In this paper we transfer a small data global existence and scattering result by Wang and Hudzik to the more general case of modulation spaces M-p,q(s)(R-d) where q = 1 and s = 0 or q ? (1, 8] and s > (d)/(q') and to the non-linear Schrodinger equation with higher-order anisotropic dispersion.

We use a method developed by Strauss to obtain global well-posedness results in the mild sense and existence of asymptotic states for the small data Cauchy problem in modulation spaces M-p,q(s)(R-d), where q = 1 and s >= 0 or q is an element of (1, infinity] and s> d/q' for a nonlinear Schrodinger equation with higher order anisotropic dispersion and algebraic nonlinearities.

We study the one dimensional nonlinear Schrodinger equation with power nonlinearity |u|(alpha-1) u for alpha is an element of [1, 5] and initial data u(0) is an element of H-1(T) + L-2(R). We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity (alpha = 2) we obtain global well-posedness in the space C(R, H-1(T)+L-2 (R)) via Gronwall's inequality.

We prove global wellposedness of the Klein-Gordon equation with power nonlinearity $|u|^{\alpha-1}u$, where $\alpha\in\left[1,\frac{d}{d-2}\right]$, in dimension $d\geq3$ with initial data in $M_{p, p'}^{1}(\mathbb{R}^d)\times M_{p,p'}(\mathbb{R}^d)$ for $p$ sufficiently close to $2$. The proof is an application of the high-low method described by Bourgain [1] where the Klein-Gordon equation is studied in one dimension with cubic nonlinearity for initial data in Sobolev spaces.

We study the one dimensional nonlinear Schrodinger equation with power nonlinearity $$\\left| u \\right| ^{\\alpha - 1} u$$\n for $$\\alpha \\in [1,5]$$\n and initial data $$u_0 \\in H^1({{\\mathbb {T}}}) + L^2({{\\mathbb {R}}})$$\n . We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity (\n $$\\alpha = 2$$\n ) we obtain global well-posedness in the space $$C({{\\mathbb {R}}}, H^1({{\\mathbb {T}}}) + L^2({{\\mathbb {R}}}))$$\n via Gronwall’s inequality.

We study the one dimensional nonlinear Schrödinger equation with power nonlinearity |u| u for α ∈ [2, 5] and initial data u0 ∈ L(R) + H(T). We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity (α = 2) we obtain unconditional global well-posedness in the space C(R, L(R)+H(T)) via Gronwall’s inequality.

where λ ∈ C and ρ > 0. Such PDEs arise in the context of high-speed soliton transmission in long-haul optical communication systems and were introduced by Karpman, see [Kar96], [DC08] and [KH94]. The case where the coefficients α, β, γ are time-dependent has been studied in [CPS15] in one dimension for the cubic nonlinearity, f(u) = |u| u with initial data in L2(R)-based Sobolev spaces. Before we state our results, we need to recall some concepts and make some definitions. The initial data u0 in our case comes from amodulation space. Modulation spaces were introduced by Feichtinger in [Fei83] (see also [Grö01] or [WH07] for a gentle introduction). Let Q0 := ( −12 , 1 2 ]d and Qk := k+Q0 for k ∈ Z . Consider any smooth partition of unity (σk) ∈ C ∞(Rd)Z d which is adapted

The original version of the chapter was inadvertently published with an error in the abstract. On line 6 of the abstract, the name ‘Okoudjou’ was misspelled as ‘Oboudjou’. The chapter has now been corrected and approved by the author.

We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, implicitly contained in [STW11], of the intersection Ms p,q(R) ∩M∞,1(R) for d ∈ N, p, q ∈ [1,∞] and s ≥ 0. We employ this algebra property to show the local wellposedness of the Cauchy problem for the cubic nonlinear Schrödinger equation in the above intersection. This improves [BO09, Theorem 1.1] by Bényi and Okoudjou, where only the case q = 1 is considered, and closes a gap in the literature. If q > 1 and s > d ( 1− 1 q ) or if q = 1 and s ≥ 0 then Ms p,q(R) ↪→ M∞,1(R) and the above intersection is superfluous. For this case we also obtain a new Hölder-type inequality for modulation spaces.

We introduce a Littlewood–Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, somehow implicitly contained in Sugimoto et al. (2011), of the intersection \(M^s_{p,q}({\mathbb {R}}^d) \cap M_{\infty , 1}({\mathbb {R}}^d)\) for \(d \in {\mathbb {N}}\), p, q ∈ [1, ∞] and s ≥ 0. We employ this algebra property to show the local well-posedness of the Cauchy problem for the cubic nonlinear Schrödinger equation in the above intersection. This improves a theorem by Bényi and Okoudjou (2009), where only the case q = 1 is considered, and closes a gap in the literature. If q > 1 and \(s > d \left (1 - \frac {1}{q}\right )\) or if q = 1 and s ≥ 0 then \(M^s_{p,q}({\mathbb {R}}^d) \hookrightarrow M_{\infty , 1}({\mathbb {R}}^d)\) and the above intersection is superfluous. For this case we also reobtain a Hölder-type inequality for modulation spaces.

We study the one dimensional nonlinear Schrödinger equation with power nonlinearity |u|α−1 u for α ∈ [1, 5] and initial data u0 ∈ L2(R) + H1(T). We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity (α = 2) we obtain global well-posedness in the space C(R, L2(R) +H1(T)) via Gronwall’s inequality.

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in the modulation space $$M_{p,q}^{s}({\mathbb {R}})$$ where $$1\le q\le 2$$ , $$2\le p<\frac{10q'}{q'+6}$$ and $$s\ge 0$$ . Moreover, for either $$1\le q\le \frac{3}{2}, s\ge 0$$ and $$2\le p\le 3$$ or $$\frac{3}{2}"\frac{2}{3}-\frac{1}{q}$$ and $$2\le p\le 3$$ or $$\frac{18}{11}""\frac{2}{3}-\frac{1}{q}$$ and $$2\le p<\frac{10q'}{q'+6}$$ we show that the Cauchy problem is unconditionally wellposed in $$M_{p,q}^{s}({\mathbb R}).$$ This improves Pattakos (J Fourier Anal Appl, 2018. https://doi.org/10.1007/s00041-018-09655-9 ), where the case $$p=2$$ was considered and the differentiation-by-parts technique was introduced to a problem with continuous Fourier variable. Here, the same technique is used, but more delicate estimates are necessary for $$p\ne 2$$ ."

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrodinger equation in one dimension with initial data mu(0) in H-s1(R) + H-s2(T), 0 <= s(1) <= s(2). In addition, we show that if mu(0) is an element of H-s(R) + H1/2+epsilon(T) Where epsilon >0 and 1/6 <= s <= 1/2 the solution is unique in H-s (R) + H1/2+epsilon(T). Our main tool is a normal form type reduction via the use of the differentiation by parts technique.

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in one dimension with initial data u_0 in H^s_1(ℝ)+H^s_2(𝕋), 0≤ s_1≤ s_2. In addition, we show that if u_0∈ H^s(ℝ)+H^1/2+ϵ(𝕋) where ϵ>0 and 1/6≤ s≤1/2 the solution is unique in H^s(ℝ)+H^1/2+ϵ(𝕋). Our main tool is a normal form type reduction via the use of the differentiation by parts technique.

We prove global existence for the one-dimensional cubic non-linear Schrödinger equation in modulation spaces $M_{p,p'}$ for $p$ sufficiently close to $2$. In contrast to known results, our result requires no smallness condition on initial data. The proof adapts a splitting method inspired by work of Vargas-Vega and Hyakuna-Tsutsumi to the modulation space setting and exploits polynomial growth of the free Schrödinger group on modulation spaces.

We show the local well-posedness of the Cauchy problem for the cubic nonlinear Schrödinger equation on modulation spaces M p,q(R) for d ∈ N, 1 ≤ p, q ≤ ∞ and s > d ( 1− 1q ) for q > 1 or s ≥ 0 for q = 1. This improves [4, Theorem 1.1] by Bényi and Okoudjou where only the case q = 1 is considered. Our result is based on the algebra property of modulation spaces with indices as above for which we give an elementary proof via a new Hölder-like inequality for modulation spaces.