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Outline ✦ Introduction, general features ✦ Exact solutions and stability ✦ Small dispersion limit ✦ Blow-up in generalized KP equations ✦ Outlook Dienstag, 29. Mai 2012
Kadomtsev-Petviashvili equation (KP) • Kadomtsev-Petviashvili 1970: transverse stability of KdV soliton (u t + uu x + u xxx ) x + u yy =0, = ±1 KPI ( = 1): strong surface tension KPII ( = 1): weak surface tension • heuristic derivation: relation of wave and advection (transport) equation, separation of left- and right-going waves, leads to (@ t + @ x )u =0 (@ t @ x )(@ t + @ x )u =0, Dienstag, 29. Mai 2012
Page 1: Numerical Study of the KP equation
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
Page 11 and 12: 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 and 66:
Dienstag, 29. Mai 2012 KP II soluti
Page 67 and 68:
Generalized KP equations • • eq
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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