On the 3 -wave Equations with Constant Boundary Conditions Georgi - PowerPoint PPT Presentation

On the 3 -wave Equations with Constant Boundary Conditions Georgi Grahovski Institute for Nuclear Research and Nuclear Energy, BAS, Sofia, Bulgaria E-mail: grah@inrne.bas.bg 12 June 2012 Geometry, Integrability and Quantization 2012, Varna (Bulgaria) Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 1 / 43

Based on the joint work with Vladimir Gerdjikov (Sofia). VSG, GGG - E-print: arXiv/1204.5346 . Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 2 / 43

Outline Introduction 1 Lax representation and Jost solutions 2 The fundamental analytic solutions of L 3 Time evolution of the scattering matrix 4 Spectral Properties of the Lax Operator 5 Conserved Quantities 6 Conclusions 7 Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 3 / 43

Introduction 3-wave resonant interaction equations The case of vanishing boundary conditions 3-wave resonant interaction model: i ∂ q 1 ∂ q 1 ∂ x + κ q ∗ ∂ t + i v 1 2 q 3 = 0 , i ∂ q 2 ∂ q 2 ∂ x + κ q ∗ ∂ t + i v 2 1 q 3 = 0 , i ∂ q 3 ∂ q 3 ∂ t + i v 3 ∂ x + κ q 1 q 2 = 0 . κ – interaction constant, v i – the group velocities of the model, q i = q i ( x , t ) , i = 1 , 2 , 3 . The 3-wave equations can be solved through the Inverse Scattering Method. Zakharov V. E., Manakov S. V., Zh. Exp. Teor. Fiz., 69 (1975), 1654–1673; ( INF preprint 74-41 , Novosibirsk (1975) (In Russian)). Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 4 / 43

Introduction ISM as a Generalised Fourier Transform Describing the Fundamental Properties of NLEEs The interpretation of the ISM as a GFT and the expansions over the so called “squared solutions” allows one to study all the fundamental properties of NLEEs: 1 the description of the whole class of NLEE related to a given spectral problem (Lax operator) solvable by the ISM; 2 derivation of the infinite family of integrals of motion; 3 the Hamiltonian properties of the NLEE’s. M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, Studies in Appl. Math. 53 (1974), n. 4, 249–315. Gerdjikov V. S., Kulish P. P., Physica D 3 (1981), 549–564. Gerdjikov V. S., Inverse Problems 2 (1986), 51–74. V.S. Gerdjikov, G. Vilasi, A.B. Yanovski, Integrable Hamiltonian Hierarchies: Spectral and Geometric Methods , Lect. Notes Phys. 748 , Springer, Berlin - Heidelberg (2008). Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 5 / 43

Introduction Describing the Fundamental Properties ... The Hamiltonian and ... going to constant boundary conditions The (canonical) Hamiltonian of 3WRI eqns. is given by: � 3 � ∞ � ∂ q ∗ H 3 − w = 1 � ∂ q k � � ∂ x − q ∗ + κ ( q 3 q ∗ 1 q ∗ 2 + q ∗ k 3 q 1 q 2 ) dx v k q k k 2 ∂ x −∞ k =1 A special interest deserves the case when some (or all) of the functions q k ( x , t ) tend to a constant as x → ±∞ . Below we choose: q 3 ( x , t ) → ρ e i φ ± , q 1 , 2 ( x , t ) → 0 , x → ±∞ . Here the constants θ = φ + − φ − and ρ are of a physical origin and play a basic role in determining the properties of 3WRI eqns. with CBC and its soliton solutions. Faddeev L. D., Takhtadjan L. A., Hamiltonian approach in the theory of solitons , Springer Verlag, Berlin (1987). Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 6 / 43

Introduction The Case of Constant Boundary Conditions “Bright” vs. “Dark” Solitons More specifically, ρ characterizes the end-points of the continuous spectrum of the Lax operator L ( λ ) . The discrete spectrum, in this case, may consist of real simple eigenvalues λ k , k = 1 , . . . , N lying in the lacuna − 2 ρ < λ k < 2 ρ . To them, there correspond the so-called “dark solitons” whose properties and behavior substantially differ from the ones of the bright solitons. The 3- and N -wave interaction models describe a special class of wave-wave interactions that are not sensitive on the physical nature of the waves and bear an universal character. Kaup D. J., Reiman A., Bers A, Rev. Mod. Phys. 51 (1979), 275–310. S. V. Manakov, Teor. Mat. Phys. 28 (1976), 172–179. Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 7 / 43

Introduction 3-waves with CBC Setting up the problem It is normal to expect that the properties 1 – 3, known for the case of vanishing boundary conditions will have their counterparts for the case of constant boundary conditions. However there is no easy and direct way to do so. For example, one may relate both cases by taking a limit ρ → 0 . Of course, in this limit most of the difficulties, related mostly with the end-points of the continuous spectrum disappear. The spectral data, the analyticity properties of the Jost solutions and the corresponding Riemann-Hilbert problem are substantially different and more difficult for ρ > 0 . Aims: 1 To study the direct scattering problem for the 3WRI eqns with CBC; 2 To study the spectral properties of the associated Lax operator. Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 8 / 43

Lax representation and Jost solutions Lax representation Algebraic Setup Consider the pair of Lax operators: � i ∂ � L ψ ≡ ∂ x + [ J , Q ( x , t )] − λ J ψ ( x , t , λ ) = 0 , � � i ∂ M ψ ≡ ∂ t + [ I , Q ( x , t )] − λ I ψ ( x , t , λ ) = 0 , with   0 q 1 q 3 J = diag ( J 1 , J 2 , J 3 ) ,  , q ∗ Q = 0 q 2  1 I = diag ( I 1 , I 2 , I 3 ) . q ∗ q ∗ 0 3 2 Q ( x , t ) , I and J are traceless matrices (i.e. the Lax operators ∈ sl (3 , C ) ) The eigenvalues of I and J are ordered as follows: J 1 > J 2 > J 3 , I 1 > I 2 > I 3 ( J 1 + J 2 + J 3 = 0 and I 1 + I 2 + I 3 = 0 ). Here λ ∈ C is a spectral parameter. Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 9 / 43

Lax representation and Jost solutions Lax representation Zero Curvature Equation The compatibility condition: i [ J , Q t ] − i [ I , Q x ] + [[ I , Q ] , [ J , Q ]] = 0 , The group velocities take the form: v 1 = I 1 − I 2 v 2 = I 2 − I 3 v 3 = I 1 − I 3 , , , J 1 − J 2 J 2 − J 3 J 1 − J 3 The interaction constant κ reads: κ = J 1 I 2 + J 2 I 3 + J 3 I 1 − J 2 I 1 − J 3 I 2 − J 1 I 3 . Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 10 / 43

Lax representation and Jost solutions Imposing Boundary Conditions Constant boundary conditions as | x | → ∞ : 3 = ρ e i φ ± . x →±∞ q 3 ( x , t ) = q ± x →±∞ q 1 ( x , t ) = lim x →±∞ q 2 ( x , t ) = 0 , lim lim For the potential matrix Q ( x , t ) one can write: ρ e i φ ±   0 0 x →±∞ Q ( x , t ) = Q ± , lim Q ± = 0 0 0   ρ e − i φ ± 0 0 The difference θ = φ + − φ − of the asymptotic phases φ ± plays a crucial rˆ ole in the Hamiltonian formulation of the 3-wave model with constant boundary conditions: its values label the leaf on the phase space M of the 3WRI model, where one can determine the class of admissible functionals, and to construct a Hamiltonian formulation. Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 11 / 43

Lax representation and Jost solutions Imposing Boundary Conditions The Asymptotic phases The two asymptotic potentials Q ± are related by Q + = Q ( θ ) Q − ( t ) Q − 1 ( θ ) , where θ = φ + − φ − and e i θ/ 2   0 0  . Q ( θ ) = 0 1 0  e − i θ/ 2 0 0 Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 12 / 43

Lax representation and Jost solutions Direct Scattering Problem for L Jost Solutions The direct and the inverse scattering problem for the Lax operator will be done for fixed t and in most of the corresponding formulae t will be omitted. The starting point in developing the DSP for 3WRI eqns are the eigenfunctions (the Jost solutions) of the auxiliary spectral problem L ( x , t , λ ) ψ ± ( x , t , λ ) = 0 , Asymptotic behavior for x → ±∞ respectively: x →±∞ ψ ± ( x , t , λ ) e iJ ( λ ) x = ψ ± , 0 ( λ ) P ( λ ) , lim where P ( λ ) is a projector: P ( λ ) = diag ( θ ( | Re λ | − 2 ρ ) , 1 , θ ( | Re λ | − 2 ρ )) and θ ( z ) is the step function. Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 13 / 43

Lax representation and Jost solutions Jost Solutions Asymptotic Lax operators P ( λ ) ensures that the continuous spectrum of L has multiplicity 3 for | Re λ | − 2 ρ > 0 and multiplicity 1, for − 2 ρ < Re λ < 2 ρ . The x and t -independent matrices ψ ± , 0 ( λ ) in diagonalize the asymptotic Lax operators: L ± ( x , t , λ ) = i ∂ ∂ x + [ J , Q ± ] − λ J . Indeed, ([ J , Q ± ] − λ J ) ψ ± , 0 ( λ ) = − ψ ± , 0 ( λ ) J ( λ ) , where J ( λ ) = − diag ( J 1 ( λ ) , J 2 ( λ ) , J 3 ( λ )) and J 1 ( λ ) = 1 � � λ 2 − 4 ρ 2 � J 2 λ + ( J 1 − J 3 ) , J 2 ( λ ) = − λ J 2 , 2 J 3 ( λ ) = 1 � � λ 2 − 4 ρ 2 λ 2 − 4 ρ 2 . � � J 2 λ − ( J 1 − J 3 ) , k ( λ ) = 2 Georgi Grahovski (INRNE) 3 -wave Equations with CBC GIQ’12 - Varna 14 / 43