Breaking mechanism from a vacuum point in the defocusing NLS - - PowerPoint PPT Presentation

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Breaking mechanism from a vacuum point in the defocusing NLS - - PowerPoint PPT Presentation

Sep 06, 2023 425 likes •775 views

Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects Breaking mechanism from a vacuum point in the defocusing NLS equation Antonio Moro Northumbria University,

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Breaking mechanism from a vacuum point in the defocusing NLS equation

Antonio Moro

Northumbria University, Newcastle upon Tyne

joint work with Stefano Trillo

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

The nonlinear Schrödinger (NLS) equation iǫψt + ǫ2 2 ψxx + σ|ψ|2ψ = 0 where ǫ > 0, σ = 1 → focusing, σ = −1 → defocusing

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

The nonlinear Schrödinger (NLS) equation iǫψt + ǫ2 2 ψxx + σ|ψ|2ψ = 0 where ǫ > 0, σ = 1 → focusing, σ = −1 → defocusing

  • Ubiquitous ↔ Universal

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

The nonlinear Schrödinger (NLS) equation iǫψt + ǫ2 2 ψxx + σ|ψ|2ψ = 0 where ǫ > 0, σ = 1 → focusing, σ = −1 → defocusing

  • Ubiquitous ↔ Universal
  • Integrable ↔ Exactly solvable (Zakharov-Shabat, ’72)

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Weak dispersive regime Study asymptotics of fast oscillating solutions ψ(x, t; ǫ) Madelung transform ψ(x, t; ǫ) =

  • u(x, t) exp
  • − i

ǫ x v(x′, t) dx′

  • Breaking mechanism from a vacuum point in dNLSE

Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Weak dispersive regime Study asymptotics of fast oscillating solutions ψ(x, t; ǫ) Madelung transform ψ(x, t; ǫ) =

  • u(x, t) exp
  • − i

ǫ x v(x′, t) dx′

  • Hydrodynamic form

ut + (uv)x = 0 vt + vvx + ux−ǫ2 4 uxx u − u2

x

2u2

  • x

= 0 ǫ << 1

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Standard Approach Look for solutions of the form uǫ = u + ǫu1 + ǫ2u2 + . . . vǫ = v + ǫv1 + ǫ2v2 + . . . . Leading (dispersionless) order → shallow water equations (SWE) ut + (uv)x = 0 vt + vvx + ux = 0 Note uj and vj, with j = 1, 2, . . . ← → transport equations

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Riemann Invariants Introduce the variables ξ = v + 2 √ u, η = v − 2 √ u such that the SWE takes the diagonal form ξt + λξx = 0 ηt + µηx = 0, where λ(ξ, η) and µ(ξ, η) are the characteristic speeds λ = 3ξ + η 4 µ = ξ + 3η 4 . Note In general, Riemann invariants break in finite time. If they break at the same point (x, t) the corresponding initial datum is said to be non-generic.

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Weak dispersive phenomenology Consider KdV equation ut + uux+ǫ2uxxx = 0

Zabusky and Kruskal, ’65

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Universality (Dubrovin, ’06)

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Universality (Dubrovin, ’06) Defocusing NLS equation near the critical point of gradient catastrophe ( generic case ) (v − vc) + (u − uc) ≃ ǫ4/7 x+ α + σU′′ ν+x+ ǫ6/7 ; ν+x− ǫ4/7

  • (v − vc) − (u − uc) ≃ ǫ2/7βU

ν+x+ ǫ6/7 ; ν+x− ǫ4/7

  • where

x± = (x − xc) + (vc ± 2√uc)(t − tc) and U(X, T) satisfies P2

I equation

X = UT − 1 6U3 + 1 24(U′2 + 2UU′′) + 1 240U(IV)

  • Breaking mechanism from a vacuum point in dNLSE

Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

A previous study: phase jump dark initial datum Solution for a jump phase-dark initial datum ψ(x, 0) = tanh (x) expiθ0, u(x, t) = |ψ(x, t)|2

Conti, Fratalocchi, Peccianti, Ruocco, Trillo, PRL 2009

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Our study: constant phase dark initial datum We consider the dark constant phase initial condition ψ(x, 0) = |tanh(x)| eiθ0

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Our study: constant phase dark initial datum We consider the dark constant phase initial condition ψ(x, 0) = |tanh(x)| eiθ0 Remark1 This case turns out to be non-generic → Universality conjecture does not apply.

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Our study: constant phase dark initial datum We consider the dark constant phase initial condition ψ(x, 0) = |tanh(x)| eiθ0 Remark1 This case turns out to be non-generic → Universality conjecture does not apply. Remark2 Vacuum points in NLS dynamics appeared in El et al, ’95, Hoefer et al, ’08

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Weak dispersive regime: comparison continuous phase jump phase A dispersive shock that opens up in a characteristic fan turns

  • ut to resolve the singularity that occurs in the range

t = 0.75 − 0.78 around the origin (x = 0)

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Dispersionless regime

  • Wave-breaking at t = tc ≃ 0.78, x ≃ 0
  • Preserved vacuum u(0, t) = 0, for t < tc
  • Gradient catastrophe scenario for vx(0, tc) ∼ ∞
  • Jump in the derivative for u(x, tc)

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects −2 −1 1 2 0.5 1 1.5 2

  • (a)

−2 −1 1 2 −2 −1.5 −1 −0.5 x

  • (b)

−0.05 0.1 0.05 −0.1

Snapshots of RIs obtained by means of numerical integration SWE at time t = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.784

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Main observations Dispersionless regime:

  • Dispersionless level: no qualitative difference
  • Riemann Invariants (RIs) simultaneously break at the

same point ↔ The critical point is non-generic Weak dispersion regime:

  • Preserved vs non preserved vacuum
  • Critical behaviour occurs around RIs breaking time

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Analytic solution of SWE Introduce the hodograph transform (x, t) ↔ (u, v) defined by vt + fu = x ut + fv = 0, where the function f(u, v) is a solution to the Tricomi-type equation fvv − ufuu = 0 Solution to IVP map: (u(x, 0), v(x, 0)) ← → f(u, v)

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Euler-Poisson-Darboux equation In Riemann invariants, the hodograph equations in reads as x − λt = wξ x − µt = wη and the Tricomi type equation is replaced by the EPD equation for the function w(ξ, η) wξη = 1 2(ξ − η) (wξ − wη)

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Differentiating the hodograph equations and solving w.r.t. ξx, ξt, ηx, ηt, we get ξx = 2(η − ξ) 2(η − ξ)wξξ + 3(wξ − wη) ηx = 2(η − ξ) 2(η − ξ)wηη + 3(wξ − wη). (4.1) The solution (ξ(x, t), η(x.t)) is said to break, i.e. it develops a gradient catastrophe singularity, if there exists a critical point (xc, tc) such that ξx and ηx are bounded for any t ∈ [0, tc) and |ξx(xc, tc)| = ∞ or |ηx(xc, tc)| = ∞.

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

A class of initial value problems General IVP solved by Geogdzhaev, ’87 - Tian-Ye, ’99 We focus on the family of initial data of the form u(x, 0) = u0(x) v(x, 0) = 0 For the sake of simplicity, u0 is assumed to be a negative hump given by an even, continuous and differentiable function centred at x = 0. Solution to the Tricomi type (EPD) equation f(u, v) = u 1

−1

g(v + 2µ √ u)

  • 1 − µ2 dµ

where g is is fixed by the initial condition

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

For the initial datum ψ(x, 0) = |tanh(x)| eiθ0 in the hydrodynamic variables u(x, 0) = tanh2(x) v(x, 0) = 0 The function f(u, v) takes the form (in Riemann Invariants) f(u(ξ, η), v(ξ, η)) =1 4 ξ r

  • (ξ − r)(r − η)

√ 4 − r 2 dr +1 4 η r

  • (ξ − r)(r − η)

√ 4 − r 2 dr

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

1 4 ξ r

  • (ξ − r)(r − η)

√ 4 − r 2 dr = − 2

  • γ

2α2(k2 − α2)(2 − ξ)(ξ − η)

  • α2E(ω)+

(α2 − k2)ω + (α4 − 2α2 + k2)Π(ω, α2) − α4 snω cnω dnω 1 − α2sn2ω

  • + γ

α4 (2 − ξ)(2 + ξ)(ξ − η)

  • −k2ω + (3k2 − α2k2 − α2)Π(ω, α2)

+(2α2k2 + 2α2 − 3k2 − α4)V2 + (α2 − 1)(α2 − k2)V3

  • Breaking mechanism from a vacuum point in dNLSE

Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

where V0 = F(φ, k) V1 = Π(φ, α2, k) V2 = 1 2(α2 − 1)(k2 − α2)

  • α2E(ω) + (k2 − α2)ω

+(2α2k2 + 2α2 − α4 − 3k2)Π(φ, α2, k) −α4 snω cnω dnω 1 − α2sn2ω

  • V3 = . . .

α2 = ξ − η 2 − η k2 = 4(ξ − η) (2 − η)(2 + ξ) γ = 2

  • (2 − η)(2 + ξ)

φ = sin−1 ξ(2 − η) 2(ξ − η)

  • ω = sn−1(φ, k2),

where F(φ, k), E(ω) and Π(φ, α2, k) are the standard notations for the elliptic integral of first kind, the complete elliptic integral

  • f second kind and the incomplete elliptic integral of the third

kind respectively and sn, cn, dn stand for the Jacobian Elliptic functions.

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Numerical vs analytic solution vt + fu = x ut + fv = 0,

1 2 0.4 0.8 u

(a)

t=0.78 t=0.7 1 2 −0.8 −0.4 x v

(b)

t=0.78 t=0.7

0.1 0.1 u x 0.1 −0.6 v x

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Near the vacuum... We compute f(u(ξ, η), v(ξ, η)) ≃ 1 32

  • ξ2 − 2

3ηξ + η2

  • (−ηξ)1/2 +

+ 1 64 (ξ − η)2 (ξ + η) sin−1 ξ + η ξ − η

  • and

x ≃ 2(−ηξ)3/2 (ξ − η)2 t ≃ − ξ + η (ξ − η)2 (−ηξ)1/2 + 1 2 sin−1

  • −ξ + η

ξ − η

  • Breaking mechanism from a vacuum point in dNLSE

Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects −2 −1 1 2 0.5 1 1.5 2

  • (a)

−2 −1 1 2 −2 −1.5 −1 −0.5 x

  • (b)

−0.05 0.1 0.05 −0.1

Snapshots of RIs obtained by means of numerical integration SWE at time t = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.784

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

On the positive half-line x > 0, the critical values such that ηx(xc, tc) → ∞ and |ξx(xc, tc)| < ∞ are xc = lim

η→0 x(ξ, η)|ξ=0 = 0,

tc = lim

η→0 t(ξ, η)|ξ=0 = lim η→0

1 2 sin−1(1) = π 4 ≃ 0.785.

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Dispersive effects: Initial vacuum point

0.2 0.4 0.6 0.8 0.01 0.02 0.03 0.04 t | (0,t)|2

Density u(0, t) = |ψ(0, t)|2 vs. time t obtained from numerical integration of NLS equation with decreasing values of dispersion ε = 5 × 10−3 (red), ε = 10−3 (green), ε = 5 × 10−4 (blue). The dashed vertical line stands for the critical time tc = π/4.

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Dispersive effects: critical phase

−0.04 0.04 0.03 0.06 x

u

(a)

0.02 0.04 −2 2 x

ux

(b)

−0.04 0.04 −0.5 0.5 x

v

(c)

0.784 0.7 0.784 0.7 0.784 0.7

Hydrodynamic variables in the neighborhood of the origin,

  • btained from numerical integration of NLS equation with

ε = 5 × 10−4. The snapshots are taken from time t = 0.76 to t = 0.784 with constant increment δt = 0.004.

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013

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Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects

Summary

  • SWEs undergo the simultaneous breaking of RIs in the

point of null density (x = 0)

  • Breaking occurs in the opposite limits x = 0± for the two

RIs

  • Dispersion starts to play a role just approaching the

breaking

  • Slow adiabatic detachment of the min density points from

zero that abruptly grows turns into a max near the breaking point for SWE Outlook

  • More general initial data
  • Analytic asymptotic description of the weakly dispersive

critical behaviour

Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013