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Generalized KP equations ⇧ t u + 6⇤u n ⇧ x u + 2 ⇧ xxx u + ⇥ ⇧ 1 x ⇧ yy u = 0, ⇤ = ±1 n = 2, = 1 Dienstag, 29. Mai 2012
Generalized KP equations • • equations (linear) and u t + u p u x + u xxx @ 1 x u yy =0 u t + u xxx @ 1 x u yy =0 u t + u p u x =0 (shocks) do not have blow up of the L 1 -norm of u. • for p4/3 blow up in finite time of ||u y || 2 for initial data with negative energy (Molinet, Saut, Tzvetkov 2002). No conjectures for KPII. Dienstag, 29. Mai 2012
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Numerical Study of the KP equation
Page 3 and 4:
Kadomtsev-Petviashvili equation (KP
Page 5 and 6:
Consequences of nonlocality R • c
Page 7 and 8:
• lump soliton (KPI), algebraical
Page 9 and 10:
(Line)Solitons, 2-soliton limit Die
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Dienstag, 29. Mai 2012 (Line)Solito
Page 13 and 14:
Hyperelliptic solutions (g=2) J. Fr
Page 15 and 16: Dienstag, 29. Mai 2012 Hyperellipti
Page 17 and 18: KP II: perturbed KdV soliton, diffe
Page 19 and 20: KP I: perturbed KdV soliton u 0 (x,
Page 25 and 26: KP I: perturbed Zaitsev solution u
Page 31 and 32: KP I: perturbed Zaitsev solution 10
Page 33 and 34: KPI: perturbed cnoidal solution ✓
Page 35 and 36: KPI: perturbed cnoidal solution =0.
Page 37 and 38: KPII: perturbed cnoidal solution =0
Page 39 and 40: Example: u 0 = sech 2 x Dienstag, 2
Page 41 and 42: Dissipative regularization u t +6uu
Page 43 and 44: Korteweg-de Vries equation u t +6uu
Page 45 and 46: Zoom in: oscillatory zone T. Grava
Page 47 and 48: Whitham equations and asymptotic so
Page 49 and 50: KdV and asymptotic solution ✏ =0.
Page 51 and 52: Dispersionless KPI (dKP, Khokhlov-
Page 53 and 54: Gradient of the dKP solution Dienst
Page 55 and 56: Dienstag, 29. Mai 2012 KP I solutio
Page 57 and 58: KP I and II solution ϵ=0.1 4 !=1 2
Page 59 and 60: Dienstag, 29. Mai 2012 KP I solutio
Page 61 and 62: Oscillatory zone shrinks with ϵ !#
Page 63 and 64: Dependence on the `weak‘ coordina
Page 65: Dienstag, 29. Mai 2012 KP II soluti
Page 69 and 70: Blow up u 0 (x, y) =6@ xx exp( (x 2
Page 71 and 72: critical exponent 4/3 ✦ no blowup
Page 73 and 74: Outlook ✦ Stability of KP II solu
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u - ACMAC

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