Ultrashort Relativistically Strong Solitons in Plasma

1. title INSTITUTE OF APPLIED PHYSICS RAS RUSSIA NIZHNY NOVGOROD ULTRASHORT RELATIVISTICALLY STRONG SOLITONS IN PLASMA Authors: Fadeev D. A.(fadey@appl.sci-nnov.ru) Mironov V. A.

2. intro introduction Initial 3d laser pulse rare plasma nonlinear response factors: relativistic electron weighting the push out of electrons from area occupied by laser pulse with ponderomotive force (transverse striction) excitation of nonlinear oscillations of electrons behind laser pulse (wake field) ions supposed to be motionless

3. wave equation nonlinear wave equation dimensionless wave equation: - complex vector potential the model consists of a single equation for complex value - co-moving time: cylindrical symmetry: initial conditions: circular polarization no multiple spectrum generation

4. history problem evolution 1.NSE research (smooth envelopes). IA.B. Borisov, A.V. Borovskii, V.V. Korobkin et al., Sov. Phys. JETP 74(4), 604 (1992) II “OPTICAL SOLITONS from fibers to photonic crystals" Y.S.Kivshar, G.P. Agrawal(2003) III “SOLITONS nonlinear pulses and beams”N.N. Akhmediev andA. Ankiewicz (1997) existence and steadiness of 3d soliton structures was determined 2.Switching to investigation of real field dynamics. IVA.A. Balakin and V.A. Mironov, JETP Lett. 75 (12), 617 (2002) The common features of electromagnetic pulses dynamics were investigated in the case of Kerr type nonlinearity.

5. limitatons limitations of the model quasi-flat model: here is the typical relativistic vector potential thus the transverse push out of electrons from the area occupied by electromagnetic laser pulse could be neglected non-refractive approach: no wake field: wake field excitation is suppressed if the duration of laser pulse is much greater than plasma wave period

6. hamiltonian formalism hamiltonian formalism integrals: energy integral quanta number integral impulse integral (could not be obtained by standard routine for hamiltonian systems) momenta evolution: centre of mass description centre of mass velocity defined by initial conditions only Laser pulse width description allow to investigate initial evolution of some types of solutions according to the set of it’s initial parameters

7. solitons solitonsI soliton general form: where: – is the free parameter solution: solution for phase equation for amplitude (to be solved numerically) Assumption of central symmetry for all 3 coordinates (two transverse spatial coordinates and co-moving time) allows to obtain quite simple differential equation for amplitude of soliton

8. solitons solitonsII two parameters: a=0.95 a=0.97 a=0.999 - selects soliton amplitude, transverse width and spectral characteristics (number of optical periods in wave packet) - scales whole soliton field along co-moving time coordinate

9. steadiness steadiness Lapunov’s method for partial derivative equations: Lapunov’s functional is combined of integrals of the equation under investigation thus it stays constant during the evolution minimization under the conditions of constant pulse here numerical investigation of steadiness: duration transverse width evolution step/30 evolution step/30

10. computationsI solitons interactionI by varying initial delay betweensolitons the relative phase before collision could be set 1 2V V 2 3 1 4 2 5 3 6 4 7 5

11. computationsII solitons interactionII results of two solitary waves interaction for different parameters of initial solitons. surfaces of amplitude after interaction on axis of the system

12. computationsIII longitudinal instability of a wave packet 986 1 6 121 156 10 23 initial gaussian pulse breaks apart on two solitary waves soltons have different velocities

13. computationsIV transverse instability of a flat wave (2D simulations)I The break-up of non perturbed super gaussian wave packet 9 31 47 0 17 120 200 72 165 nonliarity of 4th order allows to assume same processes in 3D case

14. computationsV transverse instability of a flat wave (2D simulations)II The break-up of initial super gaussian pulse on a quantity of solitons in 2D model for nonsymmetrical perturbation nonliarity of 4th order allows to assume same processes in 3D case 11 18 0 24 49 78 140 in case of 2nd order of nonliarity the break-up is slower 11 18 0 24 49 78 140

15. apendix solitons in the full wave equation For dense plasma the assumption of non refractive propagation is incorrect. In that case the full wave equation is appropriate: The last equation is obtained in new coordinates,moving with the velocity : common form for solitary wave solution: the parameter b could be obtained with substitution of the solitary wave form to the wave equation thus the soltions of full wave equation could be obtained from ones demonstrated previously (with V=1) with simple transformations: ->frequency multiplication ->longitudinal scaling for envelope

16. outro results • The mathematical approach for investigation of cold plasma and electromagnetic pulse interaction was developed. The generalization for the case of relativistic motions of plasma particles was performed. • The numerical code for the model describing nonlinear interaction of ultrashort laser pulse with plasma was developed (the code was optimized for distributed computation systems) • The soltions for the equation under investigation was obtained analytically as a generalization of NSE (Nonlinear Shroedinger equation) • The steadiness of found solitons was checked both analytically and numerically • The detailed numerical analysis of solitons interaction was performed. • The processes of transverse and longitudinal instability of wave packets accompanied by formation of soliton type structures was demonstrated in numerical simulations

17. outro thanks mail to: fadey@appl.sci-nnov.ru fadeev daniil find detailed information in article: V.A. Mironov, D.A. Fadeev, JETP, 106, 974-982 (2008)