1.1 The focusing cubic Schrödinger equation on the line

We consider the focusing cubic Schrödinger equation in one space dimension

(1.1) $$ \begin{align} \left\{ \begin{aligned} i\partial_t \psi + \partial_x^2 \psi + |\psi|^2 \psi &= 0, \quad (t,x) \in \mathbb R \times \mathbb R, \\ \psi(0) &= \psi_0. \end{aligned} \right. \end{align} $$

The Cauchy problem for (1.1) is globally well-posed in $L^2(\mathbb R)$ , see, for example, [Reference Tsutsumi83]. The focusing cubic Schrödinger equation (1.1) is a prime example of a completely integrable nonlinear dispersive equation on the line in the sense that it admits a Lax pair formulation as well as Bäcklund transformations.

We recall that (1.1) is invariant with respect to the scaling

$$ \begin{align*} \psi(t,x) \mapsto \lambda \psi(\lambda^2 t, \lambda x), \quad \lambda> 0. \end{align*} $$

Moreover, (1.1) is invariant under Galilei boosts, phase shifts, and spatial translations, which means that if $\psi (t,x)$ is a solution to (1.1), then for any $p, \gamma , \sigma \in \mathbb R$ the function

$$ \begin{align*} e^{i p x} e^{-i p^2 t} e^{i \gamma} \psi(t, x - 2pt -\sigma) \end{align*} $$

is also a solution to (1.1).

Two important features of the dynamics of the focusing cubic Schrödinger equation on the line are the modified scattering behavior of dispersive solutions to (1.1) and the existence of solitary wave solutions to (1.1).

The modified scattering behavior is related to the weak dispersion in one space dimension. More specifically, solutions to the linear Schrödinger equation on the line only decay in $L^\infty (\mathbb R)$ at the slow rate $t^{-\frac 12}$ as $t \to \infty $ . The cubic nonlinearity is therefore long-range and the asymptotics of dispersive solutions to (1.1) are bound to feature corrections with respect to the free flow.

An example of a solitary wave solution to (1.1) is given by

(1.2) $$ \begin{align} e^{it} \phi(x), \quad \phi(x) := \sqrt{2} \operatorname{\mathrm{sech}}(x). \end{align} $$

The invariance of (1.1) under scaling, Galilei boosts, phase shifts, and spatial translations then generates the following 4-parameter family of solitary waves

(1.3) $$ \begin{align} e^{i p x} e^{i (\omega - p^2) t} e^{i\gamma} \phi_\omega(x-2pt-\sigma), \quad (\omega, p, \gamma, \sigma) \in (0,\infty) \times \mathbb R^3, \end{align} $$

where we use the notation

$$ \begin{align*} \phi_\omega(x) := \sqrt{\omega} \phi(\sqrt{\omega} x) = \sqrt{2\omega} \operatorname{\mathrm{sech}}(\sqrt{\omega}x) \end{align*} $$

for the unique positive even ground state solution to the equation

(1.4) $$ \begin{align} (-\partial_x^2 + \omega) \phi_\omega = \phi_\omega^3, \quad x \in \mathbb R. \end{align} $$

A fundamental question concerning the family of solitary waves (1.3) is their stability under the flow of (1.1) with respect to small perturbations. Orbital stability in $H^1(\mathbb R)$ follows from the classical works [Reference Cazenave and Lions6, Reference Shatah and Strauss78, Reference Weinstein85, Reference Weinstein86, Reference Grillakis, Shatah and Strauss30]. Exploiting the complete integrability of (1.1), one can in fact show that orbital stability holds with respect to weaker topologies, see for instance [Reference Mizumachi and Pelinovsky67, Reference Koch and Tataru41]. The asymptotic stability of the family of solitary waves (1.3) with respect to perturbations that are small in weighted Sobolev norms was established in [Reference Cuccagna and Pelinovsky19, Reference Saalmann76, Reference Borghese, Jenkins and McLaughlin1] using complete integrability techniques. For perturbations in the energy space, asymptotic stability of (1.3) cannot hold since the family of 2-solitons constructed by the inverse scattering transform in [Reference Olmedilla71, Reference Zakharov and Shabat87] provides counter-examples, as pointed out in [Reference Martel62].

1.2 Main result

In this work we study the asymptotic stability problem for the family of solitary waves (1.3) using perturbative techniques that do not rely on the complete integrability of the focusing cubic Schrödinger equation on the line. Specifically, we combine modulation techniques with a space-time resonances approach based on the distorted Fourier transform. For technical reasons we restrict to even solutions to (1.1). Our main result establishes the asymptotic stability of the 2-parameter family of (even) solitary waves

(1.5) $$ \begin{align} e^{i \omega t} e^{i \gamma} \phi_\omega(x), \quad (\omega, \gamma) \in (0,\infty) \times \mathbb R, \end{align} $$

with respect to even perturbations that are small in a weighted Sobolev norm.

A common difficulty of asymptotic stability problems for solitary wave solutions to nonlinear Schrödinger equations on the line with low power nonlinearities is the weak dispersion in one space dimension. In the case of the focusing cubic Schrödinger equation (1.1), an additional significant challenge is the slow local decay caused by the threshold resonances of the linearized operator around the solitary waves (1.5). We uncover that in the case of the focusing cubic Schrödinger equation the worst effects of the threshold resonances are suppressed by two remarkable null structures of the quadratic nonlinearities. Beyond the intrinsic interest in this problem, we also view it as a valuable and instructive step to further develop perturbative techniques to approach related asymptotic stability problems for solitary waves of (nonintegrable) Schrödinger models on the line with threshold resonances or internal modes, where much remains to be understood in the absence of favorable structures.

The following is the main result of this work.

Theorem 1.1. For any $\omega _0 \in (0,\infty )$ there exist constants $0 < \varepsilon _0 \ll 1$ , $0 < \delta \ll 1$ , and $C_0 \geq 1$ with the following property: Let $\gamma _0 \in \mathbb R$ and let $u_0 \in H^1_x(\mathbb R) \cap L^{2,1}_x(\mathbb R)$ be even with

(1.6) $$ \begin{align} \varepsilon := \| u_0 \|_{H^1_x(\mathbb R)} + \|\langle x\rangle u_0\|_{L^2_x(\mathbb R)} \leq \varepsilon_0. \end{align} $$

Then the $H^1_x \cap L^{2,1}_x$ –solution $\psi (t,x)$ to (1.1) with initial condition

(1.7) $$ \begin{align} \psi_0(x) = e^{i \gamma_0} \big( \phi_{\omega_0}(x) + u_0(x) \big) \end{align} $$

exists globally in time and admits a decomposition

$$ \begin{align*} \psi(t,x) = e^{i \gamma(t)} \big( \phi_{\omega(t)}(x) + u(t,x) \big), \quad t \geq 0, \end{align*} $$

for some continuously differentiable paths $(\omega , \gamma ) \colon [0,\infty ) \to (0,\infty ) \times \mathbb R$ so that

(1.8) $$ \begin{align} \|u(t)\|_{L^{\infty}_x(\mathbb R)} \leq C_0\varepsilon \langle t\rangle^{-\frac12}, \quad t \geq 0. \end{align} $$

Moreover, there exists $\omega _\infty \in (0,\infty )$ such that

(1.9) $$ \begin{align} |\omega(t)-\omega_\infty| \leq C_0 \varepsilon \langle t\rangle^{-1+\delta}, \quad |\dot{\gamma}(t)-\omega_\infty| \leq C_0 \varepsilon \langle t\rangle^{-1+\delta}, \quad t \geq 0. \end{align} $$

The asymptotics of the radiation term $u(t,x)$ exhibit logarithmic phase corrections with respect to the free flow: there exist $W_-,W_+ \in L^\infty (\mathbb R)$ such that uniformly for all $x\in \mathbb R$ and all $t \geq 1$ ,

(1.10) $$ \begin{align} \big|u(t,x) - u_\infty(t,x)\big| \leq C_0 \varepsilon t^{-\frac35+\delta}, \end{align} $$

where

(1.11) $$ \begin{align} \begin{aligned} u_\infty(t,x) &:= \frac{1}{\sqrt{2t}} e^{-it\omega_\infty}e^{i \frac{x^2}{4t}} e^{-i \frac{\pi}{4}}m_{1,\omega_\infty}\Big(x,\frac{x}{2t}\Big) W_+\Big(\frac{x}{2t}\Big)e^{-i\theta_\infty(t)}e^{{\frac{i}{2}\log(t)\vert W_+(\frac{x}{2t})}\vert^2}\\ &\quad \, \, -\frac{1}{\sqrt{2t}} e^{it\omega_\infty}e^{-i \frac{x^2}{4t}} e^{i \frac{\pi}{4}}m_{2,\omega_\infty}\Big(x,-\frac{x}{2t}\Big)W_-\Big(-\frac{x}{2t}\Big)e^{i\theta_\infty(t)}e^{{-\frac{i}{2}\log(t)\vert W_-(-\frac{x}{2t})}\vert^2} \end{aligned} \end{align} $$

and

$$ \begin{align*} \theta_\infty(t) &:= \int_0^t (\dot{\gamma}(s)-\omega_\infty) \,\mathrm{d} s, \\ m_{1,\omega_\infty}(x,\xi) &:= \frac{\big( \xi + i \sqrt{\omega_\infty} \tanh(\sqrt{\omega_\infty} x) \big)^2}{\big(|\xi|-i\sqrt{\omega_\infty}\big)^2}, \\ m_{2,\omega_\infty}(x,\xi) &:= \frac{\big( \sqrt{\omega_\infty} \operatorname{\mathrm{sech}}(\sqrt{\omega_\infty} x) \big)^2}{\big( |\xi|-i\sqrt{\omega_\infty} \big)^2}. \end{align*} $$

Analogous statements hold for negative times $t \leq 0$ .

1.3 References

The literature on the asymptotic stability of solitary wave solutions to nonlinear Schrödinger equations and related models is vast. Recently, this topic has received significant attention especially in one space dimension, where the combination of weak dispersion, low power nonlinearities, and intriguing spectral features leads to a rich class of dynamics.

For problems in one space dimension, it has become customary to distinguish two notions of asymptotic stability, which complement each other. Local asymptotic stability refers to convergence of the perturbed solution locally in space to a final soliton, usually under weak assumptions such as small finite energy perturbations. Full (or detailed) asymptotic stability includes decay rates and asymptotics for the convergence to the final soliton at the expense of stronger assumptions on the initial perturbations such as smallness with respect to weighted Sobolev norms.

The analysis in this paper combines a circle of ideas and techniques related to modified scattering, modulation, and distorted Fourier theory. In what follows, we point the reader to the most relevant related articles, mostly focusing on problems in one space dimension. We refer to the surveys [Reference Cuccagna and Maeda13, Reference Kowalczyk, Martel and Muñoz46, Reference Schlag77, Reference Tao82] for a more exhaustive list of references, especially for results in higher space dimensions and for related models.

The modified scattering of small solutions to cubic Schrödinger equations on the line with or without potential has been studied in [Reference Hayashi and Naumkin31, Reference Lindblad and Soffer58, Reference Kato and Pusateri39, Reference Ifrim and Tataru36, Reference Germain, Pusateri and Rousset25, Reference Chen and Pusateri9, Reference Chen and Pusateri10, Reference Delort20, Reference Naumkin68, Reference Naumkin69, Reference Masaki, Murphy and Segata64, Reference Naumkin and Weder70]. For related results on the long-time behavior of small solutions to quadratic or cubic Klein-Gordon equations in one space dimension we refer to [Reference Delort21, Reference Lindblad and Soffer57, Reference Lindblad and Soffer56, Reference Hayashi and Naumkin32, Reference Hayashi and Naumkin33, Reference Hayashi and Naumkin34, Reference Delort22, Reference Stingo81, Reference Candy and Lindblad5, Reference Sterbenz80, Reference Lindblad and Soffer59, Reference Lindblad, Lührmann and Soffer54, Reference Lindblad, Lührmann and Soffer55, Reference Lindblad, Lührmann, Schlag and Soffer53, Reference Germain and Pusateri24]. Full asymptotic stability results for large solitary wave solutions to Schrödinger equations on the line have been obtained in [Reference Buslaev and Perelman2, Reference Buslaev and Sulem4, Reference Krieger and Schlag49, Reference Collot and Germain11, Reference Masaki, Murphy and Segata65], while local asymptotic stability results include [Reference Martel62, Reference Martel63, Reference Rialland74, Reference Rialland75, Reference Cuccagna and Maeda17, Reference Cuccagna and Maeda16]. For small solitary wave solutions to Schrödinger equations in one space dimension we refer to [Reference Mizumachi66, Reference Chen8] for full asymptotic stability results and to [Reference Cuccagna and Maeda12, Reference Cuccagna and Maeda14] for local asymptotic stability results. Closely related are the following full asymptotic stability results [Reference Kopylova and Komech42, Reference Kopylova and Komech43, Reference Germain and Pusateri24, Reference Lührmann and Schlag60, Reference Germain, Pusateri and Zhang26, Reference Kairzhan and Pusateri38, Reference Lührmann and Schlag61] and the following local asymptotic stability results [Reference Kowalczyk, Martel and Muñoz45, Reference Kowalczyk, Martel and Muñoz47, Reference Kowalczyk, Martel, Muñoz and Van Den Bosch48, Reference Kowalczyk and Martel44, Reference Li and Lührmann52, Reference Cuccagna and Maeda15, Reference Cuccagna, Maeda, Murgante and Scrobogna18, Reference Palacios and Pusateri72] for solitons in scalar field theories on the line.

References for the spectral and distorted Fourier theory as well as for linear dispersive estimates for scalar and matrix Schrödinger operators in one space dimension include [Reference Weder84, Reference Goldberg and Schlag29, Reference Goldberg28, Reference Chang, Gustafson, Nakanishi and Tsai7, Reference Krieger and Schlag49, Reference Li51, Reference Buslaev and Perelman2].

As pointed out at the beginning of the introduction, using complete integrability techniques the full asymptotic stability of the family of solitary wave solutions (1.3) to the focusing cubic Schrödinger equation was established in [Reference Cuccagna and Pelinovsky19, Reference Saalmann76, Reference Borghese, Jenkins and McLaughlin1].

Closely related articles. We conclude our literature overview with a brief discussion of the articles that are very closely related to our work. Co-dimensional asymptotic stability for solitary wave solutions to super-critical focusing Schrödinger equations $i\partial _t \psi + \partial _x^2 \psi + |\psi |^{p-1} \psi = 0$ , $p> 5$ , on the line was established in [Reference Krieger and Schlag49]. Related earlier results were obtained in [Reference Buslaev and Perelman2] under tailored assumptions on the nonlinearity and the spectrum of the linearized operator. The analysis in [Reference Krieger and Schlag49] combines modulation to take into account the symmetries of the equation with a dispersive setup largely based on Strichartz estimates, which suffices for the higher power nonlinearities considered in [Reference Krieger and Schlag49]. Moreover, [Reference Krieger and Schlag49] systematically develops the spectral and distorted Fourier theory as well as linear dispersive decay estimates for matrix linearized Schrödinger operators around solitary waves with no threshold resonance (generic case). More recently, [Reference Collot and Germain11] established full asymptotic stability of solitary waves for general nonlinear Schrödinger equations $i\partial _t \psi + \partial _x^2 \psi + F'(|\psi |^2) \psi = 0$ allowing for cubic nonlinearities under the key assumption that the matrix linearized Schrödinger operator exhibits neither nonzero eigenvalues nor threshold resonances (generic case). To deal with modified scattering dynamics related to the long-range nature of the cubic nonlinearity, the analysis in [Reference Collot and Germain11] combines modulation with a space-time resonances approach based on the distorted Fourier transform. Previously, a similar strategy had been developed in [Reference Chen8] in the related context of proving the full asymptotic stability of small solitary wave solutions to cubic Schrödinger equations on the line with a trapping potential.

The analysis in this work further develops the framework from [Reference Collot and Germain11] to prove full asymptotic stability of solitary waves of the focusing cubic Schrödinger equation on the line, where the linearized matrix operator additionally exhibits threshold resonances (nongeneric case). We also need to set up the distorted Fourier theory for this linearized operator building on the generic case treated in [Reference Krieger and Schlag49], and we establish linear dispersive decay estimates building on results from [Reference Li51]. Moreover, we use a variable coefficient normal form reminiscent of the one employed in the proof of full asymptotic stability of the sine-Gordon kink under odd perturbations in [Reference Lührmann and Schlag60], and we benefit from insights from the study of the long-time behavior of small solutions to nonlinear Klein-Gordon equations in one space dimension in the presence of a threshold resonance in the series of articles [Reference Lindblad, Lührmann and Soffer54, Reference Lindblad, Lührmann and Soffer55, Reference Lindblad, Lührmann, Schlag and Soffer53, Reference Lührmann and Schlag60, Reference Lührmann and Schlag61].

1.4 Overview of the proof

In this subsection we provide an overview of the main ideas entering the proof of Theorem 1.1.

Linearized Operator. Writing the focusing cubic Schrödinger equation (1.1) as a system for the vector $(\psi , \bar {\psi })$ and linearizing around a solitary wave $e^{i\omega t} e^{i\gamma } \phi _\omega (x)$ for some fixed $\omega \in (0,\infty )$ , $\gamma \in \mathbb R$ , we arrive at the nonselfadjoint matrix Schrödinger operator

(1.12) $$ \begin{align} \begin{aligned} \mathcal H(\omega) &:= \begin{bmatrix} -\partial_x^2 + \omega & 0 \\ 0 & \partial_x^2 - \omega \end{bmatrix} + \begin{bmatrix} -2\phi_\omega^2 & - \phi_\omega^2 \\ \phi_\omega^2 & 2\phi_\omega^2 \end{bmatrix}. \end{aligned} \end{align} $$

This operator is closed on $H^2_x(\mathbb R) \times H^2_x(\mathbb R)$ and its spectrum is well-known: $\mathcal H(\omega )$ has essential spectrum $(-\infty , -\omega ] \cup [\omega , \infty )$ with no embedded eigenvalues. The only eigenvalue is $0$ with algebraic multiplicity equal to $4$ and geometric multiplicity equal to $2$ . Its generalized eigenfunctions are generated by the symmetries, that is, the invariance of (1.1) under phase shifts, scaling, spatial translations, and Galilei boosts. They are explicitly given by

$$ \begin{align*} Y_{1,\omega} := \begin{bmatrix} i\phi_\omega \\ -i\phi_\omega \end{bmatrix}, \quad Y_{2,\omega} := \begin{bmatrix} \partial_\omega \phi_\omega \\ \partial_\omega \phi_\omega \end{bmatrix}, \quad Y_{3,\omega} := \begin{bmatrix} \partial_x \phi_\omega \\ \partial_x \phi_\omega \end{bmatrix}, \quad Y_{4,\omega} := \begin{bmatrix} ix\phi_\omega \\ -ix\phi_\omega \end{bmatrix}, \end{align*} $$

and it holds that

$$ \begin{align*} \mathcal H(\omega) Y_{1,\omega} = 0, \quad \mathcal H(\omega) Y_{2,\omega} = i Y_{1,\omega}, \quad \mathcal H(\omega) {Y_{3,\omega}} = 0, \quad \mathcal H(\omega) Y_{4,\omega} = -2i Y_{3,\omega}. \end{align*} $$

The vector-valued functions $Y_{1,\omega }, Y_{2,\omega }$ are even, while $Y_{3,\omega }, Y_{4,\omega }$ are odd. Since we only study even solutions to (1.1), we consider $\mathcal H(\omega )$ restricted to the subspace of even functions. The eigenvalue $0$ then has algebraic multiplicity $2$ and geometric multiplicity $1$ . Moreover, $\mathcal H(\omega )$ exhibits an even threshold resonance at the edge of each part of the essential spectrum. Indeed, the functions

(1.13) $$ \begin{align} \Phi_{+,\omega}(x) := \begin{bmatrix} \tanh^2(\sqrt{\omega}x) \\ -\operatorname{\mathrm{sech}}^2(\sqrt{\omega}x) \end{bmatrix}, \quad \Phi_{-,\omega}(x) := \sigma_1 \Phi_{+,\omega}(x) = \begin{bmatrix} -\operatorname{\mathrm{sech}}^2(\sqrt{\omega}x) \\ \tanh^2(\sqrt{\omega}x) \end{bmatrix} \end{align} $$

belong to $L^\infty _x(\mathbb R) \times L^\infty _x(\mathbb R) \setminus L^2_x(\mathbb R) \times L^2_x(\mathbb R)$ and satisfy

$$ \begin{align*} \mathcal H(\omega) \Phi_{+,\omega} = \omega \Phi_{+,\omega}, \quad \mathcal H(\omega) \Phi_{-,\omega} = -\omega \Phi_{-,\omega}. \end{align*} $$

In Section 3 we recall the spectral theory for $\mathcal H(\omega )$ and we systematically develop the distorted Fourier theory for $\mathcal H(\omega )$ . We point out that the distorted Fourier theory established in [Reference Buslaev and Perelman2, Reference Krieger and Schlag49] for matrix Schrödinger operators of the form (1.12) in the generic case (no threshold resonances) does not directly apply to $\mathcal H(\omega )$ . Analogously to [Reference Buslaev and Perelman2, Reference Krieger and Schlag49], our construction of the distorted Fourier transform associated with $\mathcal H(\omega )$ is based on an explicit computation of the jump of the resolvent across the essential spectrum and on a representation of the linear evolution $e^{it\mathcal H(\omega )}$ in terms of a contour integral of the resolvent (analogue of Stone’s theorem in the nonselfadjoint case). In Section 4 we collect linear decay estimates for the linear evolution associated with the operator $\mathcal H(\omega )$ .

Figure 1

Spectrum of the operator $\mathcal H(\omega )$ in (1.12).

Modulation. Our starting point for the analysis of the asymptotic stability of the family of solitary waves (1.5) is to use standard modulation techniques to decompose the perturbed even solution $\psi (t,x)$ to (1.1) into a modulated solitary wave and a radiation term that is orthogonal to directions related to the invariance of (1.1) under scaling and phase shifts. Specifically, we write

$$ \begin{align*} \psi(t,x) = e^{i \gamma(t)} \big( \phi_{\omega(t)}(x) + u(t,x) \big) \end{align*} $$

for continuously differentiable paths $\omega (t) \in (0,\infty )$ with $|\omega (t) - \omega _0| \lesssim \varepsilon $ and $\gamma (t) \in \mathbb R$ that are uniquely determined by the orthogonality conditions

(1.14) $$ \begin{align} \langle U(t), \sigma_2 Y_{1, \omega(t)} \rangle = \langle U(t), \sigma_2 Y_{2, \omega(t)} \rangle = 0, \quad U(t) := \begin{bmatrix} u(t) \\ {\overline u}(t) \end{bmatrix}, \quad t \geq 0. \end{align} $$

The resulting evolution equation for the radiation term $U(t)$ reads

(1.15) $$ \begin{align} i\partial_t U - \mathcal H(\omega)U = (\dot{\gamma} - \omega) \sigma_3 U + {\mathcal Mod} + \mathcal Q_\omega(U) + \mathcal C(U). \end{align} $$

It is coupled to the following system of first-order differential equations for the modulation parameters

(1.16) $$ \begin{align} \mathbb{M} \begin{bmatrix} \dot{\gamma} - \omega \\ \dot{\omega} \end{bmatrix} = \begin{bmatrix} \big\langle i \big( \mathcal Q_\omega(U) + \mathcal C(U) \big), \sigma_2 Y_{1,\omega} \big\rangle \\ \big\langle i \big( \mathcal Q_\omega(U) + \mathcal C(U) \big), \sigma_2 Y_{2, \omega} \big\rangle \end{bmatrix}. \end{align} $$

In the preceding identities, $\sigma _2, \sigma _3$ are the Pauli matrices defined in (2.1), ${\mathcal Mod}$ denotes relatively harmless terms that are a consequence of the time-dependence of the modulation parameters, and the quadratic and cubic nonlinearities on the right-hand sides of (1.15) and (1.16) are given by

(1.17) $$ \begin{align} \begin{aligned} \mathcal Q_\omega(U) := \begin{bmatrix} -\phi_\omega(u^2 + 2u \bar{u}) \\ \phi_\omega({\overline u}^2 + 2u {\overline u}) \end{bmatrix}, \quad \mathcal C(U) := \begin{bmatrix} -u {\overline u} u \\ {\overline u} u {\overline u} \end{bmatrix}. \end{aligned} \end{align} $$

Moreover, the matrix $\mathbb {M}$ on the left-hand side of (1.16) is given by

$$ \begin{align*} \mathbb{M} := \begin{bmatrix} 0 & c_\omega \\ c_\omega & 0 \end{bmatrix} + \begin{bmatrix} \langle U, \sigma_1 Y_{1, \omega} \rangle & \langle U, \sigma_2 \partial_\omega Y_{1, \omega} \rangle \\ \langle U, \sigma_1 Y_{2, \omega} \rangle & \langle U, \sigma_2 \partial_\omega Y_{2, \omega} \rangle \end{bmatrix}, \quad c_\omega := \frac{2}{\sqrt{\omega}}> 0, \end{align*} $$

which is clearly invertible for sufficiently small perturbations $U(t)$ .

Schematic Setup and Strategy. Now our primary task is to simultaneously establish decay and asymptotics for the radiation term $U(t)$ and to prove convergence of the modulation parameter $\omega (t)$ to an asymptotic value $\omega _\infty \in (0,\infty ) $ as $t\to \infty $ , in other words to establish sufficiently fast decay of $\omega (t) - \omega _\infty $ to zero. In view of the weak dispersion in one space dimension in conjunction with the low power nonlinearities on the right-hand sides of (1.15) and (1.16), this is a formidable challenge that gets compounded by the subtle, but severe effects of the threshold resonances on the dynamics.

In order to prove decay for the radiation term by perturbative means, we naturally want to invoke dispersive estimates for the linear evolution $i\partial _t U - \mathcal H(\omega ) U = 0$ associated with (1.15). However, the modulation parameter $\omega = \omega (t)$ is time-dependent. We therefore need to pass to a time-independent reference operator $\mathcal H({\underline {\omega }})$ on the left-hand side of (1.15) for a suitably chosen fixed scaling parameter ${\underline {\omega }}$ and then treat the difference perturbatively. Heuristically, we take ${\underline {\omega }}$ to be the asymptotic limit of $\omega (t)$ as $t \to \infty $ . The proof of Theorem 1.1 proceeds via an elaborate bootstrap argument, so that on a given finite time interval $[0,T]$ we actually chooseFootnote ¹ ${\underline {\omega }} = \omega (T)$ to be the final value of the continuously differentiable path $\omega (t)$ on that time interval. As part of our bootstrap argument we seek to show the decay estimate

(1.18) $$ \begin{align} |\omega(t) - {\underline{\omega}}| \lesssim \varepsilon \langle t\rangle^{-1+\delta}, \quad 0 \leq t \leq T, \end{align} $$

where $0 < \delta \ll 1$ is a small absolute constant. Later, we will then extract the final value $\omega _\infty $ of the path $\omega (t)$ as $t \to \infty $ from (1.18) with $T = T_n$ for an increasing sequence of times $T_n \nearrow \infty $ . The decay rate (1.18) turns out to be just about enough to treat perturbatively the error terms that arise from passing to a time-independent reference operator. We point out that since $\omega (t)-{\underline {\omega }} = -\int _t^T \dot {\omega }(s) \, \mathrm {d} s$ by the fundamental theorem of calculus, an almost twice integrable in time decay rate $s^{-2+\delta }$ for $\dot {\omega }(s)$ as $s \to \infty $ would be necessary to crudely infer the decay estimate (1.18).

For the discussion of the main aspects of the nonlinear analysis, in what follows we only consider schematic versions of (1.15) and (1.16) that retain the most relevant terms. Specifically, in the evolution equation (1.15) for the radiation term, we disregard milder modulation terms and we pass to the reference operator $\mathcal H({\underline {\omega }})$ , which produces the delicate difference term $(\omega - {\underline {\omega }}) \sigma _3 U$ on the right-hand side. Moreover, in the quadratic nonlinearity we replace $\phi _\omega (x)$ by $\phi _{\underline {\omega }}(x)$ and ignore the resulting milder error terms. In the modulation equations (1.16), we focus on the equation for $\dot {\omega }$ . To unveil its leading order structure, we only keep the quadratic nonlinearity on the right-hand side and we again replace the time-dependent parameter $\omega $ by ${\underline {\omega }}$ , disregarding the resulting milder error terms. Then we are facing the following schematic evolution equation for the radiation term

(1.19) $$ \begin{align} i\partial_t U - \mathcal H({\underline{\omega}})U = (\omega - {\underline{\omega}}) \sigma_3 U + \mathcal Q_{\underline{\omega}}(U) + \mathcal C(U) + \ldots \end{align} $$

coupled to a differential equation for the modulation parameter $\omega (t)$ of the schematic form

(1.20) $$ \begin{align} \dot{\omega} = i c_{\underline{\omega}}^{-1} \langle \mathcal Q_{\underline{\omega}}(U), \sigma_2 Y_{1,{\underline{\omega}}} \rangle + \ldots. \end{align} $$

Our overall strategy is to take a space-time resonances approach based on the distorted Fourier transform associated with the reference operator $\mathcal H({\underline {\omega }})$ to infer decay and asymptotics for $U(t)$ , while we exploit the oscillations in the equation for $\dot {\omega }$ to arrive at the decay estimate (1.18) for $\omega (t) - {\underline {\omega }}$ . As we explain next, in comparison with the generic case (no threshold resonances) considered in [Reference Collot and Germain11], this strategy is significantly compounded by the slow local decay of the radiation term caused by the threshold resonances of the linearized operator $\mathcal H({\underline {\omega }})$ .

Main Enemy: Slow Local Decay. Since the quadratic nonlinearity defined in (1.17) is spatially localized due to the rapid decay of the coefficient $\phi _{\underline {\omega }}(x)$ , the leading order behavior of $\mathcal Q_{\underline {\omega }}(U(t))$ is entirely governed by the local decay of the radiation term $U(t)$ . In view of (1.20), the same observation holds for the leading order behavior of $\dot {\omega }(t)$ .

In the absence of threshold resonances (generic case), the Schrödinger waves enjoy improved local decay, which simplifies the treatment of the terms $(\omega -{\underline {\omega }}) \sigma _3 U$ and $\mathcal Q_{\underline {\omega }}(U)$ on the right-hand side of (1.19) and the derivation of sufficient decay for $\omega (t) - {\underline {\omega }}$ . However, in our setting the threshold resonances of $\mathcal H({\underline {\omega }})$ lead to slow local decay. In fact, [Reference Li51, Theorem 1.1] by the first author quantifies how the slow local decay can be pinned down exactly to the contributions of the threshold resonances, while the bulk of the Schrödinger waves still enjoy improved local decay. Specifically, for the linear evolution of a (vector-valued) Schwartz function $F$ we have for all $t \geq 1$ that

$$ \begin{align*} \bigg\| \langle x\rangle^{-2} \bigg( e^{-it\mathcal H({\underline{\omega}})} \underline{P_{\mathrm{e}}} F - \frac{e^{-it{\underline{\omega}}}}{t^{\frac12}} \Phi_{+,{\underline{\omega}}}(x) c_+(F) - \frac{e^{it{\underline{\omega}}}}{t^{\frac12}} \Phi_{-,{\underline{\omega}}}(x) c_-(F) \bigg) \bigg\|_{L^\infty_x \times L^\infty_x} \lesssim \frac{1}{t^{\frac32}} \big\| \langle x\rangle^2 F \big\|_{L^1_x \times L^1_x}, \end{align*} $$

where $\underline {P_{\mathrm {e}}}$ denotes the Riesz projection to the essential spectrum relative to $\mathcal H({\underline {\omega }})$ , $\Phi _{\pm ,{\underline {\omega }}}(x)$ are the threshold resonances defined in (1.13), and $c_{\pm , {\underline {\omega }}}(F)$ should be thought of as projections of $F$ onto these

$$ \begin{align*} c_{+,{\underline{\omega}}}(F) := \frac{e^{-i\frac{\pi}{4}}}{\sqrt{4\pi}}\langle F,\sigma_3 \Phi_{+,{\underline{\omega}}}\rangle, \quad c_{-, {\underline{\omega}}}(F) := -\frac{e^{i\frac{\pi}{4}}}{\sqrt{4\pi}}\langle F,\sigma_3 \Phi_{-,{\underline{\omega}}}\rangle. \end{align*} $$

Thus, the leading order behavior of $\mathcal Q_{\underline {\omega }}(U(t))$ can be thought of as a linear combination of slowly decaying quadratic source terms of the schematic form

(1.21) $$ \begin{align} \frac{1}{t} \phi_{\underline{\omega}}(x) \big( e^{\mp i t {\underline{\omega}}} \Phi_{\pm, {\underline{\omega}}}(x) \big)^2 \varepsilon^2, \end{align} $$

see the expansion (5.51) for the precise expression. These source terms create resonances both in the evolution equation (1.19) for the radiation term and in the differential equation (1.20) for $\dot {\omega }$ . For more detailed explanations we refer to the discussions after (5.53), respectively after (8.14). In particular, the slow $t^{-1}$ decay of the schematic source terms (1.21) is far away from the twice integrability in time of $\dot {\omega }(t)$ pointed out earlier, which would be necessary to crudely infer the decay estimate (1.18).

Null Structures. It turns out that we do not have to address the worst effects of the resonant quadratic source terms (1.21) in our analysis thanks to two remarkable null structures in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations. These null structures appear to be closely related to the completely integrable structure of the equation, but we do not have a rigorous proof for this connection.

We refer to Lemma 5.4 and the discussion in Subsection 5.4 for a quadratic null structure in the evolution equation for the radiation term. It was already observed by the first author in [Reference Li51, Lemma 1.6] and it is reminiscent of a similar null structure for perturbations of the sine-Gordon kink, see [Reference Lührmann and Schlag60, Reference Lindblad, Lührmann, Schlag and Soffer53]. Using a variable coefficient normal form inspired by [Reference Lindblad, Lührmann and Soffer55, Reference Lührmann and Schlag60], we can then recast the quadratic nonlinearity $\mathcal Q_{\underline {\omega }}(U)$ on the right-hand side of (1.19) into a spatially localized cubic term, which is slightly better behaved than the constant coefficient cubic term $\mathcal C(U)$ .

For the details of the null structure in the modulation equations we refer to Lemma 6.3 and the discussion in Section 8, specifically the treatment of the term $I(s)$ in (8.12). Thanks to this remarkable null structure, in the proof of the decay estimate (1.18) we can integrate by parts in the time integral of the leading order quadratic contribution

$$ \begin{align*} \omega(t) - {\underline{\omega}} = - \int_t^T \dot{\omega}(s) \, \mathrm{d} s = -i c_{\underline{\omega}}^{-1} \int_t^T \big\langle \mathcal Q_{\underline{\omega}}\big(U(s)\big), \sigma_2 Y_{1,{\underline{\omega}}} \big\rangle \, \mathrm{d} s + \ldots, \end{align*} $$

which effectively turns it into the contribution of a cubic term

$$ \begin{align*} \int_t^T \big\langle \mathcal C\big(U(s)\big), \sigma_2 Y_{1,{\underline{\omega}}} \big\rangle \, \mathrm{d} s. \end{align*} $$

We show that the latter does not exhibit any time resonances, whence we can integrate by parts in time once more and turn it into the time integral of a quartic term, which enjoys the desired crude twice integrability to conclude (1.18).

Modified Scattering Analysis. Finally, we discuss the analysis of the modified scattering behavior of the radiation term. As explained above, we can use a variable coefficient normal form to recast the quadratic nonlinearity $\mathcal Q_{\underline {\omega }}(U)$ on the right-hand side of (1.19) into a better behaved spatially localized cubic term. Moreover, thanks to the sufficiently fast decay (1.18) of $\omega (t) - {\underline {\omega }}$ , we can use an integrating factor argument inspired by [Reference Collot and Germain11, Proposition 9.5] to effectively remove the term $(\omega -{\underline {\omega }}) \sigma _3 U$ from (1.19), see (5.45) and (5.46). Then we are essentially reduced to studying the asymptotic behavior of small solutions to the cubic matrix Schrödinger equation

(1.22) $$ \begin{align} i\partial_t U - \mathcal H({\underline{\omega}})U = \mathcal C(U). \end{align} $$

In analogy to the study of the long-time dynamics of small solutions to the scalar cubic Schrödinger equation with (or without) potential in one space dimension

(1.23) $$ \begin{align} i\partial_t w - H w = |w|^2 w, \quad H := -\partial_x^2 + V(x), \end{align} $$

we expect logarithmic phase corrections with respect to the free flow. As in the generic case considered in [Reference Collot and Germain11], we proceed in the spirit of the space-time resonances method for the scalar equation (1.23), see [Reference Kato and Pusateri39, Reference Germain, Pusateri and Rousset25, Reference Chen and Pusateri9, Reference Chen and Pusateri10], and we introduce the profile of the radiation term $U(t)$ relative to the reference operator $\mathcal H({\underline {\omega }})$ by filtering out the linear evolution

(1.24) $$ \begin{align} F_{\underline{\omega}}(t) := e^{it\mathcal H({\underline{\omega}})} \underline{P_{\mathrm{e}}} U(t), \end{align} $$

where we recall that $\underline {P_{\mathrm {e}}}$ denotes the projection to the essential spectrum for $\mathcal H({\underline {\omega }})$ .

Then the idea is to consider the evolution equation for the (vector-valued) distorted Fourier transform relative to $\mathcal H({\underline {\omega }})$ of the profile $\widetilde {\mathcal F}_{\underline {\omega }}[F_{\underline {\omega }}(t)](\xi ) =: ( {{\tilde {f}}_{+,{\underline {\omega }}}}(t,\xi ), {{\tilde {f}}_{-,{\underline {\omega }}}}(t,\xi ) )$ , which reads

(1.25) $$ \begin{align} \partial_t {\tilde{f}}_{\pm,{\underline{\omega}}}(t,\xi) = -ie^{\pm i t(\xi^2+{\underline{\omega}})}\widetilde{\mathcal F}_{\pm,{\underline{\omega}}}\big[ \mathcal C(U(t))\big](\xi). \end{align} $$

We uncover the fine structure of the contributions of the cubic terms on the right-hand side of (1.25) by inserting the representation formula (5.35) for the radiation term in terms of the distorted Fourier transform of its profile. Analyzing the interactions of the corresponding distorted Fourier basis elements (cubic spectral distributions), we arrive at the decomposition

(1.26) $$ \begin{align} \widetilde{\mathcal F}_{\pm, {\underline{\omega}}}\big[\mathcal C(U(t))\big](\xi) = \widetilde{\mathcal C}_{\pm,\delta_0,{\underline{\omega}}}(t,\xi) + \widetilde{\mathcal C}_{\pm,\mathrm{p.v.},{\underline{\omega}}}(t,\xi)+\widetilde{\mathcal C}_{\pm,\mathrm{reg},{\underline{\omega}}}(t,\xi) + \ldots \end{align} $$

The three terms on the right-hand side of (1.26) are trilinear expressions in terms of the components of the distorted Fourier transform of the profile involving a Dirac kernel, a Hilbert-type kernel, and a regular kernel, respectively. See Section 6.1 for the details.

In view of the linear dispersive decay estimate from Lemma 4.1, in order to recover the linear decay rate $t^{-\frac 12}$ for the radiation term, it suffices to establish slowly growing weighted energy estimates

(1.27) $$ \begin{align} \big\| \big( \partial_\xi {{\tilde{f}}_{+,{\underline{\omega}}}}(t,\xi), \partial_\xi {{\tilde{f}}_{-,{\underline{\omega}}}}(t,\xi) \big) \big\|_{L^2_\xi \times L^2_\xi} \lesssim \varepsilon \langle t\rangle^\delta, \quad t \geq 0, \end{align} $$

and uniform-in-time pointwise bounds

(1.28) $$ \begin{align} \big\| \big( {{\tilde{f}}_{+,{\underline{\omega}}}}(t,\xi), {{\tilde{f}}_{-,{\underline{\omega}}}}(t,\xi) \big) \big\|_{L^\infty_\xi \times L^\infty_\xi} \lesssim \varepsilon, \quad t \geq 0. \end{align} $$

Thanks to a simple relation between the two components ${{\tilde {f}}_{+,{\underline {\omega }}}}(t,\xi )$ and ${{\tilde {f}}_{-,{\underline {\omega }}}}(t,\xi )$ , see Lemma 3.14, it in fact suffices to derive the bounds (1.27), (1.28) for the component ${{\tilde {f}}_{+,{\underline {\omega }}}}(t,\xi )$ .

In Section 9 we establish the weighted energy estimates (1.27), where $0 < \delta \ll 1$ is the same small absolute constant as in (1.18). In Section 10 we infer the pointwise bounds (1.28) for the distorted Fourier transform of the profile via an ODE argument, in the course of which we capture the logarithmic phase corrections in the asymptotics of the radiation term. More specifically, we carry out a stationary phase analysis of the contributions of the cubic terms in (1.25), where we rely on their fine structure determined in (1.26). We find that the evolution equation for $\tilde {f}_{+,{\underline {\omega }}}(t,\xi )$ is effectively given by

(1.29) $$ \begin{align} \partial_t {{\tilde{f}}_{+,{\underline{\omega}}}}(t,\xi) = \frac{i}{2t}\vert {{\tilde{f}}_{+,{\underline{\omega}}}}(t,\xi)\vert^2{{\tilde{f}}_{+,{\underline{\omega}}}}(t,\xi) + r(t,\xi), \end{align} $$

where the leading order term stems from the cubic interactions with a Dirac kernel, and where the remainder term $r(t,\xi )$ enjoys faster decay. Using a standard integrating factor argument, we can then deduce the pointwise bounds (1.28).

Further Comments. We conclude the discussion of the main ideas of the proof of Theorem 1.1 with several more technical comments.

1. (i) Owing to the quadratic null structures uncovered in Lemma 5.4 and Lemma 6.3, the quadratic nonlinearity affects neither the asymptotics of the radiation term nor the asymptotic behavior of the modulation parameters. In the absence of favorable structures for related models with threshold resonances, this is a regime where much remains to be understood.

2. (ii) In contrast to the generic case treated in [Reference Collot and Germain11], we do not obtain integrable decay rates for $\omega (t)-\omega (T)$ and for $\dot {\gamma }(t)-\omega (t)$ . As a consequence, we have to handle a slight uncertainty in the phase shift in the proof of Theorem 1.1 and we do not extract the phase shift parameter for the final soliton. It is an interesting question how one could possibly further enhance the analysis in this paper to deduce integrable decay rates.

3. (iii) Interestingly, in the derivation of the differential equation to extract the asymptotics of the distorted Fourier transform of the profile in Lemma 10.4, the (critically decaying) cubic interactions with a Hilbert-type kernel do not contribute due to the vanishing property (6.10), see the proof of (10.23).

4. (iv) The null structure in the modulation equations discussed above is reminiscent of a similar favorable structure exploited in the modulation equations in [Reference Krieger and Schlag50, Lemma 5.12] in the context of constructing blow-up solutions to the critical focusing nonlinear Schrödinger equation on the line. The authors are grateful to Joachim Krieger for pointing out this connection.

5. (v) The tensorized product structure of the distorted Fourier basis elements is convenient in our analysis, see for instance Lemma 3.16 and the study of the fine structure of the cubic spectral distributions in Subsection 6.1.

6. (vi) We exploit the simple relationship (5.47) between the two components ${{\tilde {f}}_{+,{\underline {\omega }}}}(t,\xi )$ and ${{\tilde {f}}_{-,{\underline {\omega }}}}(t,\xi )$ of the distorted Fourier transform of the profile in the nonlinear analysis. It should be compared with the corresponding relation in the generic case, see, for example, [Reference Collot and Germain11, Lemma 5.12].

7. (vii) In the derivation of the slowly growing weighted energy estimates for the profile in Proposition 9.4, the dual local smoothing estimates for the Schrödinger evolution from Lemma 4.4 play a key role in handling all contributions of spatially localized nonlinearities with cubic-type time decay. This idea was first used in [Reference Chen and Pusateri10]. See also [Reference Lindblad, Lührmann and Soffer54, Reference Lindblad, Lührmann and Soffer55, Reference Lührmann and Schlag60] for a related approach.

8. (viii) We made use of the Wolfram Mathematica software system in the computation of the resolvent kernel in Subsection 3.2 as well as in the computation of the quadratic null structures in Lemma 5.4 and in Lemma 6.3.