Soliton and Some of Its Geophysical/Oceanographic Applications
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Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Soliton and Some of its Geophysical/Oceanographic Applications M. Lakshmanan Centre for Nonlinear Dynamics Bharathidasan University Tiruchirappalli – 620024 India Discussion Meeting, ICTS-TIFR 21 July 2016 M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Theme Soliton is a counter-intuitive wave entity arising because of a delicate balance between dispersion and nonlinearity. Its major attraction is its localized nature, ﬁnite energy and preservation of shape under collision = a remark- ably stable structure. Consequently, it has⇒ ramiﬁcations in as wider areas as hydrodynamics, tsunami dynamics, condensed matter, magnetism, particle physics, nonlin- ear optics, cosmology and so on. A brief overview will be presented with emphasis on oceanographic applications and potential problems will be indicated. M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Plan 1 Introduction 2 Historical: (i) Scott-Russel (ii) Zabusky/Kruskal 3 Korteweg-de Vries equation and solitary wave/soliton 4 Other ubiquitous equations 5 Mathematical theory of solitons 6 Geophysical/Oceanographic applications 7 Solitons in higher dimensions 8 Problems and potentialities M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Solitons Progressive wave Superposition of two waves (a) (b) ¼µ ¼µ Ü; Ü; ´ ´ Soliton as a Ù Ù counter-intuitive Ü entity Ü A wave packet Envelope Linear dispersive (c) (d) waves ¼µ ¼µ Ü; Ü; ´ ´ Ù Ù Ü Ü M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Solitons Nonlinear dispersive waves: Solitary waves/solitons Many facets of solitons: a) Tsunami as soliton? g) Magnetic soliton b) Internal soliton h) Topological soliton c) Rossbysoliton i) Planesoliton d) Boresoliton j) Lumpsoliton e) Capillarysoliton k) Dromion f) Opticalsoliton l) Lightbullet M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems John Scott-Russel John Scott-Russel (1808-82) Russel’s observations - Phenomenological relation: c = g(h + k) p M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems J. S. Russel, British Association Reports (1844) “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, preserving its original ﬁgure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles 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”. M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Solitons Soliton creation Scott Russell Aqueduct – Soliton creation – Heriot-Watt University, 1995. Korteweg-de Vries Formulation (1895) Unidirectional shallow water wave propagation in one dimension Incompressible, inviscible ﬂuid M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Solitons y l η(x,t) v a h Bed 0 x η(x, t): amplitude a << h (shallow water) h << l (long wave) = Two small parameters, ǫ = a , δ = h ⇒ h l M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Solitons Fluid motion: V~ (x, y, t)= u(x, y, t)~i + v(x, y, t)~j = Irrotational: ~ V~ =0 = V~ = ~ φ ⇒ ∇× ⇒ ∇ φ(x, y, t) : velocity potential (i) Conservation of density: dρ = ρt + ~ (ρV~ ) = 0, ρ = ρ = constant dt ∇· 0 = 2φ(x, y, t) = 0 – Laplace equation. ⇒ ∇ M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Solitons (ii) Euler’s Equation: dV~ ∂V~ = + V~ ~ V~ dt ∂t · ∇ 1 2 p = φt + ~ φ + + gy = 0 1 ⇒ 2 ∇ ρ = ~ p g~j 0 − ρ0 ∇ − (iii) Boundary conditions: φy (x, 0, t) = 0 y = h + η(x, t); φ1y = ηt + ηx φ1x p =0 = u t + u u x + v v x + gηx = 0 1 ⇒ 1 1 1 1 1 M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Solitons (iv) Taylor Expansion: ∞ φ(x, y, t)= y nφ (x, t) X n n=0 1 2 u = φ x = f y fxx + ..., 1 1 − 2 1 1 3 ∂φ0 v = φ y = y fx + y fxxx + ..., f = 1 1 − 1 6 1 ∂x (v) Introducing ǫ and δ: ′ ′ ′ u1 = ǫc0u1, v1 = ǫδc0v1, f = ǫc0f ′ ′ ′ y = h + η(x, t)= h 1+ ǫη x , t 1 M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Solitons 1 2 ηt + fx + ǫηfx + ǫf ηx δ fxxx = 0 − 6 ga 1 2 ft + ǫﬀx + 2 ηx δ fxxt = 0 ǫc0 − 2 (vi) Perturbation analysis: f = f (0) + ǫf (1) + δ2f (2) + higher order (0) (1) 1 2 (2) 1 f = η, f = η , f = ηxx −4 3 3 δ2 ηt + ηx + ǫηηx + ηxxx = 0 2 6 M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Solitons (vii) The standard contemporary form: ut 6uux + uxxx = 0 ± (ix) Cnoidal and solitary wave solution: ∂f ∂ρ ∂3f u(x, t) = 2f (ζ), ζ =(x ct) = c + 12f + = 0 − ⇒− ∂ζ ∂ζ ∂ζ3 = f (ζ)= α (α α )sn2(√α α (x ct), m) ⇒ 3 − 3 − 2 3 − 1 − c √c = sech2 (x ct + δ), δ = constant (m = 1) ⇒ 2 2 − M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Solitons 2 ¼µ Ü; 1 ´ Ù 0 -4 -2 0 2 4 Ü M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems III. FPU Problem (1955) and Zabusky-Kruskal Analysis (1965) Wave propagation in anharmonic lattices yi-1 yi yi+1 0 i-1 i i+1 N Equation of motion: 2 ∂ yi m = f (yi yi ) f (yi yi− ) , i = 1, 2,..., N 1, ∂t2 +1 − − − 1 − ∂y a2 ∂2y yi y(x, t), yi = y(x, t) + ... −→ +1 ± ∂x ± 2 ∂x2 ± = KdV equation: ut + 6uux + uxxx = 0 ⇒ M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Zabusky-Kruskal Numerical Analysis: Birth of soliton 3 2 µ 1 Ø Ü; ´ Ù 0 -1 -2 0 1 2 Ü Ù´Ü; Øµ 0.06 0.03 800 0 400 -100 0 -50 Ø 0 -400 Ü 50 100 -800 M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Soliton Solitary wave which retains its shape and speed under collision except for a phase shift – Elastic collision. M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Other ubiquitous equations Boussinesq equation 1 2 ut + uux + gηx h utxx = 0 − 3 ηt +[u(h + η)]x = 0 Benjamin-Bona-Mahoney (BBM) equation ut + ux + uux uxxt = 0 − Camassa-Holm equation ut + 2κux + 3uux uxxt = 3ux uxx + uuxxx − M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Other ubiquitous equations Kadomtsev-Petviashville (KP) equation 2 (ut + 6uux + uxxx )x + 3σ uyy = 0 (σ2 = 1: KP-I, σ2 = +1: KP-II) − Modiﬁed KdV equation 2 ut 6u ux + uxxx = 0 ± Sine-Gordon (sG) equation (in light-cone coordinates) 2 2 uxt + m sin u = 0 , m = constant M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Other ubiquitous equations Nonlinear Schr¨odinger equation 2 iqt + qxx 2 q q = 0 , q C ± | | ∈ Continuous Heisenberg ferromagnetic spin equation 2 S~t = S~ S~xx , S~ =(S , S , S ) , S~ = 1 × 1 2 3 Toda lattice equation u¨n = exp[ (un un− )] exp[ (un un)] , i = 1, 2,.., N − − 1 − − +1 − M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Mathematical theory of solitons: Cauchy IVP KdV equation = Linearization ⇒ Given: u(x, 0): Initial value (i) ψxx +[λ u(x, t)]ψ = 0, u(x, t) 0, at x − −→ | | → ∞ (ii) ψt = 4ψxxx + 3uψx + 6ux ψ − = (ψxx )t (ψt )xx = KdV ⇒ ≡ ⇒ M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Inverse Scattering Transform method Direct scattering Scattering u(x,0) data S(0) at t=0 f Time evolution o scattering data Scattering data S(t) u(x,t) at time t>0 Inverse scattering M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Inverse Scattering Transform method 1-soliton solution: u = 2κ2 sech2 κ x 4κ2t + δ − − 2-soliton solution: 2 2 2 2 2 2 κ2 cosech γ2 + κ1 sech γ1 u(x, t)= 2 κ2 κ1 − − (κ cothγ κ tanhγ )2 2 2 − 1 1 N-soliton solution M.