Modified wave operators for the defocusing cubic nonlinear Schrödinger equation in one space dimension with large scattering data

1. Problem Statement

The paper addresses the scattering theory for the one-dimensional defocusing cubic Nonlinear Schrödinger Equation (NLS) with a subcritical power perturbation:
$$i\partial_t u - H_0 u = \lambda_1 |u|^2 u + \lambda_2 |u|^{2\sigma} u, \quad x \in \mathbb{R}, t \in \mathbb{R}$$
where $H_0 = -\frac{1}{2}\partial_x^2$, $\lambda_1 > 0$ (defocusing), $\lambda_2 \in \mathbb{R}$, and $1 < \sigma < 2$.

The Core Challenge:
In 1D, the cubic nonlinearity ($|u|^2u$) is long-range. Standard scattering theory fails because solutions do not asymptotically approach free solutions ($e^{-itH_0}u_\pm$). Instead, they require modified asymptotic profiles involving a logarithmic phase correction.

• Previous Limitations: Existing results for modified scattering (both Cauchy and Final State Problems) were largely restricted to small data or relied on complete integrability (inverse scattering) or analytic function spaces.
• Goal: The authors aim to construct modified wave operators for arbitrarily large scattering data in the defocusing case ($\lambda_1 > 0$) without relying on integrability or analytic spaces. They also provide an improved explicit bound for the focusing case ($\lambda_1 < 0$).

2. Methodology

The authors introduce a novel formulation based on linearization around a prescribed asymptotic profile and a modified energy method.

A. Transformation and Linearization

1. Pseudo-Conformal Transform: The problem is transformed using $v(t, x) = T[M(t)D(t)]^{-1}u(t, x)$, where $M(t)$ and $D(t)$ are the mass and dilation operators. This maps the long-time behavior ($t \to \infty$) to the short-time behavior ($t \to 0$).
2. Asymptotic Profiles: They define a profile $v_p$ (and a refined version $\tilde{v}_p$) that captures the leading-order nonlinear phase corrections.
3. Error Equation: Let $v^* = v - v_p$. The equation for $v^*$ is derived. Crucially, the authors perform a Taylor expansion of the nonlinear term $|v|^{2\sigma}v - |v_p|^{2\sigma}v_p$ to extract the first-order term (linear in $v^*$).
4. Linearized System: Instead of treating the first-order term as part of the nonlinearity (which causes divergence for large data), they absorb it into the linear operator. This results in a system of Schrödinger equations with a non-symmetric, time-dependent, long-range potential $V(t, x)$:
   $$(i\partial_t - H(t)) \Psi = \text{Nonlinear Remainder} + \text{Error Terms}$$
   where $H(t) = H_0 + V(t, x)$ and $V(t, x) = O(t^{-1})$.

B. Modified Energy Estimate

The central technical innovation is the construction of a modified energy functional $Q_\alpha$ tailored to the linearized system.

• The Functional:
  $$Q_\alpha[\psi](t) = \frac{1}{4}\|\partial_x \psi(t)\|^2 + t^{-\alpha}\|\psi(t)\|^2 + \lambda_1 t^{-1} \||\phi|^{\sigma-1} \text{Re}[v_p(t)\psi(t)]\|^2$$
  (Note: The last term is crucial; it is positive definite when $\lambda_1 > 0$).
• The Identity: They derive a modified energy identity (Lemma 2.3) for the time derivative of a related functional $\tilde{Q}_\alpha$.
  $$\frac{d}{dt}\tilde{Q}_\alpha[\psi] = -\alpha t^{-\alpha-1}\|\psi\|^2 - \sum (\text{positive terms}) + \text{lower order error terms}$$
• The Estimate: By carefully estimating the error terms using Gagliardo-Nirenberg inequalities and Young's inequality, they prove that for sufficiently small $t$ (large physical time):
  $$Q_\alpha[U(t, s)\psi_0](t) \lesssim Q_\alpha[\psi_0](s)$$
  This establishes that the propagator $U(t, s)$ of the non-symmetric linearized system satisfies a uniform energy bound, despite the lack of symmetry in $H(t)$.

C. Fixed Point Argument

Using the energy estimate, they set up a contraction mapping in a weighted Sobolev space $X(T, \alpha, \beta, M)$. The faster decay of the "new" nonlinear remainder (due to the extraction of the linear term) allows the integral equation to be solved via a standard energy method, even for large data.

3. Key Contributions

1. Large Data Modified Scattering: Proves the existence of modified wave operators for the defocusing 1D cubic NLS with arbitrarily large scattering data in non-analytic function spaces ($H^{1+\epsilon}$). This is the first result of its kind for the Final State Problem (FSP) in this setting without integrability.
2. New Linearization Strategy: Introduces a formulation where the long-range linear part of the nonlinearity is treated as a time-dependent potential in the linear operator, rather than a perturbation. This avoids the "small data" obstruction inherent in standard contraction arguments for long-range problems.
3. Modified Energy Identity: Develops a specific energy identity for a system of Schrödinger equations with non-symmetric, long-range potentials. This allows the control of the solution without relying on the $L^2$-norm preservation (which fails for non-symmetric operators).
4. Focusing Case Improvement: For the focusing case ($\lambda_1 < 0$), they derive an explicit upper bound on the size of the scattering data ($|\lambda_1|\|\phi\|_{L^\infty}^2 < 1$) that guarantees existence, improving upon previous regularity and smallness conditions found in literature (e.g., Ozawa's work).

4. Main Results

• Theorem 1.2 (Defocusing Case): For $\lambda_1 > 0$, $\lambda_2 \in \mathbb{R}$, and $\phi \in H^{1+\epsilon}(\mathbb{R})$, there exists a unique global solution $u$ to the NLS that scatters to the prescribed modified profile $u_p$ as $t \to \infty$. The modified wave operator $W_+: \phi \mapsto u(0)$ is well-defined from $H^{1+\epsilon}$ to $H^1$.
• Theorem 1.4 (Focusing Case): For $\lambda_1 < 0$, if the scattering data satisfies $|\lambda_1|\|\phi\|_{L^\infty}^2 < 1$, a unique global solution exists with similar scattering properties.
• Uniqueness: The solutions are unique within the class of functions satisfying specific decay rates ($t^{-\beta}$) in the weighted norms.

5. Significance and Implications

• Beyond Integrability: The result demonstrates that large-data modified scattering can be achieved for non-integrable nonlinear dispersive equations using purely PDE-based energy methods, breaking the reliance on inverse scattering theory.
• Methodological Shift: The technique of "renormalizing" the non-integrable part of the nonlinearity into the linear operator (treating it as a potential) is a powerful new tool. It contrasts with methods for Hartree equations which rely on the smoothing properties of convolution; this paper shows such smoothing is not necessary if the energy structure is correctly modified.
• Future Directions: The authors suggest this method could be extended to other 1D long-range problems, systems (like Manakov systems), or potentially higher dimensions if a higher-order counterpart to the modified energy can be found.

In summary, this paper resolves a long-standing open problem regarding large-data scattering for the 1D cubic NLS by introducing a robust energy method that linearizes the problem around a modified profile, effectively neutralizing the long-range obstruction through a carefully constructed energy functional.