Soliton resolution for Calogero–Moser derivative nonlinear Schrödinger equation

In this paper we study the global behaviors of solutions, the soliton resolution, for the Calogero–Moser derivative nonlinear Schrödinger equation (CM-DNLS):

Here, $D_{+}=D\Pi_{+}=-i\partial_{x}\Pi_{+}$ is a nonlocal derivative with the projection to positive frequencies, i.e. $\Pi_{+}$ is the Cauchy–Szegő projection with the Fourier symbol, $\mathbf{1}_{\xi>0}$. (CM-DNLS) is a recently introduced nonlinear Schrödinger-type equation. (CM-DNLS) draws attention as it enjoys rich and interesting mathematical structures. To name a few, (CM-DNLS) is mass-critical (with pseudo-conformal invariance), completely integrable, and has a self-dual Hamiltonian. The purpose of this work is to show soliton resolution theorem for (CM-DNLS). We show the soliton resolution in fully general setting, without size constraints or radial symmetry. Although (CM-DNLS) is completely integrable, we do not use the integrability features, and our proof may give insights toward other non-integrable models.

(CM-DNLS) was introduced in [1] as a continuum model of the classical Calogero–Moser Hamiltonian system, which is known to be completely integrable [9, 10, 45, 47]. It is also known as continuum Calogero–Moser model (CCM) or Calogero–Moser NLS (CM-NLS). There is also a periodic counterpart, Calogero–Sutherland model [53, 54].

A rigorous mathematical analysis was initiated by Gérard and Lenzmann [26]. They verified that (CM-DNLS) is completely integrable by discovering the Lax pairs on the Hardy–Sobolev space $H_{+}^{s}(\mathbb{R})$:

The flow of (CM-DNLS) preserves the positive frequency condition, $\text{supp}\,\widehat{u}(t)\subset[0,\infty)$, and this distinctive feature is called chirality. For chiral solutions $u(t)\in H_{+}^{s}(\mathbf{\mathbb{R}})$ to (CM-DNLS), the equation admits the Lax pair structure:

with the $u$-dependent operators $\mathcal{L}_{\textnormal{Lax}}$ and $\mathcal{P}_{\textnormal{Lax}}$ defined by ¹¹1(1.2) is slightly different but equivalent to what was presented in [26]. This formulation was presented in the introduction of [40].

There is a similar Lax pairs in the full intermediate NLS [48]. Another feature of integrability on the Hardy space is an explicit formula [40] given by a holomorphic function on the upper half-plane. This is followed by the works of Gérard and collaborators in relevant models [25, 24, 27].²²2In fact, (1.1) holds true regardless of chirality. See [41, Proposition 3.5]. However, the chirality is required for the explicit formula. This is a global explicit formula for a weak solution.

We briefly recall symmetries and conservation laws. (CM-DNLS) enjoys time and space translation, and phase rotation symmetries. They are associated to the conservation laws of energy, mass, and momentum:

Moreover, it has Galilean invariance

Of particular importance are $L^{2}$-scaling symmetry

and pseudo-conformal symmetry

Associated identities to scaling and pseudo-conformal symmetries are the virial identities (1.9). We note that those symmetries are shared with the mass-critical NLS. But, we remark that $\widetilde{E}(u)$ is a complete square of first-order form. Regarding the chirality, the Galilean invariance preserves the chirality only if $c\geq 0$. The pseudo-conformal symmetry is valid for $H^{1,1}$-solutions but does not preserve the chirality.

It is well-known that stationary or static solutions play a pivotal role in the dynamics. Here, a solution to (CM-DNLS) is static if and only if it has zero energy. In [26], the authors showed that $\mathcal{R}(x)$, called the ground state or soliton,

is the unique zero energy solution (and thus static) up to scaling, phase rotation, and translation symmetries. Note that this $\mathcal{R}(x)$ is a chiral solution. More generally, any nonzero $H^{1}(\mathbf{\mathbb{R}})$ traveling wave solutions (i.e., solutions of the form $u(t,x)=e^{i\omega t}\mathcal{R}_{c,\omega}(x-2ct)$ for some $\omega,c\in\mathbf{\mathbb{R}}$) are given by $\textnormal{Gal}_{c}\mathcal{R}$ up to scaling, phase rotation, and translation symmetries [26]. Applying the pseudo-conformal transform (1.3) to $\mathcal{R}$, one obtains an explicit finite-time blow-up solution:

It is important to note that $S(t)$ is neither of finite energy ($S(t)\notin H^{1}(\mathbf{\mathbb{R}})$), nor chiral ($S(t)\notin L_{+}^{2}(\mathbf{\mathbb{R}})$).

Let us briefly discuss previously known results on (CM-DNLS). De Moura and Pilod [14] proved the local well-posedness in $H^{s}(\mathbb{R})$ for all $s>\frac{1}{2}$. It is also locally well-posed in $H_{+}^{s}(\mathbf{\mathbb{R}})$ for $s>\frac{1}{2}$ [26]. The ground state $\mathcal{R}$ is a threshold for global regularity, i.e. global existence of strong solutions. To this regard, Lax pairs provide significant information for the subthreshold $M(u)<M(\mathcal{R})=2\pi$. If $M(u)<M(\mathcal{R})$ in $H_{+}^{1}(\mathbb{R})$, then the solution is global and moreover, $\sup_{t\in\mathbf{\mathbb{R}}}\|u(t)\|_{H_{+}^{k}(\mathbf{\mathbb{R}})}\lesssim_{k}\|u_{0}\|_{H_{+}^{k}(\mathbf{\mathbb{R}})}$ for all $k\in\mathbb{N}_{\geq 1}$ [26]. Subsequently, the local well-posedness on $M(u)<M(\mathcal{R})$ was improved to $L_{+}^{2}(\mathbb{R})$ by Killip, Laurens, and Vişan [40]. At the threshold ($M(u_{0})=M(\mathcal{R})$), by adopting [43], $H^{1}$-solutions are global, as $S(t)\notin H^{1}$ [26]. The dynamics above the threshold ($M(u)>M(\mathcal{R}))$ were also studied. Gérard and Lenzmann [26] employed the Lax pair structure to construct $N$-soliton solutions of the form

where the residues $a_{k}(t)\in\mathbb{C}$ and the pairwise distinct poles $z_{k}(t)\in\mathbb{C}_{-}=\{z\in\mathbf{\mathbb{C}}:\mathrm{Im}(z)<0\}$ for $1\leq k\leq N$ solve a complexified version of the classical Calogero–Moser system. These $N$-soliton solutions blow up in infinite-time with $\|u(t)\|_{H^{s}}\sim|t|^{2s}$ as $|t|\to\infty$ for any $s>0$. Hogan and Kowalski [28], using the explicit formula, showed the existence of possibly infinite time blow-up solutions with mass arbitrarily close to the threshold $M(\mathcal{R})$. The first finite-time blow-up solutions were constructed by the authors and K. Kim [41], arising from smooth chiral data. This result addresses to the threshold problem for global regularity. Moreover, it is remarkable that although (CM-DNLS) is completely integrable, it admits finite-time blow-up solutions. Additionally, the zero dispersion limit of (CM-DNLS) was investigated by Badreddine [5]. We also refer to [4, 3] for the periodic model.

Our main theorem concerns the global dynamics beyond the threshold, so called, soliton resolution. We show the soliton resolution in a fully general setting, without radial symmetry and size constraints. We use notation for modulated functions $[f]_{\lambda,\gamma,x}$ in (2.1).

Soliton resolution is widely believed to occur in many dispersive equations. It suggests that any generic global-in-time solution asymptotically decouples into a sum of solitons (or similar solutions such as breathers) and a radiation term that goes to zero in some sense. For blow-up solutions, in various models, it is believed that each blow-up profile is a sum of modulated solitons. This type of results were conjectured for KdV equation in [23, 57] from the numerical simulations. The rigorous proof of soliton resolution was first demonstrated in various integrable PDEs via the inverse scattering method. To refer to just a few, see [22, 21] for KdV, [50] for mKdV, and [58, 52, 51, 46, 7] for 1-dimensional cubic NLS. Also refer to [37, 38] for Derivative NLS.

For non-integrable dispersive and wave equations, soliton resolution has been studied in several models. In wave equations, such as the energy-critical nonlinear wave equation (in various dimensions) and energy-critical equivariant wave maps [19, 12, 39, 17, 20, 18, 33, 11, 35]. For damped Klein-Gordon equations, soliton resolution for global solutions was established in [8, 13, 29]. For Schrödinger-type equations, soliton resolution was established for the equivariant self-dual Chern–Simon–Schrödinger equation (CSS) in [42]. In the context of parabolic equations, soliton resolution has been studied by several authors for the harmonic map heat flow [34, 36] (or references therein) and the energy-critical semilinear heat equation [2]. Among others, the authors in [36] proved a version of continuous-in-time soliton resolution without symmetry. As seen from the list, most results in non-integrable models were achieved under symmetry constraints, excluding moving solitons.
Gauge transform.

(CM-DNLS) has an extra structure, so called gauge transform. This property is shared with DNLS. Define the gauge transform

Then, new variable $v(t,x)$ solves

($\mathcal{G}$-CM) is a gauge transformed Calogero-Moser derivative NLS. All symmetries are transferred accordingly. The conservation laws of energy, mass, and momentum are given by

where $\mathcal{H}$ is the Hilbert transform. The virial identities are

It is noteworthy that ($\mathcal{G}$-CM) admits the Hamiltonian formulation

where $\nabla E(v)$ is a functional derivative with respect to $(f,g)_{r}=\mathrm{Re}\int f\overline{g}$. In other words, $-i\nabla E(v)$ is a symplectic derivative with respect to the standard symplectic form $\omega(f,g)=\mathrm{Im}\int f\overline{g}$. Moreover, as $E(v)$ is a complete square, ($\mathcal{G}$-CM) is a self-dual Hamiltonian equation. See more details in [41]. The static solution $\mathcal{R}$ of (CM-DNLS) is transformed as a static solution to ($\mathcal{G}$-CM)

Note that we chose the minus sign in the transform $v=-\mathcal{G}(u)$ to ensure that $Q$ is positive. $\mathcal{R}$ is not real-valued, and $\mathrm{Re}\mathcal{R}(x)$ and $\mathrm{Im}\mathcal{R}(x)$ exhibit different decays. However $Q$ is positive real-valued, which is a technical benefit of working with ($\mathcal{G}$-CM). In the main body of analysis, we will prove the soliton resolution for ($\mathcal{G}$-CM), and then, using the gauge transform and its inverse, we will obtain Theorem 1.1.

1. Novelty and Method. Our proof does not rely on the complete integrability. To our knowledge, this is the first non-integrable proof of soliton resolution in Schrödinger-type equations without radial symmetry and size constraints. We bypass time-sequential soliton resolution and directly prove continuous-in-time soliton resolution. The nonnegativity of energy is crucial. We believe our argument is applicable to other models with nonnegative energy, such as, wave or Schrödinger map, Chern-Simons-Schrödigner, and so on. A similar idea was used in NLS under threshold condition by Merle [43] or Dodson [16].

2. No bubble tree. (1.5) and (1.10) indicate that there is no bubble tree. i.e. any two bubbles maintain a distance larger than the scales of both. We believe this is natural. If a bubble tree existed, there would be a discontinuity in the soliton configuration in (CM-DNLS) along with the gauge transform. See more detail in Remark 5.2. As a similar result, in 1D harmonic map heat flow, there is no finite time bubble tree [56]. So far, finite-time bubble trees are not yet constructed in any model, while there are several results of infinite time bubble tree construction [55, 15, 30, 31, 32].

3. Global solutions. We prove finite-time blow-up cases first, and then take the pseudo-conformal transform to obtain results for global solutions. This is why we need to assume $u(t)\in H^{1,1}$ for global solutions, while $u(t)\in H^{1}$ suffices for finite-time blow-up solutions. After taking the pseudo-conformal transform, the translation parameter $x_{j}(t)$ becomes the velocity of the Galilean boost, $\text{Gal}_{c_{j}(t)}$, and the scaling parameter becomes $\lambda(t)\to\mathring{\lambda}(t)\coloneqq t\lambda(-t^{-1})\lesssim 1$. This results a multi-soliton configuration with moving solitons at constant velocities.

For $H^{1,1}$-solutions, thanks to the pseudo-conformal transform, the linear scattering of radiation part is easily obtained. In particular, this adresses the subthreshold problem, i.e. when $M(u)<M(\mathcal{R})$ and $u\in H^{1,1}$, then $u(t)$ has to scatter. A remaining question is whether $H^{1}$-global solutions may exhibit different dynamics other than $H^{1,1}$-solutions. At current status, even for small solution in $H^{1}$, the (linear or modified) scattering is not known. In view of results of other cubic equations in 1D, it is unclear whether the linear scattering occurs for $H^{1}$-solutions.

A multi-soliton example constructed by Gérard–Lenzmann [26] does not belong to $H^{1,1}$ due to the slow decay of solitons. Still, their examples meet our criteria in Theorem 1.1. On the other hand, even if $v(t)\in H^{1,1}$, each soliton component $\textnormal{Gal}_{c_{j}(t)}([Q]_{\lambda_{j}(t),\gamma_{j}(t),0})\notin H^{1,1}$ in the multi-soliton configuration (1.11), and thus $\varepsilon(t)\coloneqq v(t)-\sum_{j=1}^{N}\textnormal{Gal}_{c_{j}(t)}([Q]_{\lambda_{j}(t),\gamma_{j}(t),0})\notin H^{1,1}$. One might find this decomposition unsatisfactory. If one wants all components to belong to $H^{1,1}$, then one can simply truncate tails of solitons. More precisely, one can replace $[Q]_{\lambda_{j}(t),\gamma_{j}(t),0}$ with ($[Q]_{\lambda_{j}(t),\gamma_{j}(t),0})\chi_{t^{\delta}}$ such that $\lambda_{j}(t)\ll t^{\delta}$.

As mentioned above, we prove Theorem 1.2 for the finite-time blow-up solutions and then use the pseudo-conformal transform to obtain result for global solutions case. And then we use the gauge transform $\mathcal{G}$ to obtain Theorem 1.1. Thus, in main analysis we consider a finite-time blow-up solution to ($\mathcal{G}$-CM) $v(t)\in H^{1}$.

The first ingredient is the variational characterization of $Q$. In fact, $Q$ is the unique zero energy solution up to symmetries, and thus also a static solution. Furthermore, this also tells us the proximity of a small energy function to $Q$, as stated in Lemma 3.2. More specifically, if $\sqrt{E(v)}<\delta\|v\|_{\dot{H}^{1}}$, then $v=[Q+\widehat{\varepsilon}]_{\lambda,\gamma,x}$ with $\|\widehat{\varepsilon}\|_{\dot{H}^{1}}<\eta$. This is due to the nonnegativity of the energy and achieved near blow-up time. Also note that this is a stronger variational feature than that of mass-critical NLS. This allows us to extract a soliton from $v(t)$.

The second ingredient, one of our main novelty, is the energy bubbling estimate. After the first decomposition with orthogonality conditions on $\widehat{\varepsilon}$, we have the following improved estimate (Lemma 3.4):

where $R>1$ is a large parameter depending on $\|\widehat{\varepsilon}\|_{L^{2}}$ and $\chi_{R}$ is a smooth cut off on $[-R,R]$. (1.12) is motivated from a nonlinear coercivity estimate for (CSS) [42]. In (CSS), due to a special property of its nonlinearity they have a stronger estimate on the outer radiation. Then authors in [42] obtain the soliton resolution consisting of a single soliton. (1.12) benefits from the nonnegativity of energy. More importantly, we will take advantage of the energy conservation, which is located above the critical scaling.

For blow-up solutions, since $E(Q+\widehat{\varepsilon})=\lambda^{2}E(v)$ and $\lambda(t)\to 0$, the right-hand side of (1.12) is not just small but goes to zero. In particular, we have a quantitative control of the inner part of radiation $\chi_{R}\widehat{\varepsilon}$. However, in general, the outer part of radiation $\|(1-\chi_{R})\widehat{\varepsilon}\|_{\dot{H}^{1}}$ may not go to zero. Also, $M((1-\chi_{R})\widehat{\varepsilon})$ can be large. Instead, we have good control of $E((1-\chi_{R})\widehat{\varepsilon})$. Indeed, we will have a dichotomy: either

or

In fact, if the former is false, then the latter holds true sequentially in time. However, we will prove the latter holds for all $t<T$. We refer to this as the no-return property, (H1) in Lemma 4.2. We prove (H1) by observing the difference in exterior mass between sequences depending on whether $E((1-\chi_{R})\widehat{\varepsilon})\ll\|(1-\chi_{R})\widehat{\varepsilon}\|_{\dot{H}^{1}}^{2}$ holds or not. Hence, extracting multi-soliton configuration benefits from two levels of conservation laws, mass and energy. Thanks to no-return property, we do not get through time-sequential soliton resolution, but directly prove continuous-in-time soliton resolution.

In the former case of the dichotomy, we have a good quantitative estimate for $\|(1-\chi_{R})\widehat{\varepsilon}\|_{\dot{H}^{1}}$ and so we can verify that $[\widehat{\varepsilon}]_{\lambda,\gamma,x}$ converges to an asymptotic profile. This ends the soliton decomposition. In the latter case, we can reapply the decomposition to extract the second soliton from $(1-\chi_{R})\widehat{\varepsilon}$ and arrive at the above dichotomy for the outer radiation part. We can iterate this procedure. Since the mass drops by $M(Q)$ at each step, it ends at finitely many steps. At last, we arrive at the multi-soliton configuration:

The radiation $\varepsilon_{N}(t)$ with a uniform bound $\|\varepsilon_{N}(t)\|_{H^{1}}\lesssim 1$ converges to an asymptotic profile $z^{*}$ in $L^{2}$. Along the way, we also prove behaviors of the modulation parameters, $\lambda_{j}(t)\lesssim T-t$ and $\lim_{t\to T}x_{j}(t)=x_{j}(T)<\infty$. Finally, it is noteworthy that we verify a nonradial version of no bubble-tree condition, Proposition 4.4:

This is a consequence of $\|Q\widehat{\varepsilon}(t)\|_{L^{2}}\lesssim E(Q+\widehat{\varepsilon})$ in (1.12), which provides a quantitative bound on the interaction between soliton and $\widehat{\varepsilon}(t)$. We are not sure if such no-bubble tree property is a special feature of this model. One can observe this no-bubble tree condition is consistent with the gauge transform between (CM-DNLS) and ($\mathcal{G}$-CM) (Remark 5.2).

In Section 2, we introduce notation and preliminaries for our analysis. In Section 3, we review a standard variational argument and prove the energy bubbling estimate, which is a core proposition of our analysis. In Section 4, we prove the multi-soliton configuration via an induction argument. In this step, we also show the no-return property and confirm that the multi-soliton configuration holds true continuously in time. In Section 5, we complete the proofs of main theorems by applying the pseudo-conformal transform and the gauge transform.

The authors are partially supported by the National Research Foundation of Korea, NRF-2019R1A5A1028324 and NRF-2022R1A2C1091499. The authors appreciate Kihyun Kim for helpful comments on the first manuscript.

In this section, we collect notations and frequently used formulas. For quantities $A\in\mathbf{\mathbb{C}}$ and $B\geq 0$, we write $A\lesssim B\Leftrightarrow A=O(B)$ if $|A|\leq CB$ holds for some implicit constant $C$. For $A,B\geq 0$, we write $A\sim B$ if $A\lesssim B$ and $B\lesssim A$. If $C$ depends on some parameters, then we write them as a subscript. For $A=A(t)\in\mathbf{\mathbb{C}},B=B(t)>0$, we write $A=o_{t\to T}(B)$ if for any $\epsilon>0$ there exists $\delta>0$ such that if $0<T-t<\delta$, then $|A|\leq\epsilon B$. We also write simply $A\ll B$. When the quantities are function of time, the estimates are uniform in time, unless stated otherwise.

Denote a smooth cut-off function by $\chi_{R}=\chi(\frac{x}{R})$ where $\chi\in C^{\infty}$ with $\text{supp}\chi\in[-2,2]$ and $\chi\equiv 1$ on $[-1,1]$. It is possible to choose $\chi$ such that $|\chi^{\prime}|^{2}\leq C^{\prime}_{\chi}\chi$ for some $C^{\prime}_{\chi}>0$. We will also frequently use the outer cut-off

and a truncation on the centers of solitons, $\Phi_{R}=\prod_{j=1}^{N}\varphi_{R}(\cdot-x_{j})$. We also use the sharp cut-off on a set $A$ by ${\bf 1}_{A}$. We write the inhomogeneous weight by $\langle x\rangle\coloneqq(1+x^{2})^{\frac{1}{2}}$.

The Fourier transform (on $\mathbf{\mathbb{R}}$) is denoted by

with its inverse $\mathcal{F}^{-1}(f)(x)\coloneqq\tfrac{1}{2\pi}\int_{\mathbf{\mathbb{R}}}\widehat{f}(\xi)e^{ix\xi}d\xi$. We denote $|D|$ by the Fourier multiplier operator with symbol $|\xi|$, that is, $|D|\coloneqq\mathcal{F}^{-1}|\xi|\mathcal{F}$, and then $|D|=\partial_{x}\mathcal{H}=\mathcal{H}\partial_{x}$ where $\mathcal{H}$ is the Hilbert transform:

We note that $[\mathcal{H},i]=0$. We denote by $\Pi_{+}$ the Cauchy–Szegő projection from $L^{2}(\mathbb{R})$ onto the Hardy space $L_{+}^{2}(\mathbb{R})=\{f\in L^{2}(\mathbf{\mathbb{R}}):\mathrm{supp}\widehat{f}\subset[0,\infty)\}$:

Then, we have $\Pi_{+}=\frac{1}{2}(1+i\mathcal{H}).$

We use the real inner product $(\cdot,\cdot)_{r}$ given by $(f,g)_{r}=\int_{\mathbf{\mathbb{R}}}\text{Re}(\overline{f}g)dx$. For given modulation parameters, $(\lambda,\gamma,x)\in\mathbf{\mathbb{R}}_{+}\times\mathbf{\mathbb{R}}/2\pi\mathbf{\mathbb{Z}}\times\mathbf{\mathbb{R}}$, we denote a (inversely) modulated function by

Then, we have $[[f]_{\lambda,\gamma,x}]_{\lambda,\gamma,x}^{-1}=[[f]_{\lambda,\gamma,x}^{-1}]_{\lambda,\gamma,x}=f$ and $[f]_{\lambda,\gamma,x}^{-1}=[f]_{\lambda^{-1},-\gamma,-\lambda^{-1}x}$. We also note that

We have some algebraic identities with respect to $\mathcal{H}$,

1. (1)

   For $f,g\in H^{\frac{1}{2}+}$, we have

   $\displaystyle fg=\mathcal{H}f\cdot\mathcal{H}g-\mathcal{H}(f\cdot\mathcal{H}g+\mathcal{H}f\cdot g)$

   (2.2)

   in a pointwise sense.

2. (2)

   For $f\in\langle x\rangle^{-1}L^{2}$, we have

   $\displaystyle[x,\mathcal{H}]f(x)=\frac{1}{\pi}\int_{\mathbb{R}}f(y)dy.$

   (2.3)

3. (3)

   If $f\in H^{1}$ and $\partial_{x}f\in\langle x\rangle^{-1}L^{2}$, then we have

   $\displaystyle\partial_{x}[x,\mathcal{H}]f=[x,\mathcal{H}]\partial_{x}f=0,\quad\text{i.e.,}\quad[\mathcal{H}\partial_{x},x]f=\mathcal{H}f.$

   (2.4)