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; ص 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. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems

∂2 Bilinearization: u = 2 ∂x2 log F

2 Fxt F Fx Ft + Fxxxx F 4Fxxx Fx + 3F = 0 − − xx B¨acklund transformation Infinite number of conservation laws/symmetries Geometric structures/Group theoretic Hamiltonian/Bi-Hamiltonian structure Completely integrable infinite dimensional nonlinear dynamical systems

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems VI. Geophysical applications

A. Tsunami as solitons?

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Geophysical applications

Tsunami as solitons?

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Geophysical applications

Tsunami as solitons? 2004 Indian ocean / 1960 Chilean earth quake Typical λ 200 kms, h 1 to 4 kms, a 60 cms ≃ ≃ ≃ 2 Shallowness condition: ǫ = a 10−4 1, δ2 = h 10−4 h ≃ ≪ l2 ≃ Propagation of tsunami waves as KdV like soliton

Recent generalization Phase modulated solitary waves controlled by boundary conditions at the bottom. (A. Mukherjee and M.S. Janaki, Phys. Rev E(2014)). Lump solitons & Rogue waves controlled by ocean currents.

( A. Kundu, A. Mukherjee, T. Naskar, Proc. R. Soc. Lond. (2014))

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Geophysical applications

B. Internal solitons? Seafarers through strait of Malacca often note that in the Andaman sea bands of strongly increasing surface roughness occur. = Internal solitons (SAR images of ERS-1/2 satellites) ⇒ (Solitary wave like distortion of the boundary layer between warm upper sea and cold lower depths) Peculiar striations of 100 km long, separated by 6-15 kms and grouped in packets of 4-8 of the surface of Andaman and Sulu seas. = Secondary phenomena of internal solitons. ⇒ Internal solitons are traveling edges of warm water with enormous energy affecting oil rigs

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Geophysical applications

B. Internal solitons

Figure: SAR image of the Andaman Sea acquired by the ERS-2 satellite showing two internal solitary wave packets

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Geophysical applications

B. Internal solitons

g

surface soliton ρ = 0 v

h ρ 1 1

ρ h 2 2 internal solitary wave

bed

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Geophysical applications

B. Internal solitons Study of Osborne and Birch (1980) of underwater currents by an oil rig drilling at 36000 ft depth in the Andaman sea ∼ KdV solitons Other observations Theory: Consider two incompressible, immiscible fluid, densities ρ1, ρ2 and depths h1, h2 KdV Intermediate long wavelength (ILW) Benjamin-Ono

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Geophysical applications

C. Rossby solitons In the atmosphere of a rotating planet, fluid particle = ⇒ rotation rate (Coriolis freq)/latitude Motion in N/S direction inhibited by constant angular momentum

Rossby Waves: Large scale atmospheric waves caused by variation of Coriolis frequency with latitude = KdV equation can model Rossby ⇒ waves (E-W zonal flow) (long wave/incompressible fluid/β plane approximation etc) Experimental confirmation: Eight years of Topex/Poseidon altimeter observations. (Pacific dynamics) – 1998

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Geophysical applications

D. Bore solitons When a tidal wave/tsunami/storm surge hits an estuary = ⇒ hydraulic jump (step-wise perturbations like a shock wave) in the water height and speed which propagates upstream. Far less dangerous but very similar is the bore (mascaret in French), a tidal wave which can propagate in river for considerable distances.

Examples Seine river / Hooghly river 1983 Japan sea tsunami = bores ascended many rivers ⇒ IHOP of Kansas

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Geophysical applications

Figure: Tidal bore at the mouth of the Aragnau River (Brazil)

E. Capillary solitons F. Resonant three and four wave solitons G. Optical solitons

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems VII. Solitons in other situations A. Multicomponent solitons: Manakov model

2 2 iq ,z + q ,tt + 2( q + q )q = 0, 1 1 | 1| | 2| 1 2 2 iq ,z + q ,tt + 2( q + q )q = 0.

2 2 | 1| | 2| 2

Ë

Ë

¾ ¾

2

2 Ë

1.5 ½ Ë

1.5 ½

¾

¾ jÕ j

½ 1

Õ j j ½ 1 0.5 2 0.5 2 -100 1 -100 1 -5

Ë -5 ½ 0

z Ë 0 0 ½ z t 0 -1 t 5 -1

5

Ë

¾ Ë 10 -2 ¾

10 -2

Ë

¾

Ë ¾

4 4

Ë Ë ½

3 ½ 3

¾ ¾

jÕ j jÕ j ¾ ¾ 2 2 1 2 1 2 -100 -100 1 1

-5 -5 Ë

Ë 0 0 ½ ½ 0 z 0 z t t -1 -1

5 5

Ë Ë

¾ ¾ 10 -2 10 -2 M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems

Breathers and Rogue Waves

Nonlinear Schr¨odinger Equation

2 iut + uxx + q u u = 0, u C | | ∈ Coupled NLS Manakov Equation

Breather solution of NLS equation

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems

Breathers and Rogue Waves

Rogue wave solution of NLS equation

Rogue waves in BECs/Optics

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems

B. (2+1)- dimensions:

j j 3 Plane solitons 2 10 1 5 0 -10 0

-5 Ý 0 -5

Ü 5 10-10

15 -2 Lump solitons u 10 5 -1 0 2 0 y 1 0 1 x -1 -22

M. Lakshmanan Soliton and Some of its Geophysical

Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems

j j

4 0.8 Dromions 0.6 2 0.4 0 0.2 Ý -2 0 -2 -4 0

2 Ü

C. (3+1)- dimensions Localized solutions?

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems VIII. Conclusion

Solitons are very stable nonlinear entities Occur in (1+1) and (2+1) dimensions Have considerable physical relevance What happens in (2+1) and (3+1) dimensions?

M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems References

M. Lakshmanan and S. Rajasekar (2003) Nonlinear Dynamics: Integrability and Chaos, Springer, Berlin M. Lakshmanan (2009) Tsunamis and other Oceanographical Applications of Solitons To appear in Encyclopedia on Complex Systems, Springer, Berlin M. J. Ablowitz and P. A. Clarkson (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (UK) T. Dauxois and M. Peyrard (2006) Physics of Solitons, Cambridge University Press, Cambridge (UK)

M. Lakshmanan Soliton and Some of its Geophysical