Solitons and Dispersive Shock Waves

Home , Soliton

Mark J. Ablowitz

Department of Applied Mathematics University of Colorado, Boulder June 2013 Outline

I Introduction

I Water waves: Benney-Luke and Kadomtsev-Petviashvili Eq.

I Two dimensional soliton interactions

I Ocean observations

I Dispersive shock waves (DSWs)

I Conclusion Water waves and KdV Eq

I 1834 Observation of solitary wave: Russell I 1870’s – 1960 – Boussinesq, Korteweg & deVries (KdV): water waves... I 1960’s– mathematicians developed approx methods to find reduced eqs governing physical systems; KdV and nonlinear Schrödinger(NLS) eq are two important 1+1 dim’l– “universal” eqs I 1965 –computation on KdV eq. (Kruskal and Zabusky) –introduced the term Solitons I 1967 – Method to find solution of KdV (Gardner, Greene, Kuskal - photo below, Miura) KdV to IST

I 1970’s-present – KdV and further developments led to new methods & results in math physics

I Applications to many physically interesting systems

I Termed Inverse Scattering Transform (IST)– solitons are special solutions

I IST and solitons in 1+1, 2+1 dimensions –applications to NL waves Water Wave Equations

Classical equations: Deﬁne the domain D by D = < x1, x2 < , h < y < η(x, t), x = (x1, x2), t > 0 {−∞ ∞ − } The water wave equations satisfy the following system for φ(x, y, t) and η(x, t):

∆φ = 0 in D

φy = 0 on y = h − ηt + φ η = φy on y = η ∇ · ∇ 1 2 η φt + φ + gη = σ ( ∇ ) on y = η 2|∇ | ∇ · p1 + η 2 |∇ | T where g: gravity, σ = ρ : T surface tension, ρ: density WW-Nonlocal Spectral Eq

Work with A. Fokas, Z. Musslimani (JFM, ’06), reformulation: 2 eq, 2 unk: η, q = φ(x, η), rapid decay: 1 nonlocal spectral eq. and 1 PDE; ﬁxed domain Z k q dxeik·x (iη cosh[κ(η + h)] + sinh [κ(η + h)] · ∇ ) = 0 (I ) t κ

2 1 2 (ηt + q η) η qt + q + gη ∇ · ∇ = σ ( ∇ )(II ) 2|∇ | − 2(1 + η 2) ∇ · p1 + η 2 |∇ | |∇ | 2 2 2 x = (x1, x2), k = (k1, k2), κ = k1 + k2 , q(x, t) = φ(x, t, η(x, t)) WW: ﬁgure

y = η

q = φ(x, η) y = h − φ(x, y) , ρ

Water wave conﬁguration Note: η, q used by Zakharov (’68) Remarks

I (MJA,AF ZM ‘06) Derived nonlocal formulation and found:

I Conserved quantities and new integral relations I Asymptotic reductions:

I 1+1: KdV, Nonlinear Schr¨odinger(NLS) eq; i.e. ﬁnd both shallow and deep water reductions I 2+1 Benney-Luke (BL) and Kadomtsev-Petvashvili (KP) eq

I MJA and T. Haut (’08-’10):

I nonlocal eqs for waves with 1 and 2 free interfaces I asymptotic reductions: ILW-BL, ILW-KP I high order asympt. expn’s of 1-d and 2-d solitary waves Nondimensional Variables

We can make all variables nondimensional (nd): x x c x0 = 1 , x0 = γ 2 , aη0 = η , t0 = 0 t , ... 1 l 2 l l l, a are characteristic horiz. length, amplitude, and γ is a nd 0 transverse length parameter; c0 = √gh; hereafter drop

Eq are written in terms of nd variables a ε = h << 1: small amplitude h µ = l << 1: long waves γ << 1: slow transverse variation WW-Asymptotic Systems

a h Expd cosh, sinh use nd paramters: ε = h , µ = l Find: Benney-Luke (BL, ’64) eq. (nmlz’d surface tension, σ˜ = σ 1/3): − 2 2 2 qtt ∆˜ q +σµ ˜ ∆˜ q + ε(∂t ˜ q + qt ∆˜ q) = 0 (BL) − |5 | ∆˜ = ∂2 + γ2∂2 ˜ q 2 = q2 + γ2q2 x1 x2 |5 | x1 x2 If ε = µ2 = γ2 1 then BL yields KP equation; after rescaling KP eq in std form (x = (x1, x2) (x, y)) →

∂x (ut + 6uux + uxxx ) 3 sgn(˜σ)uyy = 0 − Note:σ ˜ > 0 ‘strong’ surface tension: KPI Eq σ˜ < 0 ‘weak’ surface tension: KPII Eq KP Equation: Line Solitons

KP equation in standard form with small surface tension (‘KP II’)

∂x (ut + 6uux + uxxx ) + 3uyy = 0 KP equation has line soliton solutions; simplest ones:

∂2 log F u = u = 2 N N ∂x2

Where FN is a polynomial in terms of exponentials:

η1 η1 η2 η1+η2+A12 F1 = 1 + e , F2 = 1 + e + e + e 2 2 2 2 A12 (k1−k2) −(P1−P2) ) where ηj = kj (x + Pj y (k + 3P )t), e = 2 2 − j j (k1+k2) −(P1−P2) kj , Pj are constants KP Eq–Single Line Soliton

Basic line solitons are solutions of KP (KdV) eq; they are observed η routinely: F1 = 1 + e 1 KP single line soliton –physical KP Line Soliton Solutions

Typical KP two-soliton ‘X-type’ interaction with ‘short stem’: A F2, e 12 = O(1)

Typical KP two-soliton interaction ‘X-type’ with ‘long stem’; A F2, e 12 1 KP Line Soliton Solutions–con’t

A Typical KP ‘Y-type’ interaction; F2, e 12 0 → KP Line Soliton Solutions–con’t

A Typical KP ‘H-type’ interaction; F2, e 12 1 Beaches and Line Solitons

Planar waves seen frequently. But what about ‘X’ and ‘Y’ type waves? There was one photo of ‘X-type’ with ‘long stem’: Shallow water waves oﬀ the coast of Oregon (MJA & Segur 1981) Recent Beach Photos–Short Stem X

Large amplitude X wave with short stem Short-stem X-type–con’t

Two plots and a photograph of a typical short-stem X-type interaction: eA12 = O(1) Beach Photos–con’t

Depth of the shallow water waves can be understood by noting the person walking on the beach–not noticing a nearby an X interaction! Second Beach

I X- and Y-type most I Nearly ﬂat common-occur often

I Shallow water: 2 – 7 in deep I Recent statistics

I Being near a jetty helpful I Also Shi-Shi beach I Within 2 hrs of low tide Washington State – Bernard Deconinck Recent Beach Photos–Long Stem

Long stem X –similar to Oregon photo Beach Photos–Long stem–con’t

A plot and photos of a long-stem X-type interaction: eA12 1 Beaches–Extremely Long Stem X

Extremely long stem Extreme-long-stem–con’t

A plot and photographs of an extreme-long-stem X-type interaction, eA12 1 Y-type interaction

Miles found Y-type solutions of the KP equation in 1977 and related it to Mach-stem reﬂection in gas dynamics.

A plot and photographs of a Y-type interaction: eA12 = 0 H-type interaction

Final ‘two-soliton’ H-type interaction; here the ‘stem’ has a lower rather than a higher amplitude: eA12 1 Beach Photos–con’t

Multi-interaction Beach Photos–con’t

Egret enjoying X wave – and sunset

Ref. MJA and D. Baldwin, Phys. Rev. E, 2012 Recent Statistics–Videos

Preliminary statistics–from Nuevo Vallarta Mex

I Taken March 1-3, 2013 over 6 hours

I Counted average of 65 interactions over average 2hr period

I 35%: Short -X; 25 %: Long -X; 25 % -Y; 10% -H; 5%: More complex–web-type

I Expect more statistics ... soon Videos of wave interactions can be found on MJA web page Tsunami Propagation

Tsunami prop. can be a shallow water/longL05606 wave phenomenaSONG ET AL.: MERGING TSUNAMIS DETECTED L05606

Maxworthy (’80) Song et al. (’12) Internal waves Tohoku tsunami

Figure 4. (a) ENVISAT pass at 5:25 hours after the quake, (b) Jason-1 pass at 7:30 hours after the quake, (c) Jason-2 pass at 8:20 hours after the quake, and (d–f) comparing model tsunami (black) with the satellite altimetry data along the passes, respectively. Black arrows indicate locations of merging tsunamis. Only the Jason-1 satellite, indicated by the red arrow, was at the right location and right time to catch the tsunami merge phenomenon.

different tsunami heights, which are the focus of the scien- to determine the potential of tsunami generations, but tific discovery of this study, and are shown in Figures 4a–4f. unfortunately, rapid estimate of earthquake’s magnitude has It is noted that the tsunami height observed by Jason-1 not been successful. For example, the initial estimate of the (Figure 4e) is about twice as high as those observed by March 11, 2011 earthquake was significantly smaller. Had ENVISAT (Figure 4d) and by Jason-2 (Figure 4f) along the the magnitude been correctly estimated initially, more lives same tsunami front, suggesting the amplification of the could have been saved [Ando et al., 2011]. Here, we have tsunami height is a result of merging waves. Such spatially demonstrated an alternative approach using the existing varying amplification would be difficult for a single in-situ Japanese GSI GPS network to infer seafloor displacement instrument to record because of the limited width of the jets. that determines the energy an undersea earthquake transfers to the ocean to generate a tsunami, instead of using of the 5. Discussions earthquake’s magnitude, which can more accurately and rapidly determine the tsunami power (scale). The basic steps [15] In summary, we have not only confirmed the exis- are the following: First, locate an earthquake epicenter from tence of merging tsunamis that should be the focus for far- seismometers (a few minutes after an initial quake); sec- field tsunami forecast, but also demonstrated the feasibility ondly, collect near-field GPS-derived land velocities and of using real-time GPS data for near-field tsunami early infer the seafloor motions (a few more minutes of latency warning [Hammond et al., 2011; Ohta et al., 2012]. The are possible); and thirdly, calculate the tsunami-source current tsunami warning system is based on early-estimated energy or scale based on the GPS-predicted seafloor motions earthquake-moment magnitude primarily using seismic data and local topography. If the oceanic energy scales greater

5of6 Conclusion-Water Waves

Nonlinear water waves

I Water waves interesting history and applications

I Reformulate water wave eq as a nonlocal spectral system

I Asymptotic systems: shallow water: BL, KP

I Two dimensional solitons – ocean observations

I Can also analyze 2-D solitons in BL eq via perturbation (MJA, Curtis ’11,13) Nonlinear shock waves –breaking

A simple nonlinear wave equation which exhibits breaking is the inviscid Burger’s equation

ut + uux = 0 Pulse-like initial values break in ﬁnite time into multivalued solution e.g.:

Leads to study of certain types of NL eq.–hyperbolic systems; goes back to work of Riemann, Rayleigh; large ﬁeld of study. Nonlinear shock waves

I Classical or viscous shock waves (VSWs)

I Characterized by localized steep gradient across a shock front and dissipation of energy I Viscous shock waves (VSWs) are regularized by dissipation I Many applications–e.g. gas dynamics, sonic booms, blast waves... I Dispersive shock waves (DSWs)

I Characterized by: a soliton front followed by a modulated wave train I Dispersion is the regularizing mechanism I DSW observations:

I Plasmas, Fluids, e.g. shallow water waves: undular bores, Superﬂuids: Bose-Einstein condensates, NL optics ...

I Theory goes back to studies in plasma physics 1960s-’70s Typical Equations

I Viscous shock waves (VSWs) The simplest equation describing VSWs–combines nonlinearity and dissipation is Burgers’ equation,

ut + uux = νuxx , ν > 0

I Dispersive shock waves (DSWs) The simplest equation describing VSWs –combines nonlinearity and dispersion– is the Korteweg–de Vries (KdV) equation,

2 ut + uux + ε uxxx = 0, ε > 0

Note: ν, ε typically small A single, shock-forming step evolved in time

Typical numerically computed solution of the KdV equation is in black and Burgers’ equation is in red

2.0 2.0 t=0 t=2 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 – 1.0 – 0.5 0.0 0.5 1.0 –2 –1 0 1 2 2.0 2.0 t=10 t=20 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 –10 –5 0 5 10 –20 –10 0 10 20 Background–DSW

DSW (dispersive shock wave) theory

I DSW – KdV Equation: Gurevich & Petiavskii (GP, ’74)

I GP made use of Whitham(W) theory (’65) –slowly varying waves

I DSW – NLS–W thy: Gurevich & Krylov (’87), -& El (’95), El (’02–) ,...

I Many physical applications: plasma physics, water waves, BEC, NL optic s...

I DSW interactions: W thy: Grava et al (’02), H-MJA (’07), MJA, Baldwin and H (’09)

I DSW interactions: IST: MJA, Baldwin (’13) Observations: Plasma’s–1970

See: Observations of Collisionless Shock Waves, Taylor, Baker & Ikezi, PRL, 1970 Undular Bore

Undular bore on the river Severn – Lighthill Waves in Fluids 1978 Two DSW Interaction

2.0 2.0 1.5 t = 2 1.5 t = 50 1.0 1.0 0.5 0.5 0.0 0.0 2 0 2 4 6 8 10 40 20 0 20 40 2.0 2.0 1.5 t = 10 1.5 t = 400 1.0 1.0 0.5 0.5 0.0 0.0 10 5 0 5 10 15 400 200 0 200 400 KdV eqn. Burgers’ eqn. General step-like initial data

Analyze KdV equation with the BCs

lim u(x, t) = 0 and lim u(x, t) = 6c2, c2 > 0 x→−∞ x→+∞ −

u As t ﬁnd: u(x, t) a single phase→ DSW ∞ that is → I: exponentially small for x 2c2t; − II: slowly-varying cnoidal wave for 12c2t x 2c2t; and − − x III: decaying slowly-varying oscillatory III II I solution for x 12c2t. − MJA, Baldwin (’13) Compare step-like with decaying initial data

MJA and Segur (’77): the long-time solution of the KdV equation with decaying initial u data has four regions: I: exponentially small region for x (t); ≥ O II: special similarity solution (PII) for x (t1/3); | | ≤ O III: ‘collisionless shock’ region 1/3 2/3 x for ( x) = [t (log t) ]; IV III II I and − O IV: decaying oscillatory solution for ( x) (t). − ≥ O Step-Like Data– Burgers eq: t → ∞ As t individual VSW shocks in Burgers eq. merge . → ∞

wt + wwx νwxx = 0, − 2 with limx→−∞ u(x, t) = 0 and limx→+∞ u(x, t) = c − From long time asymptotics of the sol’n via Hopf-Cole transformation (*) ﬁnd

c2 c2 c2 w(x, t) 1 + tanh x x0 + t , ∼ − 2 4ν − 2

where x0 depends on ICs

φx ( ) w = 2ν , φt = νφxx ∗ − φ MJA, Baldwin (’13) Step-Like Data–KdV eq: t → ∞

2 ut + uux + ε uxxx = 0 2 with limx→−∞ u = 0, limx→+∞ u = 6c Use IST: −

I Associate KdV with linear (Lax) pair

I One of the linear eq is the time independent Schr¨odingereq:

u(x, t) ε2v + + λ2 v = 0 xx 6

I Transform IC to scattering data

I Find time evolution of data

I Find linear integral eq to reconstruct u(x,t)

I Find long time limit from above and matching Direct problem

First transform the initial data into scattering data

I Use the limits of u to deﬁne the eigenfunctions:

I φ exp( iλx/ε) and φ¯ exp(iλx/ε) as x and ∼ − ∼ → −∞ I ψ exp(iλr x/ε) and ψ¯ exp( iλr x/ε) as x + , ∼ ∼ − → ∞ 2 2 where λr √λ c . ≡ − I The branch cut in λr is a signiﬁcant diﬀerence between the step and the decaying problem.

I The scattering data — the transmission coefficient T and the reflection coefficient R — satisfy

T (λ, λr ; t)φ = ψ¯ + R(λ, λr ; t)ψ Time evolution

Recall

T (λ, λr )φ(x; λ) = ψ¯(x; λr ) + R(λ, λr )ψ(x; λr ) The other linear equation in the Lax pair lets us evolve the transmission and reﬂection coeﬃcients in time:

2 3 2 i(4λ λr −4λ +2c λr )t/ε T (λ, λr ; t) = T (λ, λr ; 0)e

and 2 2 i(8λ λr +4c λr )t/ε R(λ, λr ; t) = R(λ, λr ; 0)e

Unlike the decaying problem, the transmission coeﬃcient T depends on time Inverse problem

To recover the solution, we use the associated Gel’Fand-Levitan-Marchenko (GLM) integral eq: Z ∞ G(x, y; t) + Ω(x + y; t) + G(x, z; t)Ω(y + z; t) dz = 0 x where

1 Z ∞ X Ω(ξ; t) = Reiλr ξ/ε dλ + C e−κ˜j ξ/ε 2επ r j −∞ j Z c √ 1 2 − c2−λ2ξ/ε + λT /λr e dλ 2επ 0 | | From G, d u(x, t) = 6c2 + 12ε2 G(x, x; t) − dx Asymptotic methods

Matched asymptotic analysis is used to ﬁnd the long time solution of the KdV eq.

I Shock front: In GLM eq. – the asymptotic expansion of the kernel (Ω) is found via steepest-descents Contribution from T is dominant near shock front The Neumann series of the GLM eq. is summed-this provides sol’n at shock front

I DSW: Multiple-scales perturbation theory is used to ﬁnd a slowly varying cnoidal-wave sol’n that matches the shock front

I Trailing edge: Employ WKB and matched asymptotics to ﬁnd the decaying solution to the left Shock front

I Near the shock front, the contribution from the transmission coeﬃcient dominates; The Neumann series is summed to ﬁnd

2 2 2 hc i u(x, t) 6c + 12c sech (ζ ζ0) ∼ − ε − 2 2 where ζ0 depends on ∂ T (λ; t) λ=0,t=0 and λ| | | 3ε ζ = x 2c2t log(6c2t x) + . − − − 4c − ···

Note: In the vanishing case, the phase of PII sol’n depends on 00 0 2 (R + [R ] )λ=0,t=0 DSW

2 I u(x, t) = 6c + g(ζ, T ) where Z δζ and T δt, 0 < δ 1− ≡ ≡ I The KdV eq. gives

2 2 ε gζζζ + ggζ 4c gζ − 2 2 3ε(3ε gζζζ + ggζ 12c gζ ) = δ − gT + , 4c(8c2T + Z) − ···

I Introduce θ(ζ, t) such that θζ κ(Z, T ) and ≡ θt ω(Z, T ) κV ; => (θζ )t = (θt )ζ => cons waves ≡ − ≡ − κT (κV )Z = 0 − I Expand 2 g(θ, Z, T ) = g0(θ, Z, T ) + δg1(θ, Z, T ) + δ g2(θ, Z, T ) + ··· I The solution of the (1) equation is O 2 g0(θ, Z, T ) = a(Z, T ) + b(Z, T ) cn 2K(θ θ0), k(Z, T ) , { − } with

κ2 = b/[48ε2k2K 2], a = 4c2 V 2b/3 + b/(3k2) − − where cn(z, k) is the Jacobian elliptic ‘cosine’, K(k(Z, T )) is the complete elliptic integral of the ﬁrst kind

I Find two secularity eq. –with cons waves this gives three 1st order PDEs for b, k, V Connection with Whitham

Can making a change of variables to transforms the three conservation equations for b, k, V into: ∂r ∂r i + v (r , r , r ) i = 0, i = 1, 2, 3, ∂T i 1 2 3 ∂Z

where vj are explicit fcns of b, k, V which in turn are fuc’ns of rj

2 g0(θ, Z, T ) = r1 r2 + r3 + 2(r2 r1) cn (2K(θ θ0), k) . − − − 2 where as t ﬁnd r1 = 0, r3 = 6c and r2 = r2(Z/T ) Above diagonal→ ∞ system was found by Whitham (’65) and used by GP in ’74 Shock tail

On the left of the DSW, there is a slowly-varying oscillatory solution:

X 1/4 A2(1 cos 2θ) u(x, t) 2A cos(θ) − + (τ −3/2), ∼ √τ − 3τ√X O where X = x/(3t), τ = 3t, − " # τ 2 A2 log(τX 3/2) θ θ X 3/2 + 0 + (τ −2) , ∼ ε 3 − 18 τ τ O

and A(X ) and θ0(X ) are related to the initial (scattering) data

√1 The sol’n in this region is O( τ ) has the same form in both the decaying and the non-decaying case (cf. MJA, Segur ’81) Shock tail–ﬁg.

u 6 c2

-6 c2 x Tail DSW Front Conclusion–DSWs

Nonlinear waves: DSWs

I Background

I Numerical calculations–merging of VSWs, DSWs

I Burgers eq.–merging of VSWs via Hopf-Cole Transf.

I KdV eq. –merging of DSWs via IST

I IST: GLM eq.: Find long time asymptotic solution to the right of DSW

I Match to soliton front: determine the phase of DSW

I Find DSW via diﬀ eq governing slowly varying elliptic fcn

I Find/match to decaying solution to the left of DSW

I Procedure similar to decaying case