# Nonlinear Waves in Physics: from Solitary Waves and Solitons to Rogue Waves

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Nonlinear Waves in Physics: From Solitary Waves and Solitons to Rogue Waves Dr. Russell Herman Mathematics & Statistics, UNC Wilmington, Wilmington, NC, USA May 13, 2020 History KdV NLPDEs Perturbations mKdV NLS Rogue Summary Outline § hydrodynamics, Soliton History § plasmas, Korteweg-deVries Equation § nonlinear optics, § biology, Other Nonlinear PDEs § relativistic ﬁeld theory, § relativity, Soliton Perturbations § geometry ... Integral Modiﬁed KdV Nonlinear Schr¨odingerEquation Rogue Waves Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 2 / 69 Summary http://www.ma.hw.ac.uk/~chris/scott_russell.html History KdV NLPDEs Perturbations mKdV NLS Rogue Summary Great Wave of Translation - 1834 I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-deﬁned heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour [14 km/h], preserving its original ﬁgure some thirty feet [9 m] long and a foot to a foot and a half [300-450 mm] in height. Its height gradually diminished, and after a chase of one or two miles [2-3km] I lost it in the windings of the channel. Such, in the month of August 1834, was my ﬁrst chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation. - John Scott Russell Union Canal, Hermiston, Scotland, http://www.ma.hw.ac.uk/~chris/canal.jpg Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 3 / 69 History KdV NLPDEs Perturbations mKdV NLS Rogue Summary John Scott Russell (1808-1882) § Engineer, Scotland § Used 30 ft tank § v 2 “ gph ` aq http://www.ma.hw.ac.uk/~chris/scott_russell.html § \Committee on Waves" § Reports: 1837, 1844 Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 4 / 69 History KdV NLPDEs Perturbations mKdV NLS Rogue Summary Years of Controversy George Biddle Airy (1801-1892). Sir George Gabriel Stokes (1819-1903). Publications on water waves: George Green (1839), Philip Kelland (1840), George Biddell Airy (1941- Tides and Waves), and Samuel Earnshaw (1847). Tried to do better than Lagrange, Laplace, Cauchy, and Poisson. Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 5 / 69 History KdV NLPDEs Perturbations mKdV NLS Rogue Summary 1870's - Nonlinear Theory from Euler's Equations Joseph Valentin Boussinesq (1842-1929). John William Strutt (Lord Rayleigh) (1842-1919). upx; tq “ 2η2 sech2pηpx ´ 4η2tqq: Rayleigh (1876) derived correct approximate solution, w/dispersion and nonlinearity, later learned of Boussinesq's (1871) work. Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 6 / 69 History KdV NLPDEs Perturbations mKdV NLS Rogue Summary Korteweg-de Vries Equation - 1895 Gustav de Vries (1866-1934) Diederik Johannes Korteweg (1848-1941) Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 7 / 69 History KdV NLPDEs Perturbations mKdV NLS Rogue Summary Fluid Equations - Mass and Momentum Conservation Bρ ` r ¨ pρvq “ 0 Bt Bv 1 ` v ¨ r ¨ v “ p´rP ` fq Bt ρ z z “ ηpx; tq z “ 0 Incompressible r ¨ v “ 0 Irrotational r ˆ v “ 0 z “ ´h x Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 8 / 69 History KdV NLPDEs Perturbations mKdV NLS Rogue Summary KdV Derivation Laplace's Equation v rφ r2φ 0 z “ ñ “ z “ ηpx; tq z “ 0 Incompressible r ¨ v “ 0 Irrotational r ˆ v “ 0 z “ ´h x Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 9 / 69 History KdV NLPDEs Perturbations mKdV NLS Rogue Summary KdV Derivation - Boundary Conditions Kinematic BCs 1 2 2 φt ` 2 pφx ` φz q ` gη “ 0 z ηt ` φx ηx ´ φz “ 0 z “ ηpx; tq z “ 0 Laplace's Equation φxx ` φzz “ 0 φ “ 0 z “ ´h z x Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 10 / 69 History KdV NLPDEs Perturbations mKdV NLS Rogue Summary KdV Derivation - Reductive Perturbation Theory Scaling τ “ 3{2t z ξ “ 1{2px ´ tq z “ ηpx; tq z “ 0 Expansions 1{2 8 n η “ n“0 ηn φ “ 1{2 8 nφ řn“0 n ř z “ ´h x 1 2η0τ ` 3η0η0ξ ` 3 η0ξξξ “ 0 Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 11 / 69 r´cf ` 3f 2 ` f 2s1 “ 0: ´cf ` 3f 2 ` f 2 “ A: ´cﬀ 1 ` 3f 2f 1 ` f 2f 1 “ Af 1: 1 2 3 1 12 ´ 2 cf ` f ` 2 f “ Af ` B: c df 1 “ Af ` B ` cf 2 ´ f 3: 2 dξ 2 c c History KdV NLPDEs Perturbations mKdV NLS Rogue Summary KdV Traveling Waves - ut ` 6uux ` uxxx “ 0 Seek solutions of form upx; tq “ f px ´ ctq ” f pξq; 1 1 Noting ux “ f pξq and ut “ ´cf pξq; we obtain ODE for f pξq: ´cf 1 ` 6ﬀ 1 ` f 3 “ 0: Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 12 / 69 ´cf ` 3f 2 ` f 2 “ A: ´cﬀ 1 ` 3f 2f 1 ` f 2f 1 “ Af 1: 1 2 3 1 12 ´ 2 cf ` f ` 2 f “ Af ` B: c df 1 “ Af ` B ` cf 2 ´ f 3: 2 dξ 2 c c History KdV NLPDEs Perturbations mKdV NLS Rogue Summary KdV Traveling Waves - ut ` 6uux ` uxxx “ 0 Seek solutions of form upx; tq “ f px ´ ctq ” f pξq; 1 1 Noting ux “ f pξq and ut “ ´cf pξq; we obtain ODE for f pξq: ´cf 1 ` 6ﬀ 1 ` f 3 “ 0: r´cf ` 3f 2 ` f 2s1 “ 0: Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 12 / 69 ´cﬀ 1 ` 3f 2f 1 ` f 2f 1 “ Af 1: 1 2 3 1 12 ´ 2 cf ` f ` 2 f “ Af ` B: c df 1 “ Af ` B ` cf 2 ´ f 3: 2 dξ 2 c c History KdV NLPDEs Perturbations mKdV NLS Rogue Summary KdV Traveling Waves - ut ` 6uux ` uxxx “ 0 Seek solutions of form upx; tq “ f px ´ ctq ” f pξq; 1 1 Noting ux “ f pξq and ut “ ´cf pξq; we obtain ODE for f pξq: ´cf 1 ` 6ﬀ 1 ` f 3 “ 0: r´cf ` 3f 2 ` f 2s1 “ 0: ´cf ` 3f 2 ` f 2 “ A: Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 12 / 69 1 2 3 1 12 ´ 2 cf ` f ` 2 f “ Af ` B: c df 1 “ Af ` B ` cf 2 ´ f 3: 2 dξ 2 c c History KdV NLPDEs Perturbations mKdV NLS Rogue Summary KdV Traveling Waves - ut ` 6uux ` uxxx “ 0 Seek solutions of form upx; tq “ f px ´ ctq ” f pξq; 1 1 Noting ux “ f pξq and ut “ ´cf pξq; we obtain ODE for f pξq: ´cf 1 ` 6ﬀ 1 ` f 3 “ 0: r´cf ` 3f 2 ` f 2s1 “ 0: ´cf ` 3f 2 ` f 2 “ A: ´cﬀ 1 ` 3f 2f 1 ` f 2f 1 “ Af 1: Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 12 / 69 c df 1 “ Af ` B ` cf 2 ´ f 3: 2 dξ 2 c c History KdV NLPDEs Perturbations mKdV NLS Rogue Summary KdV Traveling Waves - ut ` 6uux ` uxxx “ 0 Seek solutions of form upx; tq “ f px ´ ctq ” f pξq; 1 1 Noting ux “ f pξq and ut “ ´cf pξq; we obtain ODE for f pξq: ´cf 1 ` 6ﬀ 1 ` f 3 “ 0: r´cf ` 3f 2 ` f 2s1 “ 0: ´cf ` 3f 2 ` f 2 “ A: ´cﬀ 1 ` 3f 2f 1 ` f 2f 1 “ Af 1: 1 2 3 1 12 ´ 2 cf ` f ` 2 f “ Af ` B: Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 12 / 69 History KdV NLPDEs Perturbations mKdV NLS Rogue Summary KdV Traveling Waves - ut ` 6uux ` uxxx “ 0 Seek solutions of form upx; tq “ f px ´ ctq ” f pξq; 1 1 Noting ux “ f pξq and ut “ ´cf pξq; we obtain ODE for f pξq: ´cf 1 ` 6ﬀ 1 ` f 3 “ 0: r´cf ` 3f 2 ` f 2s1 “ 0: ´cf ` 3f 2 ` f 2 “ A: ´cﬀ 1 ` 3f 2f 1 ` f 2f 1 “ Af 1: 1 2 3 1 12 ´ 2 cf ` f ` 2 f “ Af ` B: c df 1 “ Af ` B ` cf 2 ´ f 3: 2 dξ 2 c c Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 12 / 69 For A “ B “ 0; c df ξ ´ ξ0 “ : 2 1 2 3 c » 2 cf ´ f upx; tq “ 2η2 sechb2pηpx ´ 4η2tqq where c “ 4η2: For A; B ‰ 0; get Jacobi Elliptic functions. - Leads to (periodic) cnoidal waves. History KdV NLPDEs Perturbations mKdV NLS Rogue Summary KdV Traveling Waves - ut ` 6uux ` uxxx “ 0 Traveling wave solution, upx; tq “ f px ´ ctq ” f pξq; c df 1 “ Af ` B ` cf 2 ´ f 3: 2 dξ 2 c c c df ξ ´ ξ0 “ : 2 1 2 3 c » Af ` B ` 2 cf ´ f b Dr. Russell Herman Nonlinear Waves in Physics May 13, 2020 13 / 69 upx; tq “ 2η2 sech2pηpx ´ 4η2tqq where c “ 4η2: For A; B ‰ 0; get Jacobi Elliptic functions. - Leads to (periodic) cnoidal waves. History KdV NLPDEs Perturbations mKdV NLS Rogue Summary KdV Traveling Waves - ut ` 6uux ` uxxx “ 0 Traveling wave solution, upx; tq “ f px ´ ctq ” f pξq; c df 1 “ Af ` B ` cf 2 ´ f 3: 2 dξ 2 c c c df ξ ´ ξ0 “ : 2 1 2 3 c » Af ` B ` 2 cf ´ f For A “ B “ 0; b c df ξ ´ ξ0 “ : 2 1 2 3 c » 2 cf ´ f b Dr.

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