Well-Posedness of the Schr\"odinger - Intermediate Long Wave system

Diego F. Correa-Casta\~neda

arxiv: 2606.22763 · v1 · pith:KRLYMLJInew · submitted 2026-06-22 · 🧮 math.AP

Well-Posedness of the Schr\"odinger - Intermediate Long Wave system

Diego F. Correa-Casta\~neda This is my paper

Pith reviewed 2026-06-26 08:10 UTC · model grok-4.3

classification 🧮 math.AP

keywords well-posednessSchrödinger equationIntermediate Long Wave equationSobolev spacesBourgain spacesgauge transformationnonlinear dispersive PDE

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The pith

The coupled Schrödinger-Intermediate Long Wave system is locally and globally well-posed in low-regularity Sobolev spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes local and global well-posedness for the initial-value problem of the Schrödinger-Intermediate Long Wave system in Sobolev spaces with low regularity. It shows that solutions exist, are unique, and depend continuously on the initial data under these conditions. A reader would care because the system models the interaction of long and short waves, and low-regularity results extend the class of physically relevant initial data for which the model remains mathematically consistent. The argument proceeds by combining energy estimates with Bourgain spaces and a gauge transformation to control the nonlinear terms.

Core claim

The authors prove that the initial value problem associated with the coupled Schrödinger and Intermediate Long Wave system is locally and globally well-posed in Sobolev spaces of low regularity. Their approach relies on the use of energy estimates, Bourgain spaces, and Tao's gauge transformation to close the necessary estimates for the coupled nonlinear system.

What carries the argument

The coupled Schrödinger-ILW system together with Bourgain spaces and Tao's gauge transformation, which together close the energy estimates at low regularity.

If this is right

Local solutions exist and are unique for initial data in the indicated low-regularity spaces.
Global solutions exist for the same data classes.
The solution map is continuous with respect to the initial data in those spaces.
The proof techniques control the nonlinear interaction between the short-wave Schrödinger component and the long-wave ILW component.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar gauge and space techniques may apply to other coupled long-short wave models that share the same structural properties.
The low-regularity threshold obtained here sets a benchmark for numerical schemes that aim to simulate the system from rough data.
Global existence opens the possibility of studying long-time asymptotic behavior such as scattering or soliton formation in this model.

Load-bearing premise

The coupled system satisfies the structural conditions that allow energy estimates, Bourgain spaces, and the gauge transformation to close at low regularity.

What would settle it

An explicit initial datum in a low-regularity Sobolev space for which either no solution exists, the solution is non-unique, or continuous dependence on data fails.

read the original abstract

The system coupled by the Schr\"odinger equation and the Intermediate Long Wave (ILW) equation is a particular model describing the interaction of long and short waves. We prove local and global well-posedness of the initial value problem associated with this system in Sobolev spaces of low regularity. Our approach relies on the use of energy estimates, Bourgain spaces, and Tao's gauge transformation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Standard application of Bourgain spaces and gauge transform to Schrödinger-ILW yields local well-posedness, but the global claim at low regularity lacks visible support for norm control.

read the letter

The paper proves local and global well-posedness for the coupled Schrödinger-ILW system in low-regularity Sobolev spaces. It applies energy estimates, Bourgain spaces, and Tao's gauge transformation to this particular long-short wave model.

The result is new for this system. The methods are established for dispersive equations, so the local part probably follows by adapting existing arguments to the coupling. That counts as honest progress on a concrete model even if the techniques themselves are not novel.

The soft spot is the global extension. Local well-posedness closes with the listed tools, but global well-posedness at low s requires an a priori bound that stops the Sobolev norm from blowing up. Conserved mass and energy usually control only L^2 or H^1, and below the energy threshold one typically needs almost-conservation or I-method estimates. The abstract gives no sign that such a bound is derived, so the global claim rests on whatever details appear in the full text. If those details are missing or insufficient, the result stops at local.

The math is direct with no circularity or free parameters. The citation pattern is normal for this subfield.

This paper is for specialists in dispersive PDEs who track well-posedness results for coupled systems. A reader already working on ILW or Schrödinger interactions could use it as a reference point.

It deserves peer review. The claim is concrete and the methods fit the problem; a referee can check whether the global step actually closes.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove local and global well-posedness of the initial-value problem for the coupled Schrödinger-Intermediate Long Wave system in low-regularity Sobolev spaces. The approach is stated to rely on energy estimates, Bourgain spaces, and Tao's gauge transformation.

Significance. If the central claims hold with complete proofs, the result would extend the low-regularity well-posedness theory for long-short wave interaction models, a topic of interest in dispersive PDEs.

major comments (1)

[Abstract] Abstract: the global well-posedness claim at low regularity is not accompanied by any indication of an a priori bound preventing Sobolev-norm blow-up. Standard conserved quantities (mass, energy) control only up to H^1; for s below the energy threshold, almost-conservation or I-method arguments are typically required, yet none are mentioned among the listed methods.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for greater clarity in the abstract regarding the global well-posedness result. We address the comment point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the global well-posedness claim at low regularity is not accompanied by any indication of an a priori bound preventing Sobolev-norm blow-up. Standard conserved quantities (mass, energy) control only up to H^1; for s below the energy threshold, almost-conservation or I-method arguments are typically required, yet none are mentioned among the listed methods.

Authors: We agree that the abstract is too brief and does not indicate how the a priori bounds are obtained to prevent Sobolev-norm blow-up at low regularity. The manuscript establishes global well-posedness by combining the listed techniques (energy estimates, Bourgain spaces, and Tao's gauge transformation) in a manner that yields the requisite control below the energy space; however, this is not made explicit in the abstract. We will revise the abstract to clarify the approach used for the global result. revision: yes

Circularity Check

0 steps flagged

No circularity: direct proof via standard analytic tools

full rationale

The paper claims local and global well-posedness via energy estimates, Bourgain spaces, and Tao's gauge transformation. No equations, parameters, or uniqueness statements reduce by construction to fitted inputs, self-citations, or renamed ansatzes. The derivation is presented as a self-contained application of existing techniques to the coupled system, with no load-bearing step that collapses to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Insufficient information from the abstract to identify any free parameters, non-standard axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5584 in / 986 out tokens · 29548 ms · 2026-06-26T08:10:27.355722+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

35 extracted references · 6 canonical work pages

[1]

AcknowledgmentsThis paper is part of my Ph.D

In the appendix, we make the energy estimates, which are used to prove local well- posedness for smooth initial data. AcknowledgmentsThis paper is part of my Ph.D. thesis at IMPA under the guidance of my advisor Felipe Linares. I want to take the opportunity to express my sincere grati- tude to him. The author was supported by CNPq-Brazil
[2]

Notation For any positive numbersaandb, the notationa≲bmeans that there exists a positive constantcsuch thata≤cb

Notation, function spaces and preliminary estimates 2.1. Notation For any positive numbersaandb, the notationa≲bmeans that there exists a positive constantcsuch thata≤cb. We also writea∼bwhena≲bandb≲a. Moreover, if α∈R,α + andα − will denote a number slightly greater and lesser thanα, respectively. Foru=u(t, x)∈S ′(R2),Fu= ˆuwill denote its space-time Fou...
[3]

The gauge transformation In the upcoming sections, we present the necessary tools for the proof of Theorem 1

Apriori estimates 3.1. The gauge transformation In the upcoming sections, we present the necessary tools for the proof of Theorem 1. Following [20], we first construct a spatial primitiveFofv(i.e.F x =v) that satisfies equation (1.20). Considerψ∈C ∞ 0 (R)such that R R ψ(y)dy= 1and define F(x, t) := Z R ψ(y) Z x y v(t, z)dz dy+G(t),(3.1) whereG=G(t)is a fu...
[4]

The nonlinear terms inω We now consider the equation (1.22)

Bilinear estimates 4.1. The nonlinear terms inω We now consider the equation (1.22). Applying the linear estimates (2.14) and (2.17) on the Duhamel formulation of (1.22), we get ∥w∥Y s, 1 2 (T) ≲∥w(0)∥ H s + χ[0,T] (t)Nδ(u, v, w) Y s,− 1 2 .(4.1) We aim to estimate each term of the right hand side of (4.1). At first, By fractional Leibniz rule, ∥w(0)∥H sx...
[5]

Proposition 2.Let ˜Nδ(v, w) :=ν 2∂xP+hi ∂−1 x wP −∂xv +ν 2∂xP+hi Ploe− ν 2 iF P−∂xv − ν2 2 i∂xP+hi e− ν 2 iF Qδv

On the other hand, in [20] the first three terms ofN δ(u, v, w)were considered. Proposition 2.Let ˜Nδ(v, w) :=ν 2∂xP+hi ∂−1 x wP −∂xv +ν 2∂xP+hi Ploe− ν 2 iF P−∂xv − ν2 2 i∂xP+hi e− ν 2 iF Qδv . Then, for all0≤s≤ 1 2 there existsκ >0such that χ[0,T] (t) ˜Nδ(v, w) Y s,− 1 2 ≲∥v∥ 2 L4 T ,x + T κ∥v∥L∞ T L2x +∥v∥ L4 T ,x +∥v∥ X −1,1 BO (T) ∥w∥ X s, 1 2 BO (T)...
[6]

For the cases= 0we use Cauchy Schwartz, and the classic Sobolev and Bourgain embedding to obtain v|u|2 L2 T L2x ≲T 1 4 ∥v∥L4 T ,x ∥u∥2 X 1 2 ,b S (T)

For allb > 1 2, χ[0,T] (t)∂xP+hi e− ν 2 iF |u|2 Y s,− 1 2 ≲∥u∥ X 1 2 ,b S (T) ∥v∥L4 T W s,4 x + ∥v∥L4 T ,x + 1 ∥u∥ X s+ 1 2 ,b S (T) .(4.6) Proof.There exists a0< ϵ < 1 2 such that ∂xP+hi e− ν 2 iF |u|2 Y s,− 1 2 (T) ≲ ∂xP+hi e− ν 2 iF |u|2 X s,− 1 2 +ϵ BO (T) .(4.7) Differentiating and using inequality (2.22), ∂xP+hi e− ν 2 iF |u|2 X s,− 1 2 +ϵ BO (T) (4...
[7]

Proof.Letβ, γ∈Cbe such thatx 2 +bx+c= (x−β)(x−γ)for allx∈R

Then Z R dx ⟨x⟩2ℓ1 ⟨x2 +bx+c⟩ ℓ2 ≲1,(4.21) where the constant is independent frombandc. Proof.Letβ, γ∈Cbe such thatx 2 +bx+c= (x−β)(x−γ)for allx∈R. Note that for anyy∈R,|x−iy|= p x2 +y 2 ≤ |x|. Then Z R dx ⟨x⟩2ℓ1 ⟨x2 +bx+c⟩ ℓ2 = Z R dx ⟨x⟩2ℓ1 ⟨(x−β)(x−γ)⟩ ℓ2 = "Z |x|≤|x−β|,|x−γ| + Z |x−β|≤|x|,|x−γ| + Z |x−γ|≤|x|,|x−β| # dx ⟨x⟩2ℓ1 ⟨(x−β)(x−γ)⟩ ℓ2 ≲1, which...
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A priori estimates for smooth solutions Let us first derive a priori estimates for smooth solutions to the system (1.19)

Well-posedness 5.1. A priori estimates for smooth solutions Let us first derive a priori estimates for smooth solutions to the system (1.19). Let(u, v) a smooth solution of (1.19) with initial data(u(0), v(0)) := (u 0, v0)andw:=w(u, v)the gauge transformation defined in (3.3). GivenT∈(0, T ∗), whereT ∗ is the maximal time of existence given in Theorem 10,...
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Energy Estimates for the Solutions of(1.1) Let(u, v)∈C [0, T] ;H s+ 1 2 (R)×H s(R) be a solution of the system (1.1), where s > 1 2 andT >0

Appendix: Energy Estimates 6.1. Energy Estimates for the Solutions of(1.1) Let(u, v)∈C [0, T] ;H s+ 1 2 (R)×H s(R) be a solution of the system (1.1), where s > 1 2 andT >0. In this case, the usual energyE s c(t)can be defined for allt∈[0, T]by Es c(t) :=∥u(t)∥ 2 L2x + D s+ 1 2 x u(t) 2 L2x +∥v(t)∥ 2 L2x + 1 2 ∥Ds xv(t)∥2 L2x . For the solutions of (1.14),...
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First, we multiply the second equation of (6.9) by2zand integrate inRto obtain ∂t∥z∥2 L2x =− ρ 2 Z R z2vxx dx+2νRe Z R zuxw dx+2ν Z R zuwx dx:=I 1+I 2+I 3.(6.18) Note that I1 ≤ |ρ| 2 ∥vx∥L∞x ∥z∥2 L2x . Using Cauchy-Schwarz and Sobolev embedding, we get I2 ≤2|ν|∥z∥ L2x∥uxw∥L2x ≤2|ν|∥z∥ L2x∥ux∥L2+ x ∥w∥L∞− x ≲∥u∥ H s+ 1 2x ∥z∥L2x∥w∥ H 1 2x , and, I3 ≤2|ν|∥z...
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In the system(1.1), suppose thatα= 1andβ=ν. For any (u0, v0)∈H s+ 1 2 (R)×H s(R), there exist a positive timeT=T ∥(u0, v0)∥H s+ 1 2 ×H s and a unique maximal solution(u, v)of the system(1.1)inC [0, T ∗], H s+ 1 2 (R)×H s(R) , withT ∗ > T, such that(u(0), v(0)) = (u 0, v0). If the maximal time of existenceT ∗ is finite, then lim t↑T ∗ ∥(u(t), v(t))∥H s+ 1 ...
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[1] [1]

AcknowledgmentsThis paper is part of my Ph.D

In the appendix, we make the energy estimates, which are used to prove local well- posedness for smooth initial data. AcknowledgmentsThis paper is part of my Ph.D. thesis at IMPA under the guidance of my advisor Felipe Linares. I want to take the opportunity to express my sincere grati- tude to him. The author was supported by CNPq-Brazil

[2] [2]

Notation For any positive numbersaandb, the notationa≲bmeans that there exists a positive constantcsuch thata≤cb

Notation, function spaces and preliminary estimates 2.1. Notation For any positive numbersaandb, the notationa≲bmeans that there exists a positive constantcsuch thata≤cb. We also writea∼bwhena≲bandb≲a. Moreover, if α∈R,α + andα − will denote a number slightly greater and lesser thanα, respectively. Foru=u(t, x)∈S ′(R2),Fu= ˆuwill denote its space-time Fou...

[3] [3]

The gauge transformation In the upcoming sections, we present the necessary tools for the proof of Theorem 1

Apriori estimates 3.1. The gauge transformation In the upcoming sections, we present the necessary tools for the proof of Theorem 1. Following [20], we first construct a spatial primitiveFofv(i.e.F x =v) that satisfies equation (1.20). Considerψ∈C ∞ 0 (R)such that R R ψ(y)dy= 1and define F(x, t) := Z R ψ(y) Z x y v(t, z)dz dy+G(t),(3.1) whereG=G(t)is a fu...

[4] [4]

The nonlinear terms inω We now consider the equation (1.22)

Bilinear estimates 4.1. The nonlinear terms inω We now consider the equation (1.22). Applying the linear estimates (2.14) and (2.17) on the Duhamel formulation of (1.22), we get ∥w∥Y s, 1 2 (T) ≲∥w(0)∥ H s + χ[0,T] (t)Nδ(u, v, w) Y s,− 1 2 .(4.1) We aim to estimate each term of the right hand side of (4.1). At first, By fractional Leibniz rule, ∥w(0)∥H sx...

[5] [5]

Proposition 2.Let ˜Nδ(v, w) :=ν 2∂xP+hi ∂−1 x wP −∂xv +ν 2∂xP+hi Ploe− ν 2 iF P−∂xv − ν2 2 i∂xP+hi e− ν 2 iF Qδv

On the other hand, in [20] the first three terms ofN δ(u, v, w)were considered. Proposition 2.Let ˜Nδ(v, w) :=ν 2∂xP+hi ∂−1 x wP −∂xv +ν 2∂xP+hi Ploe− ν 2 iF P−∂xv − ν2 2 i∂xP+hi e− ν 2 iF Qδv . Then, for all0≤s≤ 1 2 there existsκ >0such that χ[0,T] (t) ˜Nδ(v, w) Y s,− 1 2 ≲∥v∥ 2 L4 T ,x + T κ∥v∥L∞ T L2x +∥v∥ L4 T ,x +∥v∥ X −1,1 BO (T) ∥w∥ X s, 1 2 BO (T)...

[6] [6]

For the cases= 0we use Cauchy Schwartz, and the classic Sobolev and Bourgain embedding to obtain v|u|2 L2 T L2x ≲T 1 4 ∥v∥L4 T ,x ∥u∥2 X 1 2 ,b S (T)

For allb > 1 2, χ[0,T] (t)∂xP+hi e− ν 2 iF |u|2 Y s,− 1 2 ≲∥u∥ X 1 2 ,b S (T) ∥v∥L4 T W s,4 x + ∥v∥L4 T ,x + 1 ∥u∥ X s+ 1 2 ,b S (T) .(4.6) Proof.There exists a0< ϵ < 1 2 such that ∂xP+hi e− ν 2 iF |u|2 Y s,− 1 2 (T) ≲ ∂xP+hi e− ν 2 iF |u|2 X s,− 1 2 +ϵ BO (T) .(4.7) Differentiating and using inequality (2.22), ∂xP+hi e− ν 2 iF |u|2 X s,− 1 2 +ϵ BO (T) (4...

[7] [7]

Proof.Letβ, γ∈Cbe such thatx 2 +bx+c= (x−β)(x−γ)for allx∈R

Then Z R dx ⟨x⟩2ℓ1 ⟨x2 +bx+c⟩ ℓ2 ≲1,(4.21) where the constant is independent frombandc. Proof.Letβ, γ∈Cbe such thatx 2 +bx+c= (x−β)(x−γ)for allx∈R. Note that for anyy∈R,|x−iy|= p x2 +y 2 ≤ |x|. Then Z R dx ⟨x⟩2ℓ1 ⟨x2 +bx+c⟩ ℓ2 = Z R dx ⟨x⟩2ℓ1 ⟨(x−β)(x−γ)⟩ ℓ2 = "Z |x|≤|x−β|,|x−γ| + Z |x−β|≤|x|,|x−γ| + Z |x−γ|≤|x|,|x−β| # dx ⟨x⟩2ℓ1 ⟨(x−β)(x−γ)⟩ ℓ2 ≲1, which...

[8] [8]

A priori estimates for smooth solutions Let us first derive a priori estimates for smooth solutions to the system (1.19)

Well-posedness 5.1. A priori estimates for smooth solutions Let us first derive a priori estimates for smooth solutions to the system (1.19). Let(u, v) a smooth solution of (1.19) with initial data(u(0), v(0)) := (u 0, v0)andw:=w(u, v)the gauge transformation defined in (3.3). GivenT∈(0, T ∗), whereT ∗ is the maximal time of existence given in Theorem 10,...

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Energy Estimates for the Solutions of(1.1) Let(u, v)∈C [0, T] ;H s+ 1 2 (R)×H s(R) be a solution of the system (1.1), where s > 1 2 andT >0

Appendix: Energy Estimates 6.1. Energy Estimates for the Solutions of(1.1) Let(u, v)∈C [0, T] ;H s+ 1 2 (R)×H s(R) be a solution of the system (1.1), where s > 1 2 andT >0. In this case, the usual energyE s c(t)can be defined for allt∈[0, T]by Es c(t) :=∥u(t)∥ 2 L2x + D s+ 1 2 x u(t) 2 L2x +∥v(t)∥ 2 L2x + 1 2 ∥Ds xv(t)∥2 L2x . For the solutions of (1.14),...

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First, we multiply the second equation of (6.9) by2zand integrate inRto obtain ∂t∥z∥2 L2x =− ρ 2 Z R z2vxx dx+2νRe Z R zuxw dx+2ν Z R zuwx dx:=I 1+I 2+I 3.(6.18) Note that I1 ≤ |ρ| 2 ∥vx∥L∞x ∥z∥2 L2x . Using Cauchy-Schwarz and Sobolev embedding, we get I2 ≤2|ν|∥z∥ L2x∥uxw∥L2x ≤2|ν|∥z∥ L2x∥ux∥L2+ x ∥w∥L∞− x ≲∥u∥ H s+ 1 2x ∥z∥L2x∥w∥ H 1 2x , and, I3 ≤2|ν|∥z...

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