can be written We can now estimate Ω H 1 , and we estimate ∂ x Ω L 2 for example, the term Ω L 2 being similar and easier. To do this, we write To estimate all these terms in L 2 norm, we use the facts that ϕ is equal to R plus a small error term according to (2.5), that R multiplied by a term moving on the line x = v j t + x j (like r j ) is equal to R j plus a small error term according to (2.4), and finally that r j is at order e −ej t . To illustrate this, we estimate the first two terms I and II, for example, as all other terms can be treated similarly. For I, we simply remark that by the definition of γ (2.3). For II, we decompose it as Since ϕ − R H 1 Ce −4γt by (2.5), the first three terms are bounded in L 2 norm by Ce −(ej +4γ)t . Moreover, by (2.4), the next three terms are also bounded in L 2 norm by Ce −(ej +4γ)t . Finally, for the last term, we write so that, since p > 5, we can conclude similarly that II L 2 Ce −(ej +4γ)t .
Proof of Lemma 3.9.
But, if k > l, then and similarly, if k < l, then and the conclusion follows again from the definition of γ.
For the last one, we write so that and so finally ψ kt L ∞ have a similar form, it is clear that it suffices to prove the inequalities for h 2 , for example. Moreover, the first inequalities are obvious by (iii). Finally, for the last inequality, we write dropping the argument λ k for this proof, which would not be a source of confusion since there is no time derivative. Hence, we compute Developing in terms of z, we find

B Appendix
We prove here Proposition 3.10. To do this, we first need a lemma quantifying the fact that ϕ almost satisfies a transport equation similar to those satisfied by the solitons. Note finally that, since ϕ t takes values in H −1 , all integrals in this appendix may be seen as the dual bracket ·, · H 1 ,H −1 .
Remark B.2. To find the transport equation almost satisfied by ϕ, it suffices to compute an exact relation for R k with k ∈ [ [1, N ]]. In fact, as Proof of Lemma B.1. Let f ∈ H 1 and compute First note that, by (2.5), |I| C ϕ − R H 1 f H 1 Ce −4γt f H 1 . Moreover, by (2.4), we also have |III| Ce −4γt f L 2 . For the last term, we first compute and so, using ∂ 2 Therefore, by (iv) of Lemma 3.9, we also have |II| Ce −4γt f L 2 , which concludes the proof of Lemma B.1.
From the definition of H (3.14), we now compute, using integrations by parts, For I, we have to compute