A KdV soliton gas: asymptotic analysis via RiemannHilbert problems - PowerPoint PPT Presentation

Background and motivations Initial conditions Large time behaviour To be continued... A KdV soliton gas: asymptotic analysis via Riemann–Hilbert problems Manuela Girotti joint with Ken McLaughling (CSU) and Tamara Grava (SISSA-Bristol) Midwestern Workshop on Asymptotic Analysis, IU Bloomington, October 6th 2018 1 / 48

Background and motivations Initial conditions Large time behaviour To be continued... Table of contents 1 Background and motivations KdV and solitons The soliton gas and the Riemann–Hilbert problem 2 Asymptotics of the initial condition u ( x, 0) for large x ’s 3 Large time behaviour of the potential u ( x, t ) The super-critical case The sub-critical case 4 To be continued... 2 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons 1 Background and motivations KdV and solitons The soliton gas and the Riemann–Hilbert problem 2 Asymptotics of the initial condition u ( x, 0) for large x ’s 3 Large time behaviour of the potential u ( x, t ) The super-critical case The sub-critical case 4 To be continued... 3 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons The KdV equation In 1834 the Scottish engineer John Scott-Russell accidentally observed a surface water wave in the Union Canal between Edinburgh and Glasgow that appeared to be a spatially localized traveling wave, that he called “great wave of translation”. In 1895, D. J. Korteweg and G. de Vries proposed the following equation to describe this phenomenon: u t − 6 uu x + u xxx = 0 . 4 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons The one-soliton solution The simplest wave solution is: u ( x, t ) = ϕ v ( x − vt ) . With this ansatz, the PDE becomes an ODE in the variable ξ = x − vt − vϕ ′ v − 6 ϕ v ϕ ′ v + ϕ ′′′ v = 0 One solution is a rapidly decreasing, localized travelling wave (soliton): � √ v � u ( x, t ) = − v 2 sech 2 2 ( x − vt − x 0 ) 0 − 0 . 5 − 1 − 1 . 5 − 2 − 6 − 4 − 2 0 2 4 6 5 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons � √ v � u ( x, t ) = − v 2 sech 2 2 ( x − vt − x 0 ) Remark In order to have a real solution, we need v > 0 , which in turn implies that the wave-solution can move only to the right. The amplitude of the wave is proportional to the speed v , thus larger amplitude solitary waves move with a higher speed than smaller amplitude waves. 6 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons The periodic soliton solution Starting again from the ansatz: u ( x, t ) = ϕ v ( x − vt ) and imposing a periodicity condition, the solution (periodic travelling wave) can be written in terms of Jacobi elliptic functions: u ( x, t ) = β 1 − β 2 − β 3 − 2( β 1 − β 3 ) dn 2 �� � β 1 − β 3 ( x − 2( β 1 + β 2 + β 3 ) t ) + x 0 | m where dn ( z | m ) is the Jacobi elliptic function of modulus m = β 2 − β 3 β 1 − β 3 and β 1 > β 2 > β 3 . 7 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons Looking for other solutions... The Cauchy problem (Gardner-Greene-Kruskal-Miura, ’67) : � u t − 6 uu x + u xxx = 0 u ( x, 0) = q ( x ) for rapidly decaying initial data: q ( x ) → 0 as x → ±∞ . This nonlinear PDE is integrable , arising as the compatibility condition of a Lax pair of linear differential operators (Lax, ’68): d d t L = [ B , L ] with L = − d 2 B = − 4 d 3 d x 3 + 6 u d d x 2 + u, d x + 3 u x . Equivalently, the compatibility condition can be presented as the existence of a simultaneous solution to the pair of equations: L φ = Eφ, φ t = B φ where E ∈ R is the spectral parameter. 8 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons Looking for other solutions... The Cauchy problem (Gardner-Greene-Kruskal-Miura, ’67) : � u t − 6 uu x + u xxx = 0 u ( x, 0) = q ( x ) for rapidly decaying initial data: q ( x ) → 0 as x → ±∞ . This nonlinear PDE is integrable , arising as the compatibility condition of a Lax pair of linear differential operators (Lax, ’68): d d t L = [ B , L ] with L = − d 2 B = − 4 d 3 d x 3 + 6 u d d x 2 + u, d x + 3 u x . Equivalently, the compatibility condition can be presented as the existence of a simultaneous solution to the pair of equations: L φ = Eφ, φ t = B φ where E ∈ R is the spectral parameter. 8 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons Solving the Schr¨ odinger equation We start from L φ = Eφ, where L := − d 2 d x 2 + V ( x ) is the Schr¨ odinger operator with potential V ( x ) = u ( x, 0) = q ( x ) (no dependence on time... yet!). Using tools from spectral theory, GGKM calculated the scattering data , which will allow to find the solution φ to the Schr¨ odinger equation: � − λ 2 1 , . . . , − λ 2 S = n eigenvalues , c 1 , . . . , c n norming constant of the eigenfunctions , r ( λ ) reflection coefficient of the “scattering” solutions φ ± ( x ) } 9 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons Solving the Schr¨ odinger equation We start from L φ = Eφ, where L := − d 2 d x 2 + V ( x ) is the Schr¨ odinger operator with potential V ( x ) = u ( x, 0) = q ( x ) (no dependence on time... yet!). Using tools from spectral theory, GGKM calculated the scattering data , which will allow to find the solution φ to the Schr¨ odinger equation: � − λ 2 1 , . . . , − λ 2 S = n eigenvalues , c 1 , . . . , c n norming constant of the eigenfunctions , r ( λ ) reflection coefficient of the “scattering” solutions φ ± ( x ) } 9 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons Turning on time If the potential V t ( x ) = u ( x, t ) depends also on a (time) parameter t , one expects � � {− λ 2 the scattering data S = j } , { c j } , r ( λ ) to vary with t as well. If the t dependence of u ( x, t ) is given in terms of the KdV equation, u t = − u xxx + 6 uu x , then the scattering data S ( t ) evolve in a very simple and explicit manner (GGMK, ’67): 1 the discrete eigenvalues are constant: E = − λ 2 j ; 2 the norming constants have exponential behaviour: c j ( t ) = c j (0) e Aλ 3 j t ; 3 same for the reflection coefficient: r ( λ ; t ) = r ( λ ; 0) e iBλ 3 t . 10 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons Turning on time If the potential V t ( x ) = u ( x, t ) depends also on a (time) parameter t , one expects � � {− λ 2 the scattering data S = j } , { c j } , r ( λ ) to vary with t as well. If the t dependence of u ( x, t ) is given in terms of the KdV equation, u t = − u xxx + 6 uu x , then the scattering data S ( t ) evolve in a very simple and explicit manner (GGMK, ’67): 1 the discrete eigenvalues are constant: E = − λ 2 j ; 2 the norming constants have exponential behaviour: c j ( t ) = c j (0) e Aλ 3 j t ; 3 same for the reflection coefficient: r ( λ ; t ) = r ( λ ; 0) e iBλ 3 t . 10 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons Solve the Cauchy initial-value problem for KdV Recipe: u t − 6 uu x + u xxx = 0 u ( x, 0) u ( x, t ) Direct Scattering: Inverse Scattering Lax pair d d t L = [ B, L ] evolve the scattering data S (0) S ( t ) 11 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons Solve the Cauchy initial-value problem for KdV Recipe: u t − 6 uu x + u xxx = 0 u ( x, 0) u ( x, t ) Direct Scattering: Inverse Scattering Lax pair d d t L = [ B, L ] evolve the scattering data S (0) S ( t ) � � {− λ 2 Calculate the scattering data: S = j } , { c j } , r ( λ ) 11 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons Solve the Cauchy initial-value problem for KdV Recipe: u t − 6 uu x + u xxx = 0 u ( x, 0) u ( x, t ) Direct Scattering: Inverse Scattering Lax pair d d t L = [ B, L ] evolve the scattering data S (0) S ( t ) Calculate the time-evolved scattering data S ( t ), imposing u ( x, t ) to be a solution of KdV: u t = 6 uu x − u xxx . 11 / 48

Background and motivations Initial conditions Large time behaviour To be continued... KdV and solitons Solve the Cauchy initial-value problem for KdV Recipe: u t − 6 uu x + u xxx = 0 u ( x, 0) u ( x, t ) Direct Scattering: Inverse Scattering Lax pair d d t L = [ B, L ] evolve the scattering data S (0) S ( t ) Construct the inverse scattering map to obtain the solution u ( x, t ): Marchenko integral equation (Gelfand-Levitan-Marchenko, 1950’s) Riemann–Hilbert problem (Deift-Zhou, ’93; Grunert–Teschl, ’09) 11 / 48