Mixed dimensional infinite soliton trains for nonlinear Schr\"odinger equations

In this note we construct mixed dimensional infinite soliton trains, which are solutions of nonlinear Schr\"odinger equations whose asymptotic profiles at time infinity consist of infinitely many solitons of multiple dimensions. For example infinite line-point soliton trains in 2D space, and infinite plane-line-point soliton trains in 3D space. This note extends the works of Le Coz, Li and Tsai [5, 6], where single dimensional trains are considered. In our approach, spatial $L^\infty$ bounds for lower dimensional trains play an essential role.


Introduction
In this paper, we consider the nonlinear Schrödinger equation i∂ t u + ∆u + f (u) = 0, (1.1) and the solution tends to concentrate, with possible blow-up in finite time. This is the focusing effect. At the equilibrium between these two effects, one may encounter many different types of structures that neither scatter nor focus. The most common of these non-scattering global structures are the solitons, but there exist also dark solitons, kinks, etc. A generic conjecture for nonlinear dispersive PDE is the Soliton Resolution Conjecture. Roughly speaking, it says that, as can be observed in physical settings, any global solution will eventually decompose at large time into a scattering part and well separated non-scattering structures, usually a sum of solitons. Apart from integrable cases (see e.g. [13]), such conjecture is usually out of reach. Intermediate steps toward this conjecture are existence and stability results of configurations with well separated non-scattering structure, like multi-solitons, multi-kinks, infinite soliton and kink-soliton trains, etc. See [7] for a survey on these subjects. In what follows we describe results most relevant to us. Multi-solitons are solutions of (1.1) with the asymptotic profile as t → ∞, where N ≥ 2 and each R j is a soliton to be specified in (1.5). The first result of existence of multi-solitons was obtained in Zakharov and Shabat [13] in the case of the 1d focusing cubic (i.e. d = 1, f (z) = |z| 2 z) nonlinear Schrödinger equation via the inverse scattering method. Indeed, in this particular case the equation is completely integrable and one can obtain multi-solitons in a rather explicit manner. Kamvissis [4] showed that it is possible to push the inverse scattering analysis forward and obtain the existence of an infinite soliton train, i.e. a solution u of (1.1) defined as in (1.2) but with N = +∞. In fact, it is shown that, under some technical hypotheses, any solution to (1.1) with initial data in the Schwartz class will eventually decompose at large time as an infinite soliton train and a "background radiation component". There are also results for multi-dark solitons for the companion integrable Gross-Pitaevskii case, i.e. d = 1 and f (z) = (1 − |z| 2 )z, but no known results for infinite trains. In a non-integrable setting, the first existence result of multi-solitons was obtained by Merle in [10] as a by-product of the proof of existence of multiple blow-up points solutions for L 2critical (1.1), i.e. f (z) = |z| 4/d z. The techniques initiated in [10] were then developed in [2,3,8,9] for other nonlinearities. The idea, so called the energy method, is to choose an increasing sequence of time (t n ) with t n → +∞ and consider the solutions (u n ) to (1.1) which solve the equation backward in time with final data u n (t n ) = T (t n ). The sequence (u n ) is an approximate sequence for a multi-soliton. To show its convergence, two arguments are at play. First, one shows that there exists a time t 0 independent of n such that u n satisfies on [t 0 , t n ] the uniform estimates (u n − T )(t) H 1 ≤ e −µ √ ω * v * t .
Second, we have compactness of the sequence of initial data u n (t 0 ), i.e. there exists u 0 so that u n (t 0 ) → u 0 in H s for all 0 < s < 1. See also [11,12,9] for stability results under restrictive hypotheses. The energy method is very flexible and can be adapted to other situations. However, its implementation is far from being trivial when the number of solitons is infinite or when one soliton is replaced by a kink. In Le Coz, Li and Tsai [5,6], an approach based on fixed point argument has been used to construct such structures. In this approach, the large relative speed has been used to get smallness of the Duhamel term due to short interaction time. It is however delicate when the gradient of the error term is also measured. We will explain this approach in more details below, as we will use it to construct mixed dimensional infinite soliton trains.
We now make two assumptions on the nonlinearity f , which will be assumed throughout the paper.

General idea
Suppose {W j (t, x)} is a (finite or infinite) collection of solutions of (1.1). Intuitively, if these solutions are sufficiently separated from each other, then the nonlinear effects of their interactions should be negligible, and j W j should be close to a solution. We are interested in the possibility that j W j + η is a solution for some error η = η(t, x) tending to zero as t → ∞. The equation of η is hence i∂ t η + ∆η + f ( j W j + η) − j f (W j ) = 0 η| t=∞ = 0 (formally).
By Duhamel's principle, it suffices to solve the fixed point problem where , For a specific profile j W j in our construction, we will try to prove that Φ is a contraction mapping on a closed ball of some Banach space (see (1.9) and (1.10) below for examples), with the needed inequalities derived from the standard dispersive estimates and the Strichartz estimates. In doing so, the above decomposition of the source term into G and H will be convenient. Apparently, the control of G will come from our assumed control on η. On the other hand, the control of H is much more elaborate. It will rely on our assumption that different W j 's are sufficiently separated from each other. See the next section.

Infinite soliton trains
By an infinite soliton train we mean a solution of (1.1) whose asymptotic profile is of the form T = j∈N R j , where R j = R ϕ j ,ω j ,v j ,x 0 j ,γ j (t, x) are solitons as given by (1.5). We remark that the term "train" is only used in a suggestive sense. It well describes the one dimensional situation where all solitons travel in the same direction. In higher dimensions, the traveling directions of the constituting solitons can be rather arbitrary.
Consider {W j } in (1.8) to be such {R j } j∈N , with x 0 j = 0 for all j (see Remark 1.1 below). To give an idea of the kind of things to be proved, we cite two results (rephrased).
Then, for λ large enough and under suitable conditions of {ω j } and {v j }, Φ is a contraction mapping on the closed unit ball of X λ . (1.10) Suppose 0 < α 1 < 4 d+2 . Then, for suitable c ≤ 1 and large enough λ, and under suitable conditions of {ω j } and {v j }, Φ is a contraction mapping on the closed unit ball of X λ,c .
The S(t) in the statements represents the Strichartz space on the time interval [t, ∞) (relevant preliminaries will be given). What the "suitable conditions" are will be clear in due course. Roughly speaking, they concern the speeds of the following two limits: i) ω j → 0, so that the Lebesgue norms of T (or also of ∇T ) can be controlled by (1.6); ii) |v j − v k | → ∞ (as j, k → ∞, j = k), so that different solitons are sufficiently separated to each other. Remark 1.1. We assume the initial positions of all the solitons to be the origin for simplicity. This is an apparent reason why some constructions (such as the second result cited above) have to be done on time intervals [t 0 , ∞) with positive t 0 . The same situation will occur in some of our results for mixed dimensional trains.

Mixed dimensional trains
Now we consider asymptotic profiles consisting of solitons of multiple dimensions. The simplest example is where R 1;k and R 2;j are solitons in R 1 x and R 2 x respectively. For convenience, we'll call them 1D solitons and 2D solitons. (1.11) can be visualized as the profile of a line-point soliton train in R 2 x (and a plane-line soliton train in R 3 x , a space-plane soliton train in R 4 x , and so on). Similarly, we can consider a combination of eD solitons and dD solitons for 1 ≤ e < d, or even combinations involving three or more dimensions. (It turns out that there are limited realizable combinations. See the section "Main results" below.) Solutions having such kind of profiles will be called mixed (dimensional) trains. In the following we take (1.11) as an example to describe the particular difficulties in constructing mixed trains.
First, our general idea encounters a problem if we only use a 2D error η(t, x 1 , x 2 ). To see this, note that by posing a solution of the form T 1 + T 2 + η, we get Then, with x 1 being fixed, we have lim which is nonzero in general. That is H has no space decay at infinity, and hence defies any suitable estimate (we will need L p x controls of H for p ≤ 2). To resolve this problem, we will also introduce a lower dimensional error. Precisely, we will construct a solution of the form where η 1 = η 1 (t, x 1 ) is such that T 1 + η 1 is an 1D train (i.e. a solution of (1.1)). In this way, by regarding {W j } as the sequence defined by W 1 = T 1 + η 1 , and W j+1 = R 2;j for j ∈ N, we have which we will be able to estimate suitably. The main difficulty in the construction is that the 1D objects R 1;k and η 1 only allow L ∞ bounds in x 2 . There are two aspects of the effect of this restriction. 1) To estimate products involving 1D objects and η (such as |η||η 1 | α i L p x ), we must have L ∞ x 1 estimates of the 1D objects, to avoid the need of dealing with "anisotropic" estimates of η. Here by an anisotropic estimate we mean an L p 1 x 1 L p 2 x 2 estimate with p 1 = p 2 . Whether such estimates are available for η is unclear (see Appendix B). Now, for R 1;k , the L ∞ x 1 estimate is easy to obtain from (1.6). However, there is no ready result asserting an L ∞ x 1 control of η 1 . In the previous works [5,6], the authors did not concern the possibility of constructing (single dimensional) trains with L ∞ x control of the errors. (Nevertheless, (1.10) does imply such controls by Sobolev embedding. We'll discuss it in Section 4.3.) As a consequence, we will investigate this problem before going into the mixed cases.
2) On the other hand, anisotropic estimates for R 2;j (and ∇R 2;j ) are easy to obtain (also by (1.6)). In estimating products of them with the 1D objects, we will exploit such estimates. As will be seen, using anisotropic estimates does give us much better results.

Main results
We summarize our main results in the following.
We give some remarks on other possible constructions, not treated in this paper.
Remark 1.2. We focus on infinite soliton trains in this paper. Our method can apparently be used to construct trains with finitely many solitons. In that case, there is no need of any assumptions on (the finite sequences) {ω j } and {v j }, as long as (1.6) is valid for all the solitons. Remark 1.3. We may add a half kink K(t, x 1 ) (if it exists) to one side of T 1 as in [5,6], if all solitons (1D and higher dimensional) are positioned in the other side. If T 1 is finite, we may add half kinks to both sides. (See [5, Figure 1] for an illustration.) Notice that, in this case, it is still possible that there are infinitely many higher dimensional solitons. For example, consider the infinite 2D train profile T 2 , with v j = (v j1 , v j2 ) being the velocities of the solitons. To combine it with T 1 having kinks on both sides, we can arrange v j1 to make T 2 well separated from T 1 , and take |v j2 − v ℓ2 | → ∞ as j, ℓ → ∞ (j = ℓ) to make the 2D solitons to be separated from each other.
The rest of the paper is organized as follows: In Section 2, we collect some basic inequalities of the nonlinearity f . In Section 3, we construct single dimensional trains with spatial supremum control on the errors. Along the way, we give some detailed discussions as to the control of trains, which are also fundamental for mixed dimensional cases. We begin Section 4 by showing the importance of using the Strichartz estimates for constructing mixed trains. Section 4.1 gives the necessary preliminaries related to the Strichartz space. The eD-dD trains are considered in Section 4.2, and finally the 1D-2D-3D trains are constructed in Section 4.3.

Basic inequalities
In this section, we collect some inequalities that are simple consequences of Assumption (F). The only thing that can be said new is Proposition 2.3 (and Corollary 2.5), of which the flexibility in choosing the powers will be useful in some places. We first make the following Convention of notation. In this paper, a constant is called universal if it depends only on the dimension d and the nonlinearity f , in particular C 0 , α 1 , α 2 in Assumption (F) and ω * , a, D in Assumption (T) d . We will use the notation in the sense that the inequality is up to a universal multiplicative constant. The dependence on other parameters will be given explicitly, possibly as a subscript of .
For w : R d → C such that the chain rule applies to ∇f (w(x)) (e.g. w ∈ W 1,1 loc ), it's easy to check that We have the following corollary.

Proposition 2.3.
For any θ ij , φ ij ∈ [0, 1] (i = 1, 2, j ∈ N), and for any absolutely convergent series j∈N w j of complex numbers, we have Remark 2.4. It should be clear that we use j to represent j∈N , and ℓ =j (with j fixed) to represent ℓ∈N\{j} . We'll freely use such simplified notation in this paper.
Proof. The inequality is trivial if w j = 0 for all j. So assume at least one w j = 0. Let h j = |w j |/( ℓ |w ℓ |) for each j ∈ N, and let w = j w j . We have Now fix any θ ∈ [0, 1]. Notice that by Young's inequality we have This completes the proof.
Proof. The assertion follows by considering w j = 0 for j ≥ 3, and taking

Single dimensional trains with L ∞ x control of errors
In this section we investigate the possibility of constructing single dimensional trains are dD solitons as given by (1.5), with x 0 j = 0 for all j. Besides the main results (Theorem 3.7 and Theorem 3.9), many discussions in this section are also useful for next section.
By Assumption (T) d , where we used |v j |/2 + ω By the change of variable x for short). We shall however maintain the sloppy notation for simplicity. The same remark applies to T L p x and ∇T L p x , which will be considered soon. Note that as solitons do not change shapes, they can not have L s t L p x bounds for any s < ∞. Remark 3.2. Using the inequality |y| ≥ (|y 1 | + · · · + |y d |)/ √ d, we get D p ≤ D( 2 √ d ap ) d/p . Thus, for fixed p 0 > 0, p ≥ p 0 implies D p p 0 1. In particular, D p 1 if p 0 is universal. There will be times we have to consider p 0 < 1.
Proof. The first inequality is true by the following computation: Similarly, we have x ) M , which implies the second inequality.
To avoid cumbersome notation, we define In particular, T L p x ≤ D p A p , and ∇T L p x D p B p . As discussed in the Introduction, to construct solutions of the form T + η, we consider the operator where The following lemma gives more precise and complete estimates of H than those given in [ (H0) Fix any r 0 > 0. For r > s > r 0 and t ≥ 0, (H1) Fix any r 1 > 0. For r > s > r 1 and t ≥ 0, where p, q are arbitrary numbers in (0, ∞] satisfying 1 q + 1 p = 1 s . Remark 3.5. The inequalities are indeed true for all r > s > 0, only that the multiplicative constants will then depend on s. For the upper bounds given in (H0) and (H1) to be under desirable control, there are actually natural choices of r 0 and r 1 that are universal (depending only on d, α 1 , α 2 ). We'll discuss this point right after the proof.
Proof of Lemma 3.4. Each assertion is proved by the same strategy as in [5,6]: Prove the exponential decay in t of the L ∞ x norm, by singling out the soliton "nearest" to a fixed (x, t). And prove the boundedness of the L s x norm independent of t. Then the L r x estimate follows by interpolation.
Then for j = m, Thus, by (3.2) and the definition of v * , Now that the upper bound is independent of x and m, we get Next, we try to bound H L s x for finite s > r 0 > 0. By Proposition 2.3, in particular its flexibility of choosing θ ij and φ ij , Thus, for s > r 0 , by (3.5) (and Remark 3.2) By (3.9) and (3.10), for r > s > r 0 , we have

PROOF OF (H1). By (2.3) and (2.2),
Let m = m(t, x) ∈ N be as above. By (3.2) and (3.8), On the other hand, from (3.11), Hence, for s > r 1 , where p, q are any numbers in (0, ∞] satisfying 1 q + 1 p = 1 s . By (3.12) and (3.13), for r > s > r 1 , we have Now we explain how the values of A p , B p (here p is regarded as a parameter) and v * should be controlled, by adjusting {ω j } and {v j } of the profile T . As is mentioned, we need ω j → 0 and |v j − v k | → ∞. Precisely, we will need the flexibility of making v * as large as we like, and at the same time controlling the sizes of A p and B p . As to this purpose, the first obvious observation is that A p < ∞ can be true if and only if 1 Next, a little thought shows that v * > 0 and B p < ∞ can hold simultaneously only if 1 which is equivalent to α 1 < 2 and p > dα 1 2−α 1 . It turns out that these minimum requirements are sufficient. We give the relevant facts in the next lemma. For convenience, we define (3.14) Lemma 3.6.
(b) Suppose q ∈ C A , then for any constants c, Λ > 0, there exist {ω j } and {v j } such that A q ≤ c, and v * ≥ Λ. If moreover α 1 < 2 and p ∈ C B , then {ω j } and {v j } can be chosen so that B p ≤ c is also satisfied.
The proofs of these facts are elementary and are given in Appendix A. Briefly, (a) says A q A p and B q B p for q ≥ p. As a consequence, when there are several A p or B p to be controlled, it suffices to control those having smaller p. And (b) is exactly the desired control. (a) and (b) will be fundamental for the effectiveness of our estimates of G and H.
For the construction of soliton trains in this section, the needed estimates will be derived from the dispersive inequality: If p ∈ [2, ∞] and t = 0, (3.15) We now give our first main result. Then for any finite ρ > 0, there is a constant λ 0 > 0 such that the following holds: For Given ρ ∈ (0, ∞), we will prove that, for sufficiently large λ, there are {ω j }, {v j } such that Φ (defined in (3.6)) is a contraction mapping on the closed ball {η ∈ X : η X ≤ ρ}.
First, we give estimates for Φ to be a self-mapping. Given η ∈ X with η X ≤ ρ. For p ∈ [2, ∞], the dispersive inequality (3.15) implies For the first term, we have where notice that 2α 1 ∈ C A . For the second term, by (3.17), 20) where in (3.21) we use the assumption α 1 ≥ 1.
Remark 3.8. By the contraction mapping principle, for a fixed profile T such that Φ is a contraction, the error η is unique within the class we try to find it.
Before giving the next theorem, we make some comments on the choices of {ω j } and {v j }. As gradient estimate of η is not involved in the previous proof, B p does not occur, and the last part of the proof can be replaced by the following: 1) First choose {ω j } so that the coefficients E 1 ∼ E 4 are finite (equivalently, all "A p " are finite), then 2) choose λ ≥ λ 0 sufficiently large so that (3.25) -(3.28) imply that Φ is a contraction mapping. And hence 3) the construction is done for any {v j } such that (3.24) is satisfied.
It's then easy to see what choices of {ω j }, {v j } are allowable. For example, since A (α 1 +1)/2 is the A p with smallest p to be controlled in the proof, the construction is possible if and only if However, when there is B p , the Step 3) of choosing {v j } will also influence the coefficients considered in Step 1). For later considerations, we have given a proof that works even when B p is present: For every λ, Lemma 3.6 ensures that we can choose {ω j } and {v j } so that v * is large enough and all A p , B p are small. For large enough λ, Φ is hence a contraction mapping for such {ω j }, {v j }. Moreover, giving precise conditions as (3.29), though possible, would be rather cumbersome. We shall hence satisfy ourselves with such vague statement as Theorem 3.7. Suffice it to say that, once a construction is done with some choice of {ω j } and {v j }, it is done with all other choices making the present A p and B p smaller and the v * larger. One easy way to obtain such "better" choices is by rescaling, i.e. by considering {κω j } and {νv k } for suitable positive constants κ, ν. The argument is routine and we omit the details.
We now turn to our next main result. First, notice that the proof of Theorem 3.7 fails for d ≥ 2, since the dispersive inequality gives where the singularity at τ = t is not integrable. As a consequence, we consider the following alternative way: Construct trains T + η having η(t) L 2 x and ∇η(t) L r x controls for some r > d. Then the η(t) L ∞ x control follows from Sobolev embedding (Gagliardo-Nirenberg's inequality). It turns out that we still need d ≤ 3. Moreover, due to some technical benefits, we also assume the ∇η(t) L 2 x control in our construction (see Remark 3.10 after the proof). . Then for any finite ρ > 0, there are constants r > d, λ 0 > 0, and 0 < c 1 ≤ 1, such that the following holds: (3.30) If moreover d = 1 and α 1 ≥ 1, then the above assertion holds with r = ∞.
Remark. We need d ≤ 3 so that e it∆ L r ′ →L r is locally integrable in t for some r > d. We need α 1 < 2 so that C B in (3.14) is nonempty, and hence B p can be controlled for p ∈ C B .
Proof. For r > d, λ > 0, and 0 < c 1 ≤ 1, let X = X r,λ,c 1 be the Banach space of all η : [0, ∞) × R d → C with norm η X defined by the left-hand side of (3.30). By the Gagliardo-Nirenberg's inequality, for any p ∈ [2, ∞], where G d,p,r is a constant and θ = ( We will show that Φ can be a contraction mapping on the closed unit ball of X (the case of ρ = 1). Balls with other radius can be similarly treated. Moreover, we'll only give the estimates for Φ to be a self-mapping. As in the proof of Theorem 3.7, the derivations of the estimates for contractivity are no harder (and without the H parts).
Given η ∈ X with η X ≤ 1. We will first estimate Φη(t) L 2 x , and then ∇Φη(t) L r x . Finally, ∇Φη(t) L 2 x is basically a special case of ∇Φη(t) L r x .
Then from (3.32) and (3.33), the dispersive inequality gives where Part 2. Estimate of ∇Φη(t) L r x . This part is more delicate. The dispersive inequality gives To derive a suitable estimate from it, in the following we will get several conditions on the lower bounds of 1/r. The one thing to check is that they are all strictly less than 1/d, so that there is really one r > d satisfying all the conditions. Moreover, if d = 1, r can be ∞. Step Since we want r > d, we need the lower bound 1 2 − 1 d to be less than 1 d , which holds if and only if d ≤ 3. If d = 1, the lower bound is negative and we can choose r = ∞.
Step 2-1. Estimate of (1). (Only for α i > 1) Suppose with (α i − 1)q ∈ C A and p ∈ C B . Notice that (3.35) is equivalent to It's easy to check that, since d ≤ 3 and α i ≥ α 1 , the lower bound is strictly less than 1/d, and is negative if d = 1.
Step 3. Estimate of H. Suppose One can check that the lower bound is less than 1/d by d ≤ 3 and α 1 < 2, and is negative if d = 1. From (3.43), by fixing a small enough ε > 0, we have From Lemma 3.4 (H1), we get , we have α 1 q ∈ C A and p ∈ C B . From the above discussions, we get the following conclusion: Suppose r > d is sufficiently close to d, c 1 satisfies (3.40), and v * satisfies (3.45) Then we have Part 3. Estimate of ∇Φη(t) L 2 x . We have We can imitate Part 2 to obtain all the needed estimates. We summarize them below.
1. There is no need of Step 1.

For the four sub-steps in
Step 2, simply replace "r" by "2" (except those of G d,p,r and θ in using (3.31)), we have the following results: It's easy to check that p ∈ C B , and α i (1 − θ + c 1 θ) ≥ c 1 as long as (3.40) holds.

The conclusion of
Step 3 is valid with r replaced by 2. Precisely, we have where s 2 , p, q can be the same as given there.

Thus, if (3.40) and (3.45) hold, we have
where E 3 is obtained by collecting the coefficients in (2-1) -(2-4) and (3.47). The conclusions of the three Parts (namely (3.34), (3.46) and (3.48)) provide the needed estimates for Φ to be a self-mapping. Similarly we can derive the estimates for Φ to be contractive, and the theorem is true by Lemma 3.6.

Mixed dimensional trains
In this section we consider mixed trains. It would be good for the reader to recall the discussion in Section 1.3. First we point out a new problem not mentioned in Section 1.3: We can't use only the dispersive inequality (3.15) to construct mixed trains like we did in the previous section. To explain the problem, we take the 1D-2D train T 1 + η 1 + T 2 + η for example. Corresponding to this train, we have Suppose we try to find η in a Banach space X whose norm assumes the exponential decay of η(t) L p x (with possibly several p). Then we have to estimate Φη(t) L p x . To use the dispersive inequality, we can only consider p ≥ 2. Then we have to estimate G(τ ) L p ′ x , from which we will encounter (a) |η|(|T 1 which is not relevant to the problem). For (a), since the 1D objects only have L ∞ bounds in x 2 , not L q x 2 for q < ∞, we can only estimate as follows: Thus we have to also assume the exponential decay of η(t) L p ′ x for the norm of X, and hence have to estimate Φη(t) L p ′ x . Again, this can be done only if p ′ ≥ 2, and hence we must have p = p ′ = 2. Nevertheless, (b) then requires us to estimate η(τ ) L 2(α i +1) x , and the construction fails. We remark that adding some ∇η(t) L p x controls in the definition of X also results in similar problems.
Due to the above observation, we shall use the Strichartz estimate to accomplish our task. In the following section, we recall the basic definitions and facts about the Strichartz space, and then give some more specialized inequalities to be used.

Strichartz space
(4.1) Thus A is the set of all (Schrödinger) admissible pairs if d = 2. For d = 2, we take r max < ∞ to avoid the forbidden endpoint, and r max can actually be any finite number no less than 4 for our approach. We set it to be 4 for preciseness.
For τ ≥ 0, we abbreviate L q ([τ, ∞), L r (R d )) as L q t L r x (τ ), or even L q t L r x when the time interval is clear. We'll abuse notation and write L q t L r x (t), where the two "t" should not cause confusion. Define the Strichartz space x (t) ). By interpolation, Denote the dual space of S(t) by N(t). For (q, r) ∈ A, a function ξ ∈ L q ′ t L r ′ x (t) is regarded as an element in N(t) by letting In this way, we have | ξ, η N (t), For λ > 0 and t 0 ≥ 0, we define S λ,t 0 to be the class of all η ∈ S(t 0 ) such that η S λ,t 0 := sup t≥t 0 e λt η S(t) < ∞.
By definition, η S(t) ≤ η S λ,t 0 e −λt for t ≥ t 0 . In particular, since η L ∞ t L 2 In the rest of this section we prove some useful inequalities, particularly Lemma 4.4. First, we give a fact arising from a proof step of [5,Proposition 2.4]. It might be of independent interest.
where we can choose C = C(p) in such a way that C ≤ (1 − e −1 ) −1 for p ≥ 1.
Proof. We consider three cases separately.

So (4.3) is true with
3. Suppose p < q < ∞. For fixed t ≥ 0, let {t k } ∞ k=0 be a sequence satisfying t 0 = t and t k ր ∞. Then
It's enough to prove the existence of r ′ satisfying (ii) and (iv), since (ii) implies the existence of q such that (i) holds, and then (iii) also holds by m ≤ 4/d. Now (ii) and (iv) are satisfied by some r ′ if and only if 2 m + 1 ≤ 2 and r ′ max ≤ r max m + 1 , that is (4.5) Since 0 ≤ m ≤ 4/d, (4.5) is satisfied. (This is where we need r max ≥ 4 for d = 2.) Now let (q, r) ∈ A be such that ((m + 1)q ′ , (m + 1)r ′ ) is sub-admissible, then (4.4) and Corollary 4.

If d ≥ 3, there exists b satisfying (b1) and (b2) if
where the lower bound might be larger than 2. We consider two cases: (a) If 0 < m ≤ 2 d−2 , we can choose p = 2, and b any number satisfying max( we can choose p = 2(m+1)d 2(m+1)+d , and b = 1 (the only choice). It is easy to check that 2 ≤ p ≤ r max and µ > 0. . . . , x e ) and x ′′ = (x e+1 , . . . , x d ). In this subsection we construct mixed dimensional soliton trains of the form u = T e + η e + T d + η, (4.13) where

Construction of eD-dD trains
with R e;k and R d;j being eD and dD solitons as given by (1.5), with initial positions assumed to be the origin for simplicity. (The reservation of j for the indices of the dD solitons and k for those of the eD solitons will be convenient.) The eD error η e = η e (t, x ′ ) is such that T e + η e is itself an eD train (solution of (1.1)), whose existence will be provided by the previous section. And η = η(t, x) is the remaining error to be found. Denote the frequencies of R d;j and R e;k by ω j and σ k ; and the velocities by v j = (v j,1 , v j,2 , . . . , v j,d ) and u k = (u k,1 , . . . , u k,e ).
(Their corresponding bound states and phases will not be used explicitly, and hence there is no need to introduce notations for them.) R e;k is naturally regarded as a lower dimensional soliton in R d x by considering R e;k (t, x) ≡ R e;k (t, x ′ ), with velocity (u k,1 , . . . , u k,e , 0, . . . , 0). Besides the above, some more modifications of notation given in the previous section have to be made, and some anisotropic generalizations need to be introduced. We summarize them in the following.
1. We'll write A d;p for A p ({ω j }) and B d;p for B p ({ω j }, {v j }), as defined in (3.4). Similarly, we write In particular L p x = L p x ′ L p x ′′ with exactly the same norm. The following generalizations are straightforward, hence we only give them without proof. We have . By the same reason as in Remark 3.2, we'll absorb D p,q into . By a similar result of Lemma 3.3, we have 3. We need all the solitons in both sequences to be separated, hence we define where v * (T e ) and v * (T d ) are as defined by (3.7), and v * (T e , T d ) := inf j,k∈N (4.14) Here v ′ j = (v j,1 , . . . , v j,e ), the first e components of v j . The convention that we add a coefficient 1/2 in (3.7) but not in (4.14) is only to simplify some expressions.

We write C (d)
A for the original C A , and C 1 Similarly, if α 1 < 2, we have C Lemma 3.6 can also be generalized. For example, if α 1 < 2, (q 1 , q 2 ) ∈ C (e,d−e) A , (p 1 , p 2 ) ∈ C (e,d−e) B , then we can choose {ω j } and {v j } so that A d;q 1 ,q 2 and B d;p 1 ,p 2 are as small as we like, and v * as large as we like (see Appendix A). We shall not give a description of all the needed facts, but just claim that, as before, it suffices to check that all the indices of A, B appearing in our proofs lie in their corresponding controllable class C.
To construct solutions of the form (4.13), as discussed in Section 1.3, we consider the operator Φ in (1.8) with source term G + H, where , For convenience, we further divide H into H 1 + H 2 , where ,

The Strichartz estimate asserts
Estimates for H 2 (or ∇H 2 ) will be provided by Lemma 3.4. We now give our first main result. Notice that "e" here corresponds to the role of "d" in Section 3.
Remark. It's most natural to view the eD-dD trains as solutions of (1.1) in R d x , with dD solitons being "points" and eD solitons lower dimensional objects. Nevertheless, as we have mentioned in the introduction, we can also freely regard them as living in an even higher dimension, so that both e, d have nonzero codimensions to the ambient space.
Proof. We will only consider ρ = 2. The cases of other ρ can be treated similarly.
First, from the assumption, if e = 2, 3, then d ≥ 3, and hence α 1 < 2. Thus, for e = 1, 2, 3, if λ is large enough, Corollary 3.11 implies the existence of an eD train T e + η e satisfying η e (t) L 2 (4.16) It remains to prove that Φ can be a contraction mapping on the closed unit ball of S λ,t 0 . As before, we'll only give estimates for Φ to be a self-mapping. Suppose η ∈ S λ,t 0 with η S λ,t 0 ≤ 1, i.e. η S(t) ≤ e −λt for t ≥ t 0 . To estimate Φη S λ,t 0 from the Strichartz estimate (4.15), we have to estimate G N (t) , H 1 N (t) and H 2 N (t) . Since · N (t) ≤ · L 1 t L 2 x , we'll frequently just estimate · L 1 t L 2 x . Also repeatedly used is the fact η L 1 t L 2 x (t) ≤ λ −1 e −λt , obtained from (4.2) (or Corollary 4.3). Part 1. Estimate of G N (t) . We have For the first term, we have Notice that for the endpoint case α i = 4/d, the smallness of the coefficient (obtained by letting λ large) have to be provided by e −α i λt 0 . This is the reason we consider an initial time t 0 > 0. By (4.17) and (4.18) we get the needed estimate of G N (t) . H 1 N (t) . By Corollary 2.5,
We first prove the exponential decay of its L 2 x norm by interpolation.
Step 1. For s ∈ (0, ∞] and θ ∈ [0, 1], We need s/θ ∈ C , that is or equivalently We hope this can be satisfied by some s < 2, by choosing a suitable θ. A little computation shows that the minimum of the "max" is achieved by letting θ = min d e(1 + γ) , 1 . (4.21) Precisely we have the following alternatives: It's straightforward to check that, for all (e, d, α 1 ) satisfying our assumptions, the above lower bound of s is less than 2. (Here we use d − e < 4 again.) Thus, if θ is given by (4.21), there exists 0 < s 1 < 2 such that (4.20) holds with s = s 1 .
Step 2. We have We also have where recall that v ′ j = (v j,1 , . . . , v j,e ) consists of the first e components of v j . Note that for any c 1 , c 2 > 0 and w 1 , w 2 ∈ R n , Thus Taking this into (4.22), we get The number j ω γ α 1 j can be controlled as A d;p as described in Lemma 3.6. For preciseness, we can fix any p 1 ∈ C (d) Thus (4.23) gives From (4.20) (with θ given by (4.21) and s = s 1 < 2) and (4.24), we get We omit the expression of E 2 , which is obvious while cumbersome. Suppose Part 3. Estimate of H 2 N (t) . Choose s 2 ∈ ( dα 1 2(α 1 +1) , 2) (it's easy to check that the interval is nonempty). Then Lemma 3.4 (H0) implies we get From the conclusions in Part 1, Part 2, and Part 3, we are done.
Remark 4.6. Without using the anisotropic estimates for T d , our assertions will be much weaker. For example, consider (4.19) in Part 2-1. If we do not use an anisotropic estimate of T d , we can only estimate as follows: For any (q, r) ∈ A If d ≥ 4, we have γ = min(1, α 1 ) = α 1 (since α i ≤ 4/d), and (4.27) is impossible. Thus only d ≤ 3 is allowed. Moreover, even for d ≤ 3, if γ = 1, the endpoint case α 1 = α 2 = 4/d is excluded.
When 1 ≤ α 1 < 2, Theorem 3.9 implies the existence of an 1D train T 1 + η 1 such that η 1 (t) W 1,∞ x has exponential decay. This allows us to use the gradient estimate when e = 1. Precisely, we can try to construct a mixed train of the form T 1 + η 1 + T d + η (d > 1), by assuming the exponential decay of ∇η S(t) (besides η S(t) ). It turns out that we can do it only for d = 2, and under a further restriction on α 1 . The result is not only of its own interest, but also makes it possible to realize the 1D-2D-3D trains in the next section.
Proof. We will assume ρ = 2 for simplicity. For λ no less than some positive number, Theorem 3.9 implies the existence of an 1D train T 1 + η 1 satisfying In the following, we denote S λ,0 (i.e. the initial time t 0 = 0) by S λ , and let X be the Banach space of all η : [0, ∞) × R 2 → C such that We'll give estimates for Φ to be a self-mapping on the closed unit ball of X.
Suppose η ∈ X with η X ≤ 1. The estimate of Φη S λ is the same as in the proof of Theorem 4.5, except for |η| α 2 +1 N (t) . Since the value of α 2 is not restricted, we use Lemma 4.4 (N1) instead of (N0) to obtain N (t) , both (N0) and (N1) work.) We remark that here and later we use µ as a generic constant, whose value may be different in different places. Now we estimate ∇Φη S(t) . From (4.15), we have to estimate ∇G N (t) , ∇H 1 N (t) , and ∇H 2 N (t) .
We need (∞, 2) ∈ C . This is true by α 1 < 4/3. Notice that if we do not use an anisotropic estimate for ∇T 2 , the requirement becomes α 1 < 1, and the construction fails since we assume α 1 ≥ 1. Moreover, it is also due to this part that the construction is valid only for d = 2. Indeed, suppose d ≥ 3, with coordinates x = (x 1 , x ′′ ). If for some admissible (a ′ , b ′ ), Estimate (2).

By interpolation we get
where p, q are arbitrary numbers in (0, ∞] satisfying 1 q + 1 p = 1 s 3 . Since we can choose p, q such that 1 q < 1 and 1 A and p ∈ C then we get Combining all three parts, we get for some µ > 0.
(4.35) can be visualized as a plane-line-point soliton train in 3D space. It turns out to be the only mixed trains involving more than two dimensions that we can construct. To see this, we first give a discussion on the control of lower dimensional errors.
As we stressed, supremum controls in x for lower dimensional objects are necessary in constructing mixed trains. For the previous theorems on eD-dD trains, we use controls of the form η e (t) L ∞ x ≤ e −λt (4.37) established in Section 3. In fact, it is also possible to use space-time controls of the form for suitable p. In 1D space, since (4, ∞) ∈ A (1) , we can obtain η 1 L 4 t L ∞ x (t) control by constructing T 1 + η 1 such that η 1 S(t) has exponential decay in t. For e = 2, 3, since r (e) max > e (recall (4.1)), (4.38) can be obtained from the exponential decay of ∇η e S(t) and some η e L q t L 2 x (t) (e.g. (1.10)) by Gagliardo-Nirenberg's inequality. For e ≥ 4, (4.38) is not available (unless controls of even higher order derivatives of η e are considered, which we did not pursue).
There is actually no definite reason we followed a route of using (4.37) but not (4.38) in constructing eD-dD trains. As to mixed trains involving more than two dimensions, all the lower dimensional errors have to have spatial supremum controls. As a consequence, thanks to Theorem 4.7, one sees that (4.35) becomes the only possible case, where we have type (4.37) control of η 1 and type (4.38) control of η 2 . The details will be given in the proof of Theorem 4.8.
By interpolation we get the L 2 x estimate. And if a(1 − s 2 /2)v * ≥ λ, we get A . Suppose a(1 − s 3 /2)v * ≥ λ, we get Combining all three parts, we see Φη S(t) ≤ e −λt for λ large enough with suitable frequencies and velocities of the solitons.

Appendix B
Let x = (x ′ , x ′′ ) be as in Section 4.2. One would wonder if the L p x ′ L q x ′′ norm can be bounded by the L p x ∩ L q x norm. This is in general not the case. Consider a function u : R 2 → R of the form u(x, y) = 1 0<x<1 |x| ma ψ(|x| a y), where m, a are real parameters, ψ ∈ C ∞ c (R). Then for p, q ∈ (0, ∞) we have u L p xy = ψ L p ( Suppose p > q. Then if 0 < m < 1/q, there exists a > 0 such that which implies u ∈ L p xy ∩ L q xy but u L p x L q y = ∞.