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Hopf equation • hyperbolic conservation law for u(x, t), initial data u 0 (x) u t + 6uu x = 0, u(x, 0) = u 0 (x) • solution with the method of characteristics u(x, t) = u 0 ( ), x = 6tu 0 ( ) + 1 • critical time t c = min ⇥R [ 6u 0 ( )] , gradient catastrophe, t > t c : solution multivalued (shock) Dienstag, 29. Mai 2012
Example: u 0 = sech 2 x Dienstag, 29. Mai 2012
Page 1 and 2: 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
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: KPII: perturbed cnoidal solution =0
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

u - ACMAC

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