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Monotonicity properties of blow-up time for nonlinear Schr\"{o}dinger equation: numerical tests

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Christophe Besse (LPP, INRIA FUTURS), Rémi Carles (MAB), Norbert Mauser (ICP), Hans-Peter Stimming (ICP)

Analysis of PDEs · Submitted 4 Mar 2006 · 32 references

Pith's verdict: First numerical counterexamples to monotonicity of NLS blow-up time, with a solid analytic core and a blow-up-time diagnostic that needs a convergence check.

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arXiv IDmath/0603107 (version 2)
Pith IDWQPT3WL6, Pith's permanent record of this paper and its review
CitationChristophe Besse (LPP, Hans-Peter Stimming (ICP), INRIA Futurs), Norbert Mauser (ICP), R\'emi Carles (MAB). "Monotonicity properties of blow-up time for nonlinear Schr\"{o}dinger equation: numerical tests." arXiv:math/0603107 (2006). https://pith.science/paper/WQPT3WL6
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The paper

The paper claims

The paper reports numerical evidence that the blow-up time for focusing nonlinear Schrödinger equations is not a monotone function of the coupling constant, the quadratic-oscillation parameter, or the damping strength, in the L2-critical…

The reading

The paper asks whether the time at which a focusing nonlinear Schrödinger solution collapses responds monotonically to three controls: the coupling strength $\lambda$, the size $a$ of a quadratic phase oscillation in the initial data, and the damping rate $\delta$. Using two independent numerical schemes on one- and two-dimensional tests, it finds that monotonicity generally fails for initial data with two or more humps: the blow-up time can rise when the nonlinearity is strengthened, and analogous reversals occur for damping and (in the supercritical case) for the oscillation parameter. The one tested exception is the L2-critical quadratic-oscillation problem, where monotonicity is proved analytically via conformal invariance. Blow-up time controls when wave collapse happens in laser propagation, so the question matters: if these findings hold, tuning strength or damping to delay collapse can backfire for composite profiles.

What carries the argument

The carrying objects are the blow-up-time functional $T^*$, the pseudo-conformal transformation (for the critical quadratic-oscillation case), and the numerical energy-jump diagnostic. The pseudo-conformal transformation maps quadratic-oscillation initial data to the original equation with a rescaled time and yields the exact identity $T_a = aT/(a+T)$; that identity is the analytical anchor of the paper. The diagnostic defines blow-up as the first time kinetic and potential energies both increase by at least a factor of $10^4$, making $T^*$ computable. The two schemes—time-splitting spectral and relaxation—are the cross-check that the non-monotone curves are not artifacts of a single discretization.

Extended reading notes

Core claim

The central claim is that in the supercritical case $\sigma > 2/n$ the blow-up time is not monotonous with respect to $\lambda$, $a$, or $\delta$; in the critical case $\sigma = 2/n$, non-monotonicity appears for $\lambda$ and $\delta$, while monotonicity in $a$ holds and is proved analytically. The reversal is confined to multi-hump initial data; single-hump tests remain monotone. The heuristic mechanism is a race between nonlinear self-focusing at each hump and a linear phase/dispersion that merges humps: for small parameter values the humps merge before blow-up, for large values they collapse independently, and in the intermediate region the two scenarios compete, producing dips and jumps in the blow-up time. The numerical curves are reproduced by both a time-splitting spectral scheme and a relaxation scheme, and the critical quadratic-oscillation curves match the exact relation $T_a = aT/(a+T)$.

Load-bearing premise

The conclusion stands on the numerical rule that blow-up time is the first moment when kinetic and potential energy both jump by a factor of $10^4$; if that threshold responds differently across parameter values, the reported non-monotonicity would be an artifact.

Editorial extensions

If this is right

  • In the L2-supercritical regime, monotonicity-based predictions of collapse time, such as larger $\lambda$ always meaning earlier blow-up or stronger damping always delaying blow-up, are unreliable for multi-hump data.
  • Damped cubic NLS in two dimensions, the setting of the damped-NLS conjecture, exhibits non-monotone blow-up time in the reported tests.
  • The critical quadratic-oscillation case retains the exact formula $T_a = aT/(a+T)$, and the numerical and analytical curves agree there.
  • For single-hump data, monotonicity appears to hold in these tests, so the breakdown is tied to the geometry of the initial profile rather than to the equation alone.
  • The steep parameter regions where the reversal occurs coincide with a change between multi-point and single-point blow-up, suggesting that the global blow-up profile controls the timing.

Reading between the lines

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

  • If this pattern is generic, applications that control wave collapse—laser filamentation, optical pulse compression—should not assume that reducing intensity or increasing damping monotonically postpones collapse for superpositions of pulses.
  • A natural next step, not taken in the paper, is to map the blow-up time as a function of hump separation and phase mismatch; the reversal surface would show how much multi-hump structure is needed to break monotonicity.
  • Recomputing the curves with a scale-invariant blow-up criterion (for instance the first time $\max_x |u|$ passes a fixed threshold) on an adaptive mesh would test whether the energy-jump threshold is responsible for the observed reversals.
Original abstract

We consider the focusing nonlinear Schr\"{o}dinger equation, in the $L^2$-critical and supercritical cases. We investigate numerically the dependence of the blow-up time on a parameter in three cases: dependence upon the coupling constant, when the initial data are fixed; dependence upon the strength of a quadratic oscillation in the initial data when the equation and the initial profile are fixed; finally, dependence upon a damping factor when the initial data are fixed. It turns out that in most situations monotonicity in the evolution of the blow-up time does not occur. In the case of quadratic oscillations in the initial data, with critical nonlinearity, monotonicity holds; this is proven analytically.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,679 input tokens.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.79 · human participants 0.02 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.07 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We consider the focusing nonlinear Schr\"{o}dinger equation, in the $L^2$-critical and supercritical cases. We investigate numerically the dependence of the blow-up time on a parameter in three cases: dependence upon the coupling constant, when the initial data are fixed; dependence upon the strength of a quadratic oscillation in the initial data when the equation and the initial profile are fixed; finally, dependence upon a damping factor when the initial data are fixed. It turns out that in most situations monotonicity in the evolution of the blow-up time does not occur. In the case of quadratic oscillations in the initial data, with critical nonlinearity, monotonicity holds; this is proven analytically.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We consider the focusing nonlinear Schr\"{o}dinger equation, in the $L^2$-critical and supercritical cases. background 0.96 0.30 0.02 asserted 0.64
s1 We investigate numerically the dependence of the blow-up time on a parameter in three cases: dependence upon the coupling constant, when the initial data are fixed; dependence upon the strength of a quadratic oscillation in the initial data when the equation and the initial profile are fixed; finally, dependence upon a damping factor when the initial data are fixed. contribution 0.80 0.95 0.05 asserted 0.93
s2 It turns out that in most situations monotonicity in the evolution of the blow-up time does not occur. result 1.00 0.94 0.76 asserted 0.98
s3 In the case of quadratic oscillations in the initial data, with critical nonlinearity, monotonicity holds; this is proven analytically. result 1.00 0.91 0.24 asserted 1.00

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read
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  • references and citing sentences
  • relations to other papers
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not run
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  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Blow-up time with varying λ, single Gaussian data (Test 1). 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 λ blowup time critical NLS, two hump data
Figure 2. Blow-up time with varying λ (Test 2). monotonously with λ, as predicted by the heuristics for the case of a single Gaussian profile. Test 2. The next kind of data we study is (3.2) u0(x) = C  e −x 2 − 0.9e −3x 2  e −i log(e x+e−x) . The constant C is equal to 4 which leads to ku0k 2 L2 = 3.907. The difference of two Gaussian profiles results in two local maxima in the modulus of u0. The phase term has a focusing effect and its focus point does not agree with the local maxima of the modulus. The finest discretization parameters used are N p = 214 = 16384, ∆t = 2.5 · 10−6 . The…
Figure 3. Time evolution with low potential energy
Figure 4. Time evolution with high potential energy. and λ > 2.2), the blow-up time is monotonously decreasing with λ, while in between there is a region where monotonicity does not hold. Heuristically, two effects play a role in the solution with this data: the nonlinear self-focusing, which tries to focus the mass to points where the most mass is already present, and the (linear) phase influence, which tends to focus the mass at zero. In Figures 3, 4 and 5, the time evolution of the modulus of u(t, x) is shown for values of λ from the three different regions of the above curve. •
Figure 5. Time evolution in intermediate regime: non-monotonicity. 0 0.05 0.1 0.15 0.2 0.25 0 20 40 60 80 100 120 140 t λ = 2.0 kinetic energy potential energy (neg.) total energy
Figure 6. Determining blow-up time by increase of energy: (minus) total energy in comparison with kinetic and potential energy away from the blow-up nonlinear self-focusing here happens faster than phase focusing. •
Figure 7. Blow-up time with varying λ, three hump data (Test 3). factor of 104 , or slightly more, at the blow-up time
Figure 8. Blow-up time with varying λ, three hump data (Test 4). with C = 2, ku0k 2 L2 = 4.93. One of the three Gaussians has an opposite sign, so there is a constant phase shift in part of the data. The blow-up times are shown in
Figure 9. Blow-up time with varying λ, data with point symmetry (Test 5). to have either a phase that focuses away from the symmetry point between the two humps, or two humps of different heights. It turns out that non￾monotonicity can be observed for for both of these cases, too. The pertur￾bations to (3.2) tested here are small enough not to hinder non-monotonous blow-up times. Test in two space dimensions Test 7. For the test in two space dimensions, we extend (3.2) by making the phase term radially symmetric and multiplying the one-dimensional two￾hump profile by a single Gaussian in the…
Figure 10. Initial data for two dimensional case, modulus (Test 7). 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 critical two−dim. NLS λ blowup time
Figure 11. Blow-up time with varying λ, two dimensional case (Test 7). 4.1. Critical power. We use (2.7) in space dimension one, with σ = 2 with various u0(x). Test 9. Single hump: We take u0(x) = C e−x 2
Figure 12. Blow-up time with varying constant λ (Test 8). with C = 1.75. The discretization parameters in this and the two following tests are N p = 214 mesh points and ∆t = 4 · 10−6 . The discretization domain is [−8, 8] for all cases except the two largest negative a in Test 9 and 10. Here the domain is extended to [−18, 18] and the space resolution is N p = 213 resp. [−40, 40] and N p = 214 for the largest negative a of Test 11
Figure 13. Blow-up time with varying a in quadratic os￾cillations, critical power (Test 9). −4 −3 −2 −1 0 1 2 3 4 0 0.5 1 1.5 2 2.5 3 3.5 a blowup time
Figure 14. Blow-up time with varying a in quadratic os￾cillations, critical power, two hump data (Test 10). In the left figure, the dashed line is the calculated blow-up time for v and the asterisks mark the simulated blow-up times. The right figure compares the blow-up times for v and w, where w is denoted by asterisks and v by dots joined by a line. We can see that w blows up a bit earlier than v for positive a. For negative a, it blows up a bit later, but the blow-up times are rather close to those of v as in the positive a case. Again the blow-up times for v match the result of…
Figure 15. Blow-up time with varying a in quadratic os￾cillations, critical power, data with additional phase (Test 11). Test 15. Two humps placed asymmetrically: Here we use u0(x) = C  e −(3x) 2 + e −(3(x−1.5))2  . We take C = 1.8. The smallest discretization parameters used in this test are N p = 215 mesh points and ∆t = 1.5 · 10−6 . The computation domain is [−8, 8], except for the largest value of a
Figure 16. Blow-up time with varying a in quadratic os￾cillations, supercritical power. Left: blow-up times for v. Right: comparison of v and w, dots with line for v, asterisks for w (Test 12). −3 −2 −1 0 1 2 3 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 a blowup time
Figure 17. Blow-up time of v with varying a in quadratic oscillations, supercritical power (Test 13). energy level where the blow-up changes from two point blow-up to one point blow-up, and non-monotonicity can be observed. 5. Numerical test, damped NLS 5.1. Critical power. We now turn to (2.10). Test in one space dimension Test 16. For (2.10) in space dimension one, we first use the single Gaussian data (3.1). The scale C = 2 was chosen so that ku0k 2 L2 = 5.013, the smallest discretization parameters used are N p = 212 mesh points and ∆t = 2.5 · 10−6 . The blow-up time with respect to…
Figure 18. Blow-up time of v with varying a in quadratic oscillations, supercritical power, data with additional phase (Test 14). 0 0.5 1 1.5 2 2.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 a Blow−up time
Figure 19. Blow-up time for w with varying a in quadratic oscillations, supercritical power, asymmetric data (asterisks and circles). Solid line: Blow-up time for v. (Test 15). The blow-up time with respect to changing δ is shown in
Figure 20. Blow-up time with varying damping constant δ (Test 16). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 δ blowup time critical NLS with damping
Figure 21. Blow-up time with varying damping constant δ (Test 17). 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 δ blowup time two−dim. NLS with damping
Figure 22. Blow-up time with varying damping constant δ (Test 18)
Figure 23. Blow-up time with varying damping constant δ, supercritical power (Test 19). 5.2. Supercritical Power. Test 19. We tested equation (2.10) in space dimension one with σ = 3. The data are chosen as (3.2) with C = 3.8, so ku0k 2 L2 = 3.53. The discretization parameters used in this test are N p = 212 meshpoints and ∆t = 5.0 · 10−6 or ∆t = 10−5 . The discretization domain is [−8, 8]. The blow-up times with respect to the damping constant δ are shown in
Figure 24. Blow-up time with varying damping constant δ, supercritical power (Test 20). • Dependence upon the coupling constant λ, for fixed n, σ and u0: i∂tu + ∆u = −λ|u| 2σu , (t, x) ∈ R+ × R n ; u|t=0 = u0 . • Dependence upon the magnitude of a quadratic oscillation intro￾duced in the initial data: only a varies in the following equation, i∂tw + ∆w = −|w| 2σw ; w|t=0 = u0(x)e −i |x| 2 4a . • Dependence upon the strength of the damping δ ≥ 0 in i∂tψ + ∆ψ = −|ψ| 2σψ − iδψ ; ψ|t=0 = u0 . In the L 2 super-critical case σ > 2 n , the tests we performed highly suggest that in either of the above…

Figures reproduced from arXiv:math/0603107 by the authors.

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