The global Cauchy problem for the NLS with higher order anisotropic dispersion

We use a method developed by Strauss to obtain global wellposedness results in the mild sense for the small data Cauchy problem in modulation spaces $M_{p,q}^s(\mathbb{R}^d)$, where $q=1$ and $s\geq0$ or $q\in(1,\infty]$ and $s>\frac{d}{q'}$ for a nonlinear Schr\"odinger equation with higher order anisotropic dispersion and algebraic nonlinearities.


introduction and main results
We are interested in the following Cauchy problem where (t, x) = (t, Such PDE arise in the context of high-speed soliton transmission in long-haul optical communication system, see [5]. The case where the coefficiets α, β, γ are time dependent has been studied in [2] in one dimension for the cubic nonlinearity, f (u) = |u| 2 u, with initial data in L 2 (R)-based Sobolev spaces. In [1] it is proved that (1) with nonlinearity f (u) = |u| p u where is globally wellposed in L 2 (R d ) via Strichartz estimates and the charge conservation equation In the same paper the case of initial data is equipped with the norm In this paper we consider the Cauchy problem (1) with initial data u 0 in modulation spaces M s p,q (R d ). Modulation spaces were introduced by Feichtinger in [6] and since then, they have become canonical for both time-frequency and phase-space analysis. They provide an excellent substitute for estimates that are known to fail on Lebesgue spaces. To state the definition of a modulation space we need to fix some notation. We will denote by S ′ (R d ) the space of tempered distributions. Let Q 0 be the unit cube with center the origin in R d and its translations Q k := Q 0 + k for all k ∈ Z d . Consider a partition of unity , and define the isometric decomposition operators where F denotes the Fourier transform in R d . Then the norm of a tempered distribution where we denote by k = 1 + |k| the Japanese bracket. It can be proved that different choices of the function σ 0 lead to equivalent norms in M s p,q (R d ) (see e.g. [3, Proposition 2.9]). When s = 0 we denote the space M 0 p,q (R d ) by M p,q (R d ). In the special case where p = q = 2 we have M s 2,2 (R d ) = H s (R d ) the usual Sobolev spaces. For α ∈ R we define the weighted mixed-norm space Let us denote by π(u m+1 ) any (m + 1)-time product of u andū, where m ∈ Z + . Define also the quantity The main results are the following theorems.
For q = 1, let s ≥ 0 and for q > 1, let s > d q ′ . Then there exists a δ > 0 such that for The restriction on the power of the nonlinearity described in Theorem 1 is explained in remark 9.
There exists δ > 0 such that for any Remark 3. For q < ∞, the solution from Theorem 1 and 2 is a continuous function with values in the corresponding modulation space, i.e. indeed a mild solution. For the more delicate situation q = ∞ see [9].
The idea of studying the Cauchy problem (1) with such time-decay norm is inspired by [13], where the authors considered the NLS and the NLKG equations. As mentioned there, this idea goes back to the work of Strauss, see [11]. Their results were improved in [7] and [8] where the author considered the nonlinear higher order Schrödinger equation where φ( √ −∆) = F −1 φ(|ξ|)F and φ is a polynomial, with initial data u 0 in a modulation space. , if γ = 0. In both cases m 0 > 3.
Remark 5. In [13, Theorem 1.1 and Theorem 1.2], the authors only considered modulation spaces M s p,q (R d ) with q = 1. But, by Theorem 6, their crucial estimate (6.6) also holds for q ∈ (1, ∞] and s > d q ′ . Hence, the statements of their theorems is true in this case too. 1.1. Preliminaries. It is known that for s > d/q ′ (where q ′ is the conjugate exponent of q) and p, q ∈ [1, ∞], the embedding is continuous. The same is true for the embedding which holds for any s 1 , s 2 ∈ R and any p 1 , p 2 , q 1 , q 2 ∈ [1, ∞] satisfying p 1 ≤ p 2 and either q 1 ≤ q 2 and s 1 ≥ s 2 or q 2 < q 1 and s 1 > s 2 + d q 2 − d q 1 (see [6, Proposition 6.8 and Proposition 6.5]).
We are going to use the following Hölder type inequality for modulation spaces which appeared in [3,Theorem 4.3] (see also [4]).
Theorem 6. Let d ≥ 1 and 1 ≤ p, p 1 , p 2 , q ≤ ∞ such that 1 . The propagator of the homogeneous Schrödigner equation with higher order anisotropic dispersion is given by For the rest of the paper, A B shall mean that there is a constant C > 0 such that A ≤ CB. The next dispersive estimate is from [1, Theorem 1.1]: Using this, we claim the following Theorem 8. Consider s ∈ R, p ∈ [2, ∞] and q ∈ [1, ∞]. Then is as in Equation (12) and the implicit constant is independent of the function f and the time t.
Proof. The operators k and W (t) commute and hence we immediately arrive at by invoking Theorem 7. Moreover, as p ∈ [2, ∞], we have for any k ∈ Z d and any t ∈ R. Above, we used the Hausdoff-Young inequality for the first and last estimate and the fact that supp(σ k ) ⊆ B √ d (k) for the second inequality. Taking the minimum of the right-hand sides of (14) and (15) shows Multiplying by the weight k s and taking the l q (Z d )-norm on both sides we arrive at the desired estimate.

proofs of the main theorems
In this section we present the proofs of the main theorems.
Proof of Theorem 1. For the sake of brevity, let us shorten the notation by setting By the Banach fixed-point theorem, it suffices to show that the operator defined by is a contractive self-mapping of the complete metric space for some R ∈ R + . We begin with the self-mapping property and observe that Notice, that µ(d, γ, m + 2) = 2 γ m,d and hence, by the dispersive estimate (13), one obtains  Hölder's inequality for modulation spaces from Theorem 6 is applicable (due to the assumptions on s, q) and yields