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On a system of Schr\"odinger equations with general quadratic-type nonlinearities

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Pith's one-line read This paper proves a sharp dichotomy for quadratic Schrödinger systems: global existence versus finite-time blow-up is decided by ground states of an associated elliptic system.

desk verdict Solid general framework for quadratic Schrödinger systems, but the n=5 sharp threshold as printed uses the wrong energy and needs correction. read the letter →

arxiv 1908.04159 v1 pith:Q6CGKCMJ submitted 2019-08-12 math.AP

Norman Noguera , Ademir Pastor This is my paper · ORCID
classification math.AP MSC 35Q5535B4435B3535A15
keywords quadraticSchrödingersystemsglobalwell-posednessblow-upgroundstatesvirialidentityconcentration-compactnessstabilityofstandingwavessharpthreshold
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Schrödinger systems with quadratic nonlinearities appear as models of second-harmonic generation and three-wave mixing in nonlinear optics, where the basic question is whether an optical beam propagates forever or collapses. This paper establishes that, for a broad class of such systems -- no explicit form of the nonlinearity is assumed, only structural conditions -- the dividing line between global existence and finite-time blow-up is drawn by ground states of an associated elliptic system. In dimension 4 the criterion is charge below the ground-state charge; in dimension 5, under an energy bound, the product of charge and kinetic energy must stay below the ground-state value. At the threshold in dimension 4 the paper constructs explicit blow-up solutions, showing the criterion is sharp. It also proves that ground states are stable in dimensions 1-3 and unstable in 4 and 5.

What carries the argument

The engine is the Weinstein functional $J(u)=Q(u)^{3/2-n/4}K(u)^{n/4}/|P(u)|$, with $Q$ the charge, $K$ the kinetic energy, and $P$ the nonlinear potential. Minimizing $J$ over nonzero-potential configurations produces non-negative radially symmetric ground states $\psi$ of the elliptic system; the minimum value supplies the sharp constant in the Gagliardo-Nirenberg inequality $|P(u)|\le C_{\mathrm{op}} Q(u)^{3/2-n/4}K(u)^{n/4}$, with $C_{\mathrm{op}}$ measured by $Q(\psi)$. The virial identity then controls the variance $V(t)=\sum_k \frac{\alpha_k^2}{\gamma_k}\|xu_k(t)\|_{L^2}^2$, whose second derivative is linear in $K(u)$ in the critical dimension and therefore turns the threshold inequalities into a positivity condition that decides blow-up. In dimension 4, pseudo-conformal invariance maps ground states into explicit solutions that blow up at exactly $Q(u_0)=Q(\psi)$. Concentration-compactness at fixed charge transfers the variational picture into orbital stability of the ground-state set.

What would settle it

For a concrete system satisfying (H1)-(H8), take n = 4 and initial data with Q(u0) below Q(psi): the theorem predicts a global H1 solution, so observing finite-time blow-up for such data would refute the criterion. Alternatively, at exactly Q(u0) = Q(psi), the pseudo-conformal construction gives an explicit blow-up solution; direct differentiation of its phase and nonlinear terms would confirm or refute the claimed sharpness.

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Extended reading notes

Core claim

Let $\psi$ be a ground state of the elliptic system $-\gamma_k\Delta\psi_k + \frac{\alpha_k^2}{\gamma_k}\psi_k = f_k(\psi)$. The central discovery is a pair of sharp threshold theorems. For $n=4$, every $H^1$ solution with $Q(u_0)<Q(\psi)$ is global, and the bound is sharp: solutions with $Q(u_0)=Q(\psi)$ obtained from $\psi$ by the pseudo-conformal transformation blow up in finite time. For $n=5$, assuming $Q(u_0)E(u_0)<Q(\psi)E(\psi)$ (with the energy evaluated at $\beta=0$), the inequality $Q(u_0)K(u_0)<Q(\psi)K(\psi)$ forces global existence, while the reverse inequality $Q(u_0)K(u_0)>Q(\psi)K(\psi)$ forces finite-time blow-up for initial data with finite variance or radial symmetry. The proof rests on a sharp Gagliardo-Nirenberg inequality whose best constant is expressed through the ground-state charge, together with the virial identity $V''(t)=2nE_0-2nL(u)+2(4-n)K(u)$, which converts the threshold into convexity of the variance.

Load-bearing premise

The dichotomy rests on the gauge condition (H4), which says the nonlinear potential is unchanged when each component is rotated by its own characteristic phase; if that condition fails, the conserved charge, the virial identity, and the pseudo-conformal symmetry used to prove sharpness all break down.

Editorial extensions

If this is right

  • In dimension 4, initial data with charge below the ground-state charge never collapse; the equality case contains finite-time blow-up, so the condition $Q(u_0)<Q(\psi)$ is both sufficient and sharp.
  • In dimension 5, within the energy window $Q(u_0)E(u_0)<Q(\psi)E(\psi)$, the sign of $Q(u_0)K(u_0)-Q(\psi)K(\psi)$ decides between global existence and finite-time blow-up.
  • For dimensions $1\le n\le 3$, the set of ground states is orbitally stable in $H^1$; for $n=4,5$, the standing waves built from ground states are unstable, since arbitrary nearby data can blow up.
  • The sharp Gagliardo-Nirenberg constant for a general quadratic-type nonlinearity is given in terms of the ground-state charge, so the threshold can be computed once ground states are known.
  • The structural hypotheses allow the same threshold theory to cover different quadratic models without writing the nonlinearity in an explicit algebraic form.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Dropping the gauge condition, as occurs in non-resonant quadratic systems, removes the conserved charge and the virial identity that drive this criterion; the threshold would then have to be sought by a different mechanism.
  • The pseudo-conformal blow-up at the threshold in dimension 4 has an explicit form with amplitude growing like $(T-t)^{-2}$, so the sharp equality case can be simulated in concrete quadratic-media models.
  • The n = 5 theorem proves global existence but not scattering; by analogy with known scattering results below the ground state for mass-resonance quadratic systems, extending the threshold to asymptotic scattering is a natural next step.
  • The variational scheme -- minimizing the Weinstein functional and applying concentration-compactness -- may yield analogous thresholds for other super-modular nonlinearities beyond the quadratic class.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies a system of N coupled Schrödinger equations with quadratic-type nonlinearities, with general interaction terms encoded by a potential F satisfying assumptions (H1)-(H8). The main results are: local and global well-posedness in L2 and H1 in dimensions 1≤n≤6; existence of ground states for the associated elliptic system via a Weinstein-type minimization; and a sharp global-existence versus blow-up dichotomy in the L2-critical dimension n=4 and in the L2-supercritical/H1-subcritical dimension n=5, expressed in terms of charge/energy/momentum of the ground states. The paper also proves stability of the ground-state set for n=1,2,3 by concentration-compactness and instability for n=4,5. The proofs are mostly self-contained and rely on Strichartz estimates, virial identities, pseudo-conformal invariance (n=4), and a sharp Gagliardo-Nirenberg inequality whose constant is obtained from the constructed ground states.

Significance. If the statements are made precise, the paper provides a useful general framework for quadratic Schrödinger systems with a wide class of nonlinearities, going beyond the specific three-wave systems studied earlier. The construction of ground states by minimizing the Weinstein functional, the explicit relation between the sharp constants and ground states, and the stability/instability analysis are valuable and are carried out in considerable detail. The paper is commendable for not assuming an explicit form of the nonlinearity and for proving the key variational identities. The main reservation is that the sharp n=5 dichotomy and the n=4 sharpness statement are formulated in a way that appears to require the restrictive hypothesis β=0 (or β_k=α_k^2/γ_k in the n=4 sharpness transfer), while the manuscript presents the results without that restriction in some statements. This does not destroy the core ideas, but it affects the advertised scope of the central theorem.

major comments (3)
  1. [§5.2–5.3 (Theorems 5.9(ii) and 5.16)] The n=5 half of the sharp dichotomy is internally inconsistent as stated. Theorem 5.9(ii) defines E as the energy (3.5) with β=0, but the proof of (5.14) uses the conservation law from Lemma 3.13. For β>0, the β=0 energy is not conserved: L(u(t))=Σβ_k‖u_k(t)‖² is not constant, since only the weighted charge Q is conserved under (H4). Thus the bound K(u(t))≤E0+2CopQ0^{1/4}K(u(t))^{5/4} cannot be derived with E0 equal to the β=0 energy unless β=0 in the system. Conversely, Theorem 5.16, presented for the general system (1.1), uses the full virial identity V''(t)=10E(u0)-10L(u(t))-2K(u(t)) and then invokes the ground-state identity E(ψ)=K(ψ)/5. That identity is valid for ψ∈G(1,0) only when E is the β=0 energy; with the full energy E=K+L-2P it is false when L(ψ)>0. Therefore either the whole n=5 sharp dichotomy is restricted to β=0, in which case this restriction must be stated prominently, or the hypotheses of Theorem 5.16 must be reworked with the correct ground-state energy values for nonzero β. This is a load-bearing issue for the paper's central claim.
  2. [§5.3.1 (Theorem 5.14, Corollary 5.15)] The sharpness of the n=4 threshold in Theorem 5.9(i) is only proved for a restricted set of parameters. Lemma 5.12 establishes pseudo-conformal invariance for β_k=0, and Corollary 5.15 uses Remark 5.13 to transfer the blow-up construction to the case β_k=α_k^2/γ_k. No construction is given for other values of β_k. The statement of Theorem 5.9(i) and the abstract's claim of a sharp criterion should therefore be qualified: at the threshold Q(u0)=Q(ψ), blow-up is demonstrated only for the β=0 system and, after the gauge shift, for β_k=α_k^2/γ_k. Without this qualification the reader may infer a sharpness that the paper does not establish for general β.
  3. [§5.2, proof of Theorem 5.9 and §4, Theorem 4.11] The proof of Theorem 5.9 states without proof that 'ξ0 in (3.8) and ξ1 in (4.14) are the same' and then uses Corollary 4.12. This identification is not immediate from Theorem 4.11 as printed, because ξ0 is defined with |P(u)| and the functional Re∫F, while ξ1 is defined on the positive cone with P=∫F. The equality is true when F satisfies (H6) and (H7) and b_k=α_k^2/γ_k, but it should be stated and proved as a lemma before it is used in the sharp constant computation. This is a missing supporting step rather than an error.
minor comments (5)
  1. [Abstract and §1] The abstract contains a grammatical error: 'in terms on the ground state solutions' should be 'in terms of the ground state solutions'.
  2. [§2, assumption (H2)] The two displayed inequalities in (H2) appear identical in the manuscript. Please ensure that the second one is the estimate for ∂/∂z̄_m and that the notation for the conjugate derivative is clear, since Lemma 2.4 relies on both types of derivatives.
  3. [§5.1, Corollary 5.3] The notation Q(xu(t)) is used although Q is defined in (3.4) for vector functions u=(u1,...,ul). Please clarify that x u means (x u1,...,x ul) and that Q(xu)=Σ(α_k^2/γ_k)‖x u_k‖².
  4. [§4, Lemma 4.13; §5, Theorem 5.11; §6, Lemma 6.18] Several results are stated with proofs deferred to 'standard arguments' or omitted entirely: Lemma 4.13 (regularity and exponential decay of elliptic solutions), Theorem 5.11 (blow-up for E0<0 or E0=0, P0<0), and Lemma 6.18 (properties of the functional T and the set M). These results are used in the instability proofs, and for a self-contained paper they should either be proved or accompanied by precise references with the adaptation to systems spelled out.
  5. [References] The paper cites [31] and [33] as prior specific models; it would be helpful for the reader if the introduction explicitly stated that the present assumptions (H1)-(H8) cover those models but are not restricted to them, and if the precise overlap with [21] and [20] on non-resonant systems were discussed in light of the gauge condition (H4).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sharp thresholds are expressed in terms of ground states that are constructed in the same paper by variational methods, not assumed as inputs.

full rationale

The paper's derivation chain is self-contained with respect to its central claims. The thresholds in Theorems 5.9, 5.15, and 5.16 are stated in terms of the charge, energy, and kinetic energy of ground states ψ of the elliptic system (5.10). These ground states are not imported from prior work or fitted to data; they are constructed in Section 4 by minimizing the Weinstein-type functional J (Theorem 4.11), and the sharp Gagliardo-Nirenberg constant is derived from that construction (Corollary 4.12). The global-existence and blow-up arguments then use conservation laws, virial identities, and Lemma 3.17, all proved in the paper under the stated hypotheses. The gauge condition (H4) is an explicit assumption that restricts the class of systems, not a result derived from the desired dichotomy. Self-citations [31] and [33] are used for contextual prior specific models and for standard technical ideas; they are not load-bearing substitutes for the general proofs here. The skeptic's concern about Theorem 5.9(ii) involving the β=0 energy while the proof uses the full energy is a consistency/correctness issue, not a circularity: it does not make the theorem's conclusion an input to its derivation. No fitted parameter is renamed as a prediction, and no uniqueness or ansatz is smuggled in via self-citation. Therefore the paper receives score 0.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

The central results rest on the structural hypotheses (H1)-(H8), especially the gauge condition (H4) and homogeneity (H5). These are stated assumptions, not derived facts. The other axioms are standard background from dispersive PDE theory.

assumptions (10)
  • standard math Strichartz estimates for the linear propagator e^{itα_k^{-1}(γ_kΔ-β_k)}.
    Used in Section 3 to close the contraction arguments for local well-posedness in L2 and H1. Cited to Cazenave [7, Theorem 2.3.3].
  • standard math Sobolev multiplication law H^{s1}×H^{s2}→H^s under the conditions of Lemma 2.14.
    Used in Lemma 2.15 to prove f_k maps H1 into H^{-1} continuously for n≤6.
  • standard math Compact embedding H^1_rd(R^n) into L^3(R^n) for 1≤n≤5.
    Essential for the compactness argument in the ground-state existence theorem (Theorem 4.11) and the concentration-compactness analysis (Lemma 6.21).
  • standard math Concentration-compactness principle of Lions.
    Framework for the stability proof in Section 6.1, used to rule out vanishing and dichotomy for minimizing sequences.
  • standard math Strong maximum principle for elliptic equations.
    Used to show the constructed ground state is positive componentwise in Theorem 4.11.
  • domain assumption (H4) Gauge condition: ReF(e^{iα_k/γ_k θ}z_k)=ReF(z).
    Powers conservation of charge, virial identities, and pseudo-conformal invariance. Restricts to mass-resonance-like systems.
  • domain assumption (H5) F is homogeneous of degree 3.
    Gives homogeneity degree 2 for the nonlinearities and the scaling used throughout.
  • domain assumption (H6) |Re∫F(u)|≤∫F(|u|).
    Used to show J(|ψ|)≤J(ψ), needed in the minimization of the Weinstein functional.
  • domain assumption (H7) F is real-valued on R^l and f_k≥0 on the positive cone.
    Used to identify f_k=∂F/∂x_k on real vectors and to apply the maximum principle for positivity.
  • ad hoc to paper (H8) F decomposes into super-modular terms vanishing on hyperplanes.
    Technical condition guaranteeing P(ψ*)≥P(ψ) under symmetric-decreasing rearrangement; not physically motivated and not satisfied by all quadratic-type nonlinearities.

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Pith. "Pith review of On a system of Schr\"odinger equations with general quadratic-type nonlinearities." pith.science (2026). https://pith.science/paper/Q6CGKCMJ

@misc{pith2026190804159,
  author       = {Pith},
  title        = {Pith review of: On a system of Schr\"odinger equations with general quadratic-type nonlinearities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q6CGKCMJ}},
  note         = {Machine review of arXiv:1908.04159}
}
read the original abstract

In this work we study a system of Schr\"odinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms on the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.

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Works this paper leans on

41 extracted references · 40 canonical work pages

  1. [1]

    Adams, J.J.F

    R.A. Adams, J.J.F. Fournier, Sobolev Spaces , Second Edition, Pure and Applied Mathematics, Else- vier/Academic Press, Amsterdam, 2003

    work page 2003

  2. [2]

    Albert, Concentration compactness and stability of solitary-wave s solutions to nonlocal equations , Contemp

    J.P. Albert, Concentration compactness and stability of solitary-wave s solutions to nonlocal equations , Contemp. Math. 221 (1999), 1–30

    work page 1999

  3. [3]

    Bégout, Necessary conditions and sufficient conditions for global ex istence in the nonlinear Schrödinger equation, Adv

    P. Bégout, Necessary conditions and sufficient conditions for global ex istence in the nonlinear Schrödinger equation, Adv. Math. Sci. Appl. 12 (2002), 817–827

    work page 2002

  4. [4]

    Buryak and Y.S

    A.V. Buryak and Y.S. Kivshar, Solitons due to second harmonic generation , Phys. Lett. A. 197 (1995), no. 5/6, 407–412

    work page 1995

  5. [5]

    Buryak, P

    A.V. Buryak, P. Di Tranpani, D.V. Skryabin, and S. Trillo , Optical solitons due to quadratic nonlinearities: From basic physics to futuristic applications , Phys. Rep. 370 (2002), no. 2, 63–235

    work page 2002

  6. [6]

    Burchard and H

    A. Burchard and H. Hajaiej, Rearrangement inequalities for functionals with monotone integrands, J. Funct. Anal. 233 (2006), no. 2, 561–582

    work page 2006

  7. [7]

    Cazenave, Semilinear Schrödinger Equations , American Mathematical Society, 2003

    T. Cazenave, Semilinear Schrödinger Equations , American Mathematical Society, 2003

    work page 2003

  8. [8]

    Colin, T

    M. Colin, T. Colin, and M. Otha, Stability of solitary waves for a system of nonlinear Schröd inger equations with three waves interations , Ann. Inst. H. Poincaré Anal. Non Linéaire. 26 (2009), no. 6, 2211–2226

    work page 2009

Show all 41 references
  1. [9]

    Colin, L

    M. Colin, L. Di Menza, and J.C. Saut, Solitons in quadratic media , Nonlinearity. 29 (2016), no. 3, 1000– 1035

    2016

  2. [10]

    Corcho, S

    A. Corcho, S. Correia, F. Oliveira, and J.D. Silva, On a nonlinear Schrödinger system arising in quadratic media, arXiv 1703.10509v2

    arXiv

  3. [11]

    Dinh, Strong instability of standing waves for a NLS system with qu adratic interaction , arXiv:1810.00676v1

    V.D. Dinh, Strong instability of standing waves for a NLS system with qu adratic interaction , arXiv:1810.00676v1

    arXiv

  4. [12]

    Esfahani and A

    A. Esfahani and A. Pastor, Sharp constant of an anisotropic Gagliardo-Nirenberg-typ e inequality and applications, Bull. Braz. Math. Soc. 48 (2017), no. 1, 171–185

    2017

  5. [13]

    Gilbarg and N

    D. Gilbarg and N. Trundiger, Elliptic Partial Differential Equations of Second Order , Classics in Mathe- matics, Springer, 1998

    1998

  6. [14]

    Hajaiej, On the necessity of the assumptions used to prove Hardy-Litt lewood and Riesz rearrangement inequalities, Arch

    H. Hajaiej, On the necessity of the assumptions used to prove Hardy-Litt lewood and Riesz rearrangement inequalities, Arch. Math. 96 (2011), no. 3, 273–280

    2011

  7. [15]

    Hamano, Global dynamics below the ground state for the quadratic Sch ödinger system in 5d , arXiv:1805.12245

    M. Hamano, Global dynamics below the ground state for the quadratic Sch ödinger system in 5d , arXiv:1805.12245

    arXiv

  8. [16]

    Hamano, Scattering for the quadratic nonlinear Schrödinger system in R5 without mass-resonance condition, arXiv:1903.05880v1

    M. Hamano, Scattering for the quadratic nonlinear Schrödinger system in R5 without mass-resonance condition, arXiv:1903.05880v1

    1903 arXiv

  9. [17]

    Hayashi, C

    N. Hayashi, C. Li, and P.I. Naumkin, On a system of nonlinear Schrödinger equations in 2D , Differential Integral Equations. 24 (2011), no. 5/6, 417–434. 50 N. NOGUERA AND A. PASTOR

    2011

  10. [18]

    Hayashi, T

    N. Hayashi, T. Ozawa, and K. Tanaka, On a system of nonlinear Schrödinger equations with quadrat ic interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire. 30 (2013), no. 4, 661–690

    2013

  11. [19]

    Holmer and S

    J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonli near Schrödinger equation, Commun. Math. Phys. 282 (2008), no. 2, 435–467

    2008

  12. [20]

    T. Inui, N. Kishimoto, K. Nishimura, Blow-up of radially symmetric solutions for the quadratic n onlinear Schrödinger system without mass-resonance , arXiv:1810.09153v2

    arXiv

  13. [21]

    T. Inui, N. Kishimoto, K. Nishimura, Scattering for a mass critical NLS system below the ground st ate with and without mass-resonance , arXiv:1810.07904v3

    arXiv

  14. [22]

    Kavian, A remark on the blowing-up solutions to the Cauchy problem fo r Nonlinear Schrödinger Equa- tions, Trans

    O. Kavian, A remark on the blowing-up solutions to the Cauchy problem fo r Nonlinear Schrödinger Equa- tions, Trans. Am. Math. Soc. 299 (1987), no. 1, 193–203

    1987

  15. [23]

    Kivshar, et al., Multi-component optical solitary waves , Phys

    Y.S. Kivshar, et al., Multi-component optical solitary waves , Phys. A 288 (2000), no. 1/4, 152–173

    2000

  16. [24]

    Leoni, A first course in Sobolev spaces , Graduate Studies in Mathematics, Vol

    G. Leoni, A first course in Sobolev spaces , Graduate Studies in Mathematics, Vol. 105, American Math- ematical Society, Providence, 2009

    2009

  17. [25]

    Li and N

    C. Li and N. Hayashi, N, Recent porgress on nonlinear Schrödinger system with quadr atic interactions , The Scientific World Journal. 2014 (2014)

    2014

  18. [26]

    Li, On a system of nonlinear Schrödinger equations and scale inv ariant spaces in 2D , Differential Integral Equations

    C. Li, On a system of nonlinear Schrödinger equations and scale inv ariant spaces in 2D , Differential Integral Equations. 28 (2015), no. 3/4, 201–220

    2015

  19. [27]

    Linares and G

    F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations , Universitex, Springer, New York, 2009

    2009

  20. [28]

    Lieb and M

    E. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, RI, 2001

    2001

  21. [29]

    Lions, The concentration compactness principle in the calculus of variations

    P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré Anal. Non Linéaire. 1 (1984), no 2, 109 –145

    1984

  22. [30]

    Lions, The concentration compactness principle in the calculus of variations

    P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case, II , Ann. Inst. H. Poincaré Anal. Non Linéaire. 1 (1984), no. 4, 2 23–283

    1984

  23. [31]

    Noguera and A

    N. Noguera and A. Pastor, On the dynamics of a quadratic Schrödinger system in dimensi on n = 5 , arXiv:1810.01390

    arXiv

  24. [32]

    Pastor, Weak concentration and wave operator for a 3D coupled nonlin ear Schrödinger system, J

    A. Pastor, Weak concentration and wave operator for a 3D coupled nonlin ear Schrödinger system, J. Math. Phys. 56 (2015), no. 2, 021507-1– 021507-18

    2015

  25. [33]

    Pastor, On a three wave interaction Schrödinger systems with quadra tic nonlinearities: Global well- posedness and standing waves , Commun

    A. Pastor, On a three wave interaction Schrödinger systems with quadra tic nonlinearities: Global well- posedness and standing waves , Commun. Pure Appl. Anal. 18 (2019), no. 5

    2019

  26. [34]

    Ogawa and Y

    T. Ogawa and Y. Tsutsumi, Blow-up of H 1 solutions for the Nonlinear Schrödinger Equation , J. Differ- ential Equations 92 (1991), no. 2, 317–330

    1991

  27. [35]

    Ozawa and Y

    T. Ozawa and Y. Tsutsumi, Remarks on proofs of conservation laws for nonlinear Schröd inger equations, Calc. Var. Partial Differential Equations 25 (2006), no. 3, 4 03–408

    2006

  28. [36]

    Tao, Multilinear Weighted convolutions L2 function, and applications to nonlinear dispersive equati ons, Amer

    T. Tao, Multilinear Weighted convolutions L2 function, and applications to nonlinear dispersive equati ons, Amer. J. Math. 123 (2001), no. 5, 839–908

    2001

  29. [37]

    Weinstein, Nonlinear Schrödinger equations and sharp interpolation e stimates, Commun

    M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation e stimates, Commun. Math. Phys. 87 (1983), no. 4, 567–576

    1983

  30. [38]

    Yew, A.R

    A.C. Yew, A.R. Champneys, and P.J MacKenna, Multiple solitary waves due to second-harmonic gener- ation in quadratic media. , J. Nonlinear Sci. 9 (1999), no. 1, 33–52

    1999

  31. [39]

    Yew, Multipulses of nonlinearly coupled Schrödinger equation , J

    A.C. Yew, Multipulses of nonlinearly coupled Schrödinger equation , J. Differential Equations. 173 (2001), no. 1, 92–137

    2001

  32. [40]

    Zhang, Local well-posednees for a system of quadratic nonlinear Sc hrödinger equations in one or two dimensions, Commun

    H. Zhang, Local well-posednees for a system of quadratic nonlinear Sc hrödinger equations in one or two dimensions, Commun. Math. Meth. App. Sci. 39 (2016), no. 14, 4257–4267

    2016

  33. [41]

    L. Zhao, F. Zhao, and J. Shi, Higher dimensional solitary waves generated by secodn-har monic generation in quadratic Media , Calc. Var. 54 (2015), no. 3, 2657–2691. IMECC-UNICAMP, Rua Sérgio Buarque de Holanda, 651, 13083-8 59, Campinas-SP, Brazil E-mail address : nnoguera57@...

    2015

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