1 / 38

Wave Collapse in Nonlocal Nonlinear Schrödinger Equations

Wave Collapse in Nonlocal Nonlinear Schrödinger Equations. İ. BAKIRTAŞ İTÜ DEPARTMENT OF MATHEMATICS M. J. ABLOWITZ *, B. ILAN ** * CU DEPARTMENT OF APPLIED MATHEMATICS ** UC MERCED DEPARTMENT OF APPLIED MATHEMATICS. Ablowitz et al. Physica D 207 (2005) 230-253. COLLAPSE.

vicki
Download Presentation

Wave Collapse in Nonlocal Nonlinear Schrödinger Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Wave Collapse in Nonlocal Nonlinear Schrödinger Equations İ. BAKIRTAŞ İTÜ DEPARTMENT OF MATHEMATICS M. J. ABLOWITZ *, B. ILAN ** * CU DEPARTMENT OF APPLIED MATHEMATICS ** UC MERCED DEPARTMENT OF APPLIED MATHEMATICS Ablowitz et al. Physica D 207 (2005) 230-253 Nonlinear Physics. Theory and Experiment IV 2006

  2. COLLAPSE • The solutions of nonlinear wave equations often exhibit important phenomena such as stable localized waves (e.g. solitons), self similar structures, chaotic dynamics and wave singularities such as shock waves (derivative discontinuities) and/or wave collapse (i.e, blow up) where the solution tends to infinity in finite time or finite propagation distance. • Nonlinear wave collapse is a matter of interest in many areas of physics, hydrodynamics and optics. • A prototypical equation that arises in cubic media, such as Kerr media in optics, is the (2+1)D focusing cubic nonlinear Schrödinger equation NLS Nonlinear Physics. Theory and Experiment IV 2006

  3. Nonlinear Schrödinger Equation & Collapse • Kelley (1965) carried out direct numerical simulations of cubic NLS that indicated the possibility of wave collapse. • Vlaslov et al. (1970) proved that the solutions of the cubic NLS satisfy the Virial Theorem (Variance Identity) Hamiltonian: They also concluded thatthe solution of the NLS can become singular in finite time (or distance) because a positive quantity could become negative for initial conditions satisfying . Nonlinear Physics. Theory and Experiment IV 2006

  4. Subsequently many researchers have studied the NLS in detail: • Weinstein (1983) showed that when the power is sufficiently small, i.e., The solution exists globally. Therefore, the sufficient condition for collapse is While the necessary condition for collapse is Weinstein also found that the ground state of the NLS also plays an important role in the collapse theory. The ground state is a “stationary” solution of the form Nonlinear Physics. Theory and Experiment IV 2006

  5. Papanicolaou et al. (1994) studied the singularity structure near the collapse point and showed asymptotically and numerically that colapse occurs with a (quasi) self-similar profile. • Merle and Raphael (1996) elaborated on the behavior of blow up phenomena of NLS. • Gaeta et al. (2000) carried out detailed experiments which reveal the nature of the singularity formation and showed that collapse occurs with a self-similar profile. Nonlinear Physics. Theory and Experiment IV 2006

  6. There are considerably fewer studies of the wave collapse that arise in nonlinear media whose governing equations have quadraticnonlinearities, such as water waves and nonlinear optics. The derivation of the NLSM system is based on an expansion of the slowly-varying wave amplitude in the first and second harmonics of the fundamental frequency, as well as a mean term that corresponds to the zeroth harmonic. This leads to a system of equations that describes the nonlocal-nonlinear coupling between a dynamic field that is associated with the first harmonic and a static field associated with the mean term. Nonlinear Physics. Theory and Experiment IV 2006

  7. For the physical models considered in this study, the general nonlinear Schrödinger-mean (NLSM)system can be written in the following form • These equations are also sometimes referred to as Benney-Roskes • or Davey-Stewartson type and are nonlocal because the second equation can be solved for Which corresponds to a strongly-nonlocal function Nonlinear Physics. Theory and Experiment IV 2006

  8. NLSM EQUATION FROM WATER WAVES • NLSM equations were originally obtained by Benney and Roskes (1969) in their study of the instability of wave packets in multidimensional water wave packets in water of finite depth, without surface tension. • Davey and Stewartson (1974) derived a special form of NLSM equations in the study of water waves, near the shallow water limit. • Djordjevic and Redekopp (1977) extended the results of Benney and Roskes to include the surface tension. • Ablowitz and Segur(1979) analyzed the Benney- Roskes equations and showed that the singularity exists in some parameter regimes.They further introduced the Hamiltonian of NLSM system. • Existence and well-posedness of solutions to NLSM equations was studied by Ghidaglia and Saut (1990) Nonlinear Physics. Theory and Experiment IV 2006

  9. Derivation of NLSM in water waves Free-surface gravity-capillary water waves NLSM results from a weakly nonlinear quasi-monchromatic expansion of velocity potential as : direction of propagation : transverse direction : time : measure of the weak nonlinearity : coefficients of the zeroth, first, second harmonics Substituting the wave expansion into Euler’s equations with a free surface and assuming slow modulations of the field in and directions results a nonlinearly coupled system for and . Nonlinear Physics. Theory and Experiment IV 2006

  10. In the context of water waves,Ablowitz and Segur (1979), studied the NLSM (Benney-Roskes) Equations in the following form where Dimensionless coord., are the wave numbers in the directions, are suitable functions of : group velocity wave number, dispersion coefficients and surface tension where normalized water depth Nonlinear Physics. Theory and Experiment IV 2006

  11. By rescaling the variables, previous systemcan be transformed to (Elliptic-elliptic case), this system admits Collapse, requires large surface tension For Nonlinear Physics. Theory and Experiment IV 2006

  12. Hamiltonian & Virial Theorem • Ablowitz&Segur (1979) defined the Hamiltonian Each bracket, { }, in H is positive definite, and the second bracket vanishes in the linear limit of Benney Roskes equations. Clearly H<0 is possible. Furthermore, they showed that the Virial Theorem holds As can be seen if H <0, the moment of inertia vanishes at a finite time and no global solution exists after this time. This indicates a rapid development of singularity by which we mean the FOCUSING. Nonlinear Physics. Theory and Experiment IV 2006

  13. NLSM EQUATION FROM OPTICS • In isotropic (Kerr) media, where the nonlinear response of thematerial depends cubically on the applied field, the dynamics of aquasi-monochromatic optical pulse is governed by the NLS equation. • Generalized NLS systems with coupling to a mean term also appearin various physical applications. These equations are denoted asNLSM type equations. NLSM type equations arise in nonlinear opticsby studying materials with quadratic nonlinear response. • Ablowitz, Biondini and Blair (1997, 2001) found that NLSM type equations describe the evolution of the electromagnetic field in the quadratically polarized media. Both scalar and vector NLSM systems, in three space + one time dimension, were obtained. • Numerical calculations of NLSM type equations in case of nonlinear optics were carried out by Crasovan, Torres et al.(2003) Indications of wave collapse were found in certain parameter regime. Nonlinear Physics. Theory and Experiment IV 2006

  14. Derivation of NLSM in optics The electric polarization field of intense laser beams propagating in optical media can be expanded in powes of the electric field as (*) :Electric field vector : Susceptibility tensor coefficients of the medium Quasi monochromatic expansion of the component of the electromagnetic Field with the fundamental harmonic, second harmonic and a mean term is Using a polarization field of the form (*) in Maxwell’s equations leads to NLSM Type equations for non zero Nonlinear Physics. Theory and Experiment IV 2006

  15. Ablowitz, Biondini and Blair (1997) For scalar system, if the time dependencein these equations is neglected and problem is reconsidered for thematerials belong to a special symmetry class thenit can be seen that these equations are NLSM type equations. In optics, U is the normalized amplitude of the envelope of theoptical beam and V is the normalized static field, ρis thecoupling constant which comes from the combined opticalrectification- electro optic effect and is the asymmetry parameter comes from the anisotropy of the material. This system is recently Investigated by Crasovan et al.(2003) Nonlinear Physics. Theory and Experiment IV 2006

  16. Integribility of NLSM 1- When derivatives with respect to y can be neglected (e.g., in a narrow canal) the second equationcan be integrated immediately, and one recovers the one- dimensional nonlinear Schrödinger equation which can be solved by the inverse scattering transform (IST). M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (1981) 2.In deep water limit, the mean flow vanishes and NLSM equations reduce to (2+1)-dimensional NLS equation: Contrary to the one-dimensional case, this equation is likely not solvable by IST. Also, for various choices of parameters the solutions can blow up in finite time. Nonlinear Physics. Theory and Experiment IV 2006

  17. 3-A different scenario arises in the opposite limit,that is shallow water. In this case, after rescaling, the equations can be written as : with or • This system, usually called the Davey-Stewartson (DS I or DS II) equations, • is of IST type, and thus completely integrable. • For the Davey-Stewartson system, several exact solutions are available. • In particular, stable localized pulses, often called dromions are known • to exist. • Existence and well-posedness of solutions to NLSM type equations was • studied by Ghidaglia and Saut (1990). • Behavior of the blow up singularity was analyzed by Papanicolaou (1994). Nonlinear Physics. Theory and Experiment IV 2006

  18. Global existence and collapse for NLSM Papanicolaou et al. (1994) Power Hamiltonian Thus, in optics case, the coupling to the mean field corresponds to a self- defocusing mechanism, while in water waves case, it corresponds to a self- focusing mechanism =>focusing in water waves case is easier to attain. Virial Theorem holds Nonlinear Physics. Theory and Experiment IV 2006

  19. NLS Ground State NLS stationary solutions, which are obtained by substituting into the NLS equation, satisfy The ground state of the NLS can be defined as a solution in H1 of this equationhaving the minimal power of all the nontrivial solutions. The existence and uniqueness of the ground state have been proven. Ground state is radially symmetric, positive and monotonically decaying. Solution exists globally for where Nonlinear Physics. Theory and Experiment IV 2006

  20. NLSM Ground State NLSM stationary solutions, which are obtained by substituting into the NLSM equation, satisfy The ground state of the NLSM can be defined as a nontrival solution (F, G) in H1 such that F has the minimal power of all the nontrivial solutions. The existence of the ground state has been proven by Cipolatti (92). In the same spirit as for NLS, Papanicolaou et al. (94) extended the global existence theory to the NLSM and proved that Solution exists globally for where where Nonlinear Physics. Theory and Experiment IV 2006

  21. AIM OF THE STUDY • Investigating the blow up structure of NLSM type equations for both opticsand water waves problem, in the context of : ♦ Hamiltonian approach which was introduced by Ablowitz and Segur (79) ♦ Global existence theory ♦ Numerical methods • Obtaining the ground state mode : Nonlinear Physics. Theory and Experiment IV 2006

  22. Numerical method & Initial Conditions for Optics and Water Waves Cases • Ground state mode is obtained by using a fixed point numerical procedure similar to what was used by Ablowitz and Musslimani (2003) in dispersion-managed soliton theory. • For Hamiltonian approach and direct simulation, a symmetric Gaussian type of inital condition is used is the input power where Hamiltonian Nonlinear Physics. Theory and Experiment IV 2006

  23. Threshold power for which H=0 , given by Such that when then and, therefore, the solution collapses at finite distance. Alternatively, Such that when then and collapse is guaranteed by the Virial Theorem. Nonlinear Physics. Theory and Experiment IV 2006

  24. Critical power for collapse as a function of for H<0 N<Nc Nonlinear Physics. Theory and Experiment IV 2006

  25. The regions in thecorresponding to collapse and global-existence N<Nc N<Nc H<0 H<0 (a) Nonlinear optics (b) Water waves Nonlinear Physics. Theory and Experiment IV 2006

  26. NLSM MODE Nonlinear Physics. Theory and Experiment IV 2006

  27. The on-axis amplitudes of the ground state & Contour plots OPTICS NLS TOWNES Water waves optics For Nonlinear Physics. Theory and Experiment IV 2006

  28. The astigmatism of the ground state F(x,y) (a) ν= 0.5 with -1 ≤ ρ ≤ 1 (b) ρ = -0.2 (dashes) and ρ= 0.2(solid) with 0 ≤ ν ≤ 1 Nonlinear Physics. Theory and Experiment IV 2006

  29. Input Astigmatism~Astigmatic initial conditions For optics case: As input beam becomes narrower along the x-axis, the critical power for collapse increases, making the collapse more difficult to attain. Nonlinear Physics. Theory and Experiment IV 2006

  30. WATER WAVES • The focusing factor of the NLSM solutions • The corresponding astigmatism of the solution as a function of the focusing factor (Input power is taken as N=1.2 Nc(ν= 0.5, ρ= -1)≈12.2) OPTICS NLS TOWNES Nonlinear Physics. Theory and Experiment IV 2006

  31. Self-similarity of the collapse profile In order to study the self-similarity of the collapse process, the modulation function is recovered from the solution as The rescaled amplitude of the solution of the NLSM, i.e is compared with ground state and In order to show that the collapse process is quasi-self similar with the corresponding ground state, the rescaled amplitude is shown to converge pointwise to as Nonlinear Physics. Theory and Experiment IV 2006

  32. Convergence of the modulated collapse profile (dashes) to the NLSM ground state (solid) Along x axis (top) and along y axis (bottom)with (ν, ρ)= (0.5,1) Nonlinear Physics. Theory and Experiment IV 2006

  33. Convergence of the modulated collapse profile (dashes) to the NLSM ground state (solid) Along x axis (top) and along y axis (bottom)with (ν, ρ)= (0.5,-1) Nonlinear Physics. Theory and Experiment IV 2006

  34. Convergence of the modulated collapse profile (dashes) to the NLSM ground state (solid) Along x axis (top) and along y axis (bottom)with (ν, ρ)= (4,- 4) Nonlinear Physics. Theory and Experiment IV 2006

  35. Convergence of the modulated collapse profile (dashes) to the NLSM ground state (solid) Along x axis (top) and along y axis (bottom)with (ν, ρ)= (4,- 4) (semi-log plot) Nonlinear Physics. Theory and Experiment IV 2006

  36. Collapse Arrest Nonlinear Physics. Theory and Experiment IV 2006

  37. Related NLSM Type System Consider the NLSM system without the cubic term Hamiltonian Virial Theorem is not changed and collapse is possible for negative Substituting the initial conditions into the Hamiltonian, the threshold power for zero Hamiltonian Nonlinear Physics. Theory and Experiment IV 2006

  38. CONCLUSIONS • Direct numerical simulation results are consistent with the Virial Theorem and Global Existence Theory. This is in the same spirit as the results of classical NLS equation. • In contrast to the NLS case, stationary solutions of NLSM are not radially symmetric but elliptic. • Ground state profile is astigmatic and therefore, the collapse profile is astigmatic. • The singularity occurs in water waves more quickly than in optics. • As z approaches to zc (collapse distance) numerical simulations show that the nature of singularity for both optics and water waves, is described by a self-similar collapse profile given in terms of the ground state profile. • From the experimental perspective, self similar collapse in quadratic-cubic media remains to be demonstrated in either free-surface waves and nonlinear optics. Nonlinear Physics. Theory and Experiment IV 2006

More Related