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
Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects The nonlinear Schrödinger (NLS) equation i ǫψ t + ǫ 2 2 ψ xx + σ | ψ | 2 ψ = 0 where ǫ > 0 , σ = 1 → focusing , σ = − 1 → defocusing Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013
Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects The nonlinear Schrödinger (NLS) equation i ǫψ t + ǫ 2 2 ψ xx + σ | ψ | 2 ψ = 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
Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects The nonlinear Schrödinger (NLS) equation i ǫψ t + ǫ 2 2 ψ xx + σ | ψ | 2 ψ = 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
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 � x � − i � � v ( x ′ , t ) dx ′ ψ ( x , t ; ǫ ) = u ( x , t ) exp ǫ Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013
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 � x � − i � � v ( x ′ , t ) dx ′ ψ ( x , t ; ǫ ) = u ( x , t ) exp ǫ Hydrodynamic form u t + ( uv ) x = 0 v t + vv x + u x − ǫ 2 u − u 2 � u xx � x = 0 ǫ << 1 2 u 2 4 x Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013
Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects Standard Approach Look for solutions of the form u ǫ = u + ǫ u 1 + ǫ 2 u 2 + . . . v ǫ = v + ǫ v 1 + ǫ 2 v 2 + . . . . Leading ( dispersionless ) order → shallow water equations (SWE) u t + ( uv ) x = 0 v t + vv x + u x = 0 Note u j and v j , with j = 1 , 2 , . . . ← → transport equations Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013
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 ξ t + λξ x = 0 η t + µη x = 0 , where λ ( ξ, η ) and µ ( ξ, η ) are the characteristic speeds λ = 3 ξ + η µ = ξ + 3 η . 4 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
Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects Weak dispersive phenomenology Consider KdV equation u t + uu x + ǫ 2 u xxx = 0 Zabusky and Kruskal, ’65 Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013
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
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 − v c ) + ( u − u c ) ≃ ǫ 4 / 7 � x + α + σ U ′′ � ν + x + ǫ 6 / 7 ; ν + x − �� ǫ 4 / 7 � ν + x + ǫ 6 / 7 ; ν + x − � ( v − v c ) − ( u − u c ) ≃ ǫ 2 / 7 β U ǫ 4 / 7 where x ± = ( x − x c ) + ( v c ± 2 √ u c )( t − t c ) and U ( X , T ) satisfies P 2 I equation � 1 � 6 U 3 + 1 1 24 ( U ′ 2 + 2 UU ′′ ) + 240 U ( IV ) X = UT − Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013
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 ) exp i θ 0 , 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
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 ) | e i θ 0 Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013
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 ) | e i θ 0 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
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 ) | e i θ 0 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
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 out 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
Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects Dispersionless regime • Wave-breaking at t = t c ≃ 0 . 78, x ≃ 0 • Preserved vacuum u ( 0 , t ) = 0, for t < t c • Gradient catastrophe scenario for v x ( 0 , t c ) ∼ ∞ • Jump in the derivative for u ( x , t c ) Breaking mechanism from a vacuum point in dNLSE Antonio Moro, Cagliari 2nd September 2013
Introduction Universality Vacuum point Zero dispersion IVP Numerical vs Analytic solution Critical Point Dispersive effects 2 1.5 1 � 0.1 0.5 0 (a) − 0.05 0 0 − 2 − 1 0 1 2 0 (b) 0 − 0.5 − 0.1 0 0.05 � − 1 − 1.5 − 2 − 2 − 1 0 1 2 x 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
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
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 + f u = x ut + f v = 0 , where the function f ( u , v ) is a solution to the Tricomi-type equation f vv − uf uu = 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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