Integrable dynamics of soliton gases Gennady EL II Porto Meeting on Nonlinear Waves 2-22 June 213
Outline INTRODUCTION KINETIC EQUATION HYDRODYNAMIC REDUCTIONS CONCLUSIONS
Motivation & Background Main premise: nonlinear dispersive wave systems can demonstrate complex behaviour demanding a statistical description; Experimental/observational evidence of the presence of turbulent regimes in physical systems well described by integrable equations. e.g. Osborne; Onorato; Pelinovsky (1995 213) water waves; Suret, Randoux, Picozzi (211) fibre optics; Theoretical framework: Outlined in V.E. Zakharov, Turbulence in integrable systems, Stud. Appl. Math. (29). Two fundamental problems: Wave turbulence: collective dynamics of incoherent weakly nonlinear waves; Soliton turbulence: collective dynamics of incoherent solitons.
What is a soliton gas? Soliton gas: a semiclassical incoherent soliton ensemble characterised by a distribution function f(λ, x, t) over the (IST) spectrum. Microscopic dynamics: integrable systems (KdV, NLS...) well understood. Macroscopic dynamics: evolution of f(λ, x, t) plus moments u, u 2, u(x)u(x +τ), etc. far less explored. A closely related object: a dispersive shock wave (DSW) a semiclassical coherent soliton ensemble. Soliton gas can be viewed as a fully randomized DSW. The nonlinear wave field u(x, t) of the soliton gas represents an integrable soliton turbulence.
Soliton gas formation: KdV/defocusing NLS The semi-classical limit of the Zabusky-Kruskal (1965) problem: u t + 6uu x +ǫ 2 u xxx =, u(x, ) = u (x), u (x+ L) = u (x) ǫ 1 Left: Z-K experiment; Right: the KdV DSW (ǫ 1) The gradient catastrophe at each period is resolved via a (single-phase) DSW. The DSWs expand, overlap and interact yielding the complex multiphase (multiperiodic) dynamics. In the semi-classical (ǫ ) limit the dynamics are described by the hyperbolic Whitham equations. The number of oscillating phases N = N(x, t) grows with time at each point so that N as t : an integrable Landau-Hopf scenario of transition to (soliton) turbulence (Gurevich, Zybin & El, 1999).
Soliton gas formation: focusing NLS Long-time behaviour in the NLS "box problem" (El, Khamis & Tovbis) in progress iψt + ψxx + 2 ψ 2 ψ =, ψ(x, ) = A[H(x ) H(x L)], L 1. The bifurctaion / genus growth mechanism is now nonlinear modulational instability (the Whitham equations are elliptic). 4 12 35 1 3 t2 6 15-4 -2 x 2 4 t= 6 4 2 t=5 6 4 2 t=1 6 4 2 t=15 4 6 4 2 t=2 2 6 4 2 t=25 8 4 6 1 5 density 8 25 6 4 2 t=3 4 2 x 2 4 6 Previous results: The genus one bifurcation: El, Gurevich, Khodorovskii & Krylov (1993); Kamchatnov (1997); Jenkins & K. McLaughlin(211). Peregrine breathers (rogue waves) at the genus 2 breaking curves (caustics): Bertola & Tovbis (28); Hoefer & Tovbis (211); Grimshaw & Tovbis (213).
Rarefied soliton gas (Zakharov, JETP 1971). The KdV equation u t + 6uu x + u xxx =. Consider an infinite sequence of the KdV solitons randomly distributed on R with small density ρ 1. The IST spectrum S = { η 2 n ; n = 1, 2,...}. W.l.o.g. assume η n [, 1]. Due to isospectrality, the only dynamical process that takes place is the phase-shift in the two-soliton collisions. Introduce the continuous spectral distribution function f(η) for η n such that the number of solitons with η n [η,η + dη] in the interval [x, x + dx] is f(η)dηdx. Then f(η, x, t) satisfies the kinetic equation: f t +(sf) x =, (1) s(η) = 4η 2 + 1 1 ln η +µ η η µ f(µ)[4η2 4µ 2 ]dµ. (2) Here f(η) f(η, x, t); s(η) s(η, x, t). Important: system (1), (2) is obtained in the small-density approximation, ρ = 1 f(η)dη 1;
Beyond the rarefied gas approximation: the thermodynamic limit of the Whitham equations (El, 23) The idea (KdV): Soliton gas formation via multiple-phase DSW interactions; In the semiclassical limit, the DSW is described by the Whitham modulation equations: t λ j + V j (λ) x λ j =, j = 1, 2,...,2N + 1, where λ j are the endpoints of the spectral (semi-classical IST) gaps and N is the number of phases in the DSW. One can introduce the integrated density of states ρ(λ) associated with the Whitham spectrum {λ j }. Consider the (continuum) limit of the Whitham system as N such that ρ(λ) = O(1): the thermodynamic-type limit. This implies an asymptotic structure of the (band/gap) spectrum: gaps N 1, bands e N for N 1. The structure of the spectrum as N is consistent with the infinite-soliton limit. Semi-classical incoherent soliton ensemble (El, Krylov, Molchanov & Venakides 21).
Soliton-gas kinetic equation The result: a nonlinear integro-differential equation for the spectral measure f(η, x, t) the kinetic equation for a dense soliton gas. f t +(fs) x =, (1) s(η) = 4η 2 + 1 η 1 ln η +µ η µ f(µ)[s(η) s(µ)]dµ. (2) Here f(η) f(η, x, t) is the spectral distribution of solitons in the gas and s(η) s(η, x, t) is the (macroscopic) gas velocity. The small-density ( 1 fdη 1) expansion of (2), yields s(η) = 4η 2 + 1 η 1 ln η +µ η µ f(µ)[4η2 4µ 2 ]dµ+o(ρ 2 ) (3) Zakharov s (1971) kinetic equation for a rarefied soliton gas. So Eqs. (1), (2) represent the generalisation of Zakharov s kinetic equation to the case of the gas of finite density.
Generalised kinetic equations for soliton gases with elastic collisions (El & Kamchatnov, PRL (25)) Two ingredients: (i) the speed of a free soliton S(η) and (ii) the phase shift x η,µ = G(η,µ) due to the soliton-soliton collision. Then the self-consistent definition of the soliton velocity s(η) in a dense soliton gas with the spectral distribution f(η) is given by the integral equation s(η) = S(η)+ where f(η) f(η, x, t), s(η) s(η, x, t). G(η,µ)[s(η) s(µ)]f(µ)dµ, Isospectrality implies the conservation equation for the spectral distribution function f(η, x, t): f t +(sf) x =.
Example: focusing NLS soliton gas iu t + u xx + 2 u 2 u =. The NLS bright soliton is characterised by a complex eigenvalue λ = α+iγ, < α <, < γ < in the ZS scattering problem, where 2γ is the solution amplitude and 4α the speed. The phase shift due to collision of λ-soliton (λ = α+iγ) with µ-soliton (µ = ξ + iη) is given by G(α,ξ,γ,η) = 1 2 2γ ln λ µ λ µ. Then the kinetic equation for the gas of bright NLS solitons is f t +(sf) x =, s(α,γ) = 4α+ 1 ln λ µ 2 2γ λ µ f(ξ,η)[s(α,γ) s(ξ,η)]dξdη. Here f f(α,γ; x, t), s s(α,γ; x, t)
Multicomponent cold-gas hydrodynamic reductions We introduce the cold gas multi-flow ansatz: f(η, x, t) = N f i (x, t)δ(η η i ), i=1 which reduces the (integro-differential) kinetic equation to a system of N hydrodynamic conservation laws: t u i = x (u i v i ), i = 1,...,N, where u i = η i f i (x, t) and v i = s(η i, x, t). The densities u i and velocities v i = v(η i, x, t) are related algebraically: v i = ξ i + m i ǫ im u m (v m v i ), ǫ ik = ǫ ki, ξ i = S(η i ), ǫ ik = 1 η i η k G(ηi,η k ) >, i k.
Hydrodynamic reductions: N = 2 For N = 2 the system of hydrodynamic laws assumes the form t u 1 = x (u 1 v 1 ), t u 2 = x (u 2 v 2 ) u 1 = 1 ǫ 12 v 2 ξ 2 v 1 v 2, u2 = 1 ǫ 12 v 1 ξ 1 v 2 v 1. Passing to the Riemann invariants we obtain v 1 t = v 2 v 1 x, v 2 t = v 1 v 2 x. (1) The system (1) is strictly hyperbolic (hence modulational stability) and linearly degenerate (i.e. its characteristic velocities do not depend on their own Riemann invariants). Hence no wave breaking. What about N > 2?
Integrability of hydrodynamic reductions Theorem (El, Kamchatnov, Pavlov & Zykov, J.Nonlin.Sci. 211) N-component hydrodynamic type system t u i = x (u i v i ), i = 1,...,N, v i = ξ i + i k ǫ ik u k (v k v i ), ǫ ik = ǫ ki, where ξ 1,ξ 2,...,ξ N are constants and ˆǫ is a constant symmetric matrix, ǫ ik = ǫ ki, is: diagonalizable ( {r j (u)} : r i t = V i (r)r i x, i = 1,...,N) linearly degenerate, ( i V i =, i = 1,...,N) semi-hamiltonian (i.e. integrable by the generalised hodograph transform (Tsarev 1985, 1991)), for any N.
Riemann invariants: explicit construction Pavlov, Taranov& El, Theor. Math, Phys. (212) Let ˆǫ = [ǫ mn ] N N be a symmetric matrix, ǫ ik = ǫ ki ; and ǫ ii = r i (u). Theorem 1 Algebraic system v i = ξ i + N solution: u i = m=1 ǫ im u m (v m v i ) admits parametric N β mi, v i = 1 u i m=1 N m=1 ξ m β mi, where symmetric functions β ik (r) are the elements of the matrix ˆβ = [β mn ] N N such that ˆβˆǫ = 1. ( ) Theorem 2 Under the parametric representation (*) the N-flow reduction of the kinetic equation assumes the Riemann form r i t = v i (r)r i x ( ) The Riemann invariant representation (**) makes possible effective construction of exact solutions to the kinetic equation.
Toy Problem 1 Propagation of a trial soliton through a one-component soliton gas. Consider the KdV equation u t + 6uu x +ǫ 2 u xxx =, ǫ 1. (1) with the initial condition in the form of a randomized soliton lattice plus a trial soliton. u(x, ) = + i= 2µ 2 i sech2 ǫ 1 [µ i (x 4µ i t x i )]+2η 2 1 sech2 ǫ 1 [η 1 (x 4η 1 t)], (2) where µ i is a Gaussian random value with the mean η and a narrow variance σ η ; x i is a Poisson random value distributed on the x-axis with some finite density κ. The amplitude of the trial soliton 2η 2 1 > 2η2. Question: what will the average speed of the trial soliton be while it propagates through a stochastic soliton lattice?
Solution Consider the hydrodynamic reduction of the kinetic equation, using the two-flow ansatz. f = f (x, t)δ(η η )+f 1 (x, t)δ(η η 1 ) Find expression for the average velocity of the second component: where s 1 = 4η 2 1 4(η2 1 η2 )α f 1 α f α 1 f 1, (1) α,1 = 1 ln η +η 1 η,1 η η 1 >. The speed of the trial "η 1 - soliton is obtained from (1) by setting f (x, t) = κ, f 1 (x, t) =. Then s 1 = 4(η2 1 η2 α κ ) 1 α κ > 4η 2 1,
Direct numerical simulations (Carbone, Dutykh & El, in progress) Propagation of a trial soliton through a homogeneous one-component cold soliton gas (disordered soliton lattice). High accuracy geometric numerical scheme: (Dutykh, Chhay & Fedele 213); 1.8 1.6 1.4 1.2 u(x,) 1.8.6.4.2 5 1 15 2 25 3 35 x Left: initial conditions. The spectral parameter of the soliton gas η is distributed by Gaussian with the mean < η >=.35 and variance σ =.2. The density of the soliton gas κ =.61. (2 solitons ), the trial soliton spectral parameter η 1 is taken in the interval.45.9. Right: Comparison for the dependence of the averaged speed of the trial soliton s 1 on its spectral parameter η 1. Solid line: analytical solution; symbols: numerical data.
Toy Problem 2. Collision of two cold soliton gases: focusing NLS No continuous solution. Two hyperbolic conservation laws available: ρ 1 t + (s 1ρ 1 ) x =, ρ 2 t + (s 2ρ 2 ) x =, s 1 = s 1 (ρ 1,ρ 2 ) s 2 = s 2 (ρ 1,ρ 2 ) Weak solution: three constant states separated by two strong discontinuities. ρ 1 ρ 1c+ρ2c ρ1c ρ2c ρ 2 Detailed solution in: (El & Kamchatnov, PRL 25). c - t c + t x
Wave field moments in a soliton gas: the KdV case For the two first moments we obtain u(x, t) = 4 ηf(η, x, t)dη, u 2 (x, t) = 16 3 η 3 f(η, x, t)dη Important restriction: σ = u 2 u 2 (1) For instance, for a one-component soliton gas f = f (x, t)δ(η η ), so u = 4η f (x, t), u 2 = 16 3 η3 f (x, t). i.e. the variance σ = 16f η( 2 η 3 f ). Now one can see that (1) imposes a restriction on the possible density values f for a one-component soliton gas with a given η : f < f cr = η 3. Critical density for a single-component soliton gas.
Conclusions & Perspectives A consistent procedure of the derivation of the kinetic equations for soliton gases in integrable systems is proposed. The N-component cold-gas hydrodynamic reductions of the kinetic equation for soliton gas represent linearly degenerate integrable systems of hydrodynamic type for any N. Exact solutions are available via the generalised hodograph transform. Integrability of the hydrodynamic reductions is a strong indication to the integrability of the full kinetic equation in the sense yet to be understood. An analytical theory of integrable Landau-Hopf turbulence is yet to be constructed. It is an especially interesting problem in the context of the focusing NLS equation as the genus growth bifurcations could lead to the formation of rogue waves (Peregrine breathers of high amplitude) at the caustics. Critical phenomena in soliton gases (e.g. when the variance σ = u 2 (u) 2 vanishes). Behaviour of higher moments (skewness, kurtosis) in soliton gases.
References El, G.A., Krylov A.L., Molchanov, S.A. and Venakides, S., "Soliton turbulence as a thermodynamic limit of stochastic soliton lattices", Physica D 152/153, 653-664 (21). El, G.A., The thermodynamic limit of the Whitham equations, Phys. Lett. A 311, 374-383 (23). G.A. El and A.M. Kamchatnov, Phys. Rev. Lett. 95 (25) Art. No 2411 G.A. El, A.M. Kamchatnov, M.V. Pavlov, S.A. Zykov, J. Nonl. Sci. 21 (211) 151-191 V.E. Zakharov, Stud. Appl. Math. 122 (29) 219-234. E.N. Pelinovsky et al., Phys. Lett. A 377 (213) 272-275.