She is the editor of IntechOpen's book "Perturbation Methods with Applications in Science and Engineering". Antar is a member of the Scientific Committee of TUMTMK (Turkish National Committee for Theoretical and Applied Mechanics).
Introduction
The reduction to the KP model allows us to construct approximate soliton solutions of the nonlocal NLS, which are presented in Section 3. We also present the dark hump solitons, as well as the dark ring and antidark soliton solutions of the NLS, which are numerically found to propagate undistorted in the framework of the NLS model.
Nonlocal NLS and multiscale analysis 1 Introduction of the model and linear regime
The nonlinear regime – asymptotic analysis
Similarly, Φ0 in the polar case can be expressed as a superposition of a radially expanding wave (depending on ϱ) and a radially contracting wave (depending on ~ϱ). Similarly, Φ0 in the polar case can be expressed as a superposition of a radially expanding wave (depending on ϱ) and a radially contracting wave (depending on ϱ).
Θ ! ffiffiffiffiffiffiffiffi
Approximate soliton solutions
- Antidark stripe solitons and their interactions
- Dark lump solitons
- Ring dark and antidark solitons
We start with Cartesian geometry and look in particular at the simplest soliton solutions of the KP equation, the so-called line solitons. Similar to the case of solitons with a dark stripe, once Eq. 48) gives rise to the estimated dark knobby soliton of the non-local NLS.
Conclusions and discussion
Color online) development of dark (top) and antidark (bottom) ring solitons in Figure 9. The obtained results were based on the formal reduction of the nonlocal NLS model to the KP (cKP) equation.
An N-soliton solution to the DNLS equation based on a revised inverse scattering transform
- The revised inverse scattering transform and the Zakharov-Shabat equation for DNLS equation with VBC
- The typical examples for one- and two-soliton solutions
We give two concrete examples – the one- and two-soliton solutions as illustrations of the general explicit soliton solution. Finally, we state that the exact N-soliton solution to the DNLS equation can be converted to that of the MNLS equation by a gauge-like transformation.
A simple method to derive and solve Marchenko equation for DNLS equation
- The lax pair and its Jost functions of DNLS equation
- Marchenko equation for DNLSE and its demonstration
- A multi-soliton solution of the DNLS equation based upon pure Marchenko formalism
- The special examples for one- and two-soliton solutions
Thus, the time-independent part of Cnis is insignificant and can only be absorbed or normalized by redefining the soliton center and initial phase. When there are N simple poles λ1,λ2,⋯,λN in the first quadrant of the complex plane λ, the Marchenko equation will give an N-sol solution to the DNLS equation with VBC in the no-reflection case.
Soliton solution of the DNLS equation based on Hirota’s bilinear derivative transform
- Fundamental concepts and general properties of bilinear derivative transform
- Bilinear derivative transform of DNLS equation
- Soliton solution of the DNLS equation with VBC based on HBDT .1 One-soliton solution
The above equations contain all the information needed to find a soliton solution of the DNLS equation with VBC. Finally, we reach the one-soliton solution of the DNLS equation with VBC.
Breather-type and pure N-soliton solution to DNLS + equation with NVBC based on revised IST
The resulting one-soliton solution can naturally tend towards the well-defined termination of the VBC asρ!0 case [17-20] and the pure one-soliton solution in the degenerate case. The connection between the solution and the Jost functions of the DNLS+equation The asymptotic behavior of the Jost solutions in the limit ej !λ ∞ can be.
- Introduction of time evolution factor
For the Jost functions to satisfy the second Lax equation, the time evolution factor h t , zð Þ must be introduced using a standard procedure [21, 22] in the Jost functions and scattering data. Nevertheless, in what follows, the time variable in the Jost functions will be excluded because it has no bearing on the treatment of the Z-S equation.
Verification of standard form and the explicit breather-type multi-soliton solution
- The explicit N-soliton solution to the DNLS + equation with NVBC
Due to the limitation of space, the asymptotic behavior of the N-soliton solution is just similar to that of the pure N-soliton solution in Ref. The one- and two-soliton solutions for DNLS+comparison with NVBCWe give two concrete examples – the one- and two-breath type soliton solutions.
The one and two-soliton solutions to DNLS + equation with NVBC We give two concrete examples – the one and two breather-type soliton solu-
- Explicit pure N-soliton solution to the DNLS + equation with NVBC
Formula (130) includes the one-soliton solution of the DNLS equation with VBC as the limiting case. The evolution of the breathing-type two-soliton solution with respect to time and space is given in Figure 3.
Space periodic solutions and rogue wave solution of DNLS equation DNLS equation is one of the most important nonlinear integrable equations in
- Bilinear derivative transformation of DNLS equation
- Solution of bilinear equations
Space-time evolution of the modulus of the rogue wave solution with ρ¼1 and σ¼pffiffiffi2. Space-time evolution of the modulus of the second-order periodic spatial solution with.
Concluding remarks
In this section, the periodic solutions of the 1st order and the 2nd order space of the KN equation are derived by means of HBDT. Meanwhile, based on the longwave limit, the simplest rogue wave model has been obtained according to the periodic solution of 1st order space.
Appendices
Atomic spin chain
- Atomic spin chain with magnetic dipole: dipole interaction
- Atomic spin chain with light-induced dipole-dipole interaction In fact, the site-to-site interaction can also be tuned by laser field, and the
A strong static magnetic field B is applied to polarize the spin orientations of the ground state of the atomic chain along the quantized axis. The Jmagmn parameter describes the strength of the site-to-site spin coupling caused by static MDDI.
Magnetic soliton in atomic spin chain 1 Magnetic soliton under the MDDI
- Magnetic soliton under the LDDI
For two adjacent sites, A2¼Jmag02 =Jmag01 measures the relative strength of the NNN spin coupling to the NN spin coupling. Thus for two adjacent sites, A2¼J02=J01 measures the relative strength of the NNN spin coupling to the NN spin coupling.
Conclusion
Modulation instability of nonlinear spin waves in an atomic chain of spinor Bose-Einstein condensates. Controllable magnetic soliton excitations in an atomic chain of spinor Bose-Einstein condensates confined in an optical lattice.
Switched 2 Micron Solid-State Lasers and Their Applications
Flash-pumped 2 micron solid-state pulsed lasers
For flash-pumped EO Q-switched 2 μm solid-state lasers, the highest single pulse energy of 520 mJ was achieved with a corresponding pulse width of 35 ns, which provided the highest pulse peak power from the flash-pumped 2 μm. In the future, flash-pumped 2 μm solid state lasers still have great potential for various applications that require high energy.
Diode-pumped actively modulated 2 micron solid-state lasers 1 AO modulated 2 micron solid-state lasers
- EO modulated 2 micron solid-state lasers
At a repetition rate of 5 kHz, the maximum pulse energy was 7 mJ with a corresponding pulse width of 75 ns [24]. A maximum pulse energy of 550 mJ and a minimum pulse width of 14 ns at a repetition rate of 1 Hz were obtained [42].
Diode-pumped SAs modulated 2 micron solid-state lasers
- Two-dimensional nanomaterial modulated 2 micron solid-state lasers In recent years, two-dimensional (2D) material based SAs have been widely
- Cr 2+ -doped crystal modulated 2 micron solid-state lasers
- Gain crystal modulated 2 micron solid-state lasers
A maximum output pulse energy of 1,766 mJ at a repetition frequency of 685 Hz was obtained with a pulse duration of 15.4 ns, which indicated that a Cr:CdSe crystal could be a promising SA in 2 μm passive Q-switched lasers [89]. At a repetition rate of 42.1 kHz, a minimum pulse duration of 48 ns and a maximum pulse energy of 2 mJ were obtained.
Applications of pulsed 2 micron solid-state lasers
As mentioned above, the popular SAs used in the 2 μm Q-switched solid-state lasers are mainly low-dimensional nanomaterials and Cr-doped crystals. A pulsed 2 μm laser with high peak power is also a promising pump source for OPOs in the wavebands such as 3~5 and 8~12 μm.
Summary and outlook
Passive Q-switching and Q-switched mode-locking operations of 2 μm Tm:CLNGG laser with MoS2 saturable absorbing mirror. Ho:SSO solid-state saturable-absorber Q-switch for pulsed Ho:YAG laser resonantly pumped by a Tm:YLF laser.
The analytical model
QP¼I X;ð Y;ZÞ �ðαPXPþαPXPð1�αPÞð1�αWÞðh tð Þ �h tð �t0ÞÞ ¼QPðr;θ;φÞ (2) where t0 is the exposure time, αPis powder absorption coefficient and αClear the workpiece absorption coefficient. QP¼I X;ð Y;ZÞ �ðαPXPþαPXPð1�αPÞð1�αWÞðh tð Þ �h tð �t0ÞÞ ¼QPðr;θ;φÞ (2) where t0 is exposure time, αPis is powder absorption coefficient and αW is workpiece absorption coefficient.
Simulations and comments
For electron beam processing [7] you can refer to Katz and Penfolds absorption law [8] and also Tabata-Ito-Okabe absorption law [9]. Accordingly, we can consider that Figures 1–7 make sense even if Au, Ag or Al substrates are used.
Conclusions
From Maxwell's equations, it is possible to obtain the equation of motion of the electric and magnetic fields, the solution of which describes the propagation of electromagnetic waves. In other words, the mode of response of the medium to electromagnetic excitation is contained in the mean polarization due to the propagation of the electromagnetic wave.
Modeling the optical lattice
The optical sensor can choose the propagation of the optical rays through the natural conduction band of the photonic crystal, since the function of our model, the so-called optical potential, describes the optical light propagation through the crystal [2]. effects such as Raman and Brillouin effects. The optical sensor can choose the propagation of the optical rays through the natural conduction band of the photonic crystal, since the function of our model, the so-called optical potential, describes the optical light propagation through the crystal [2].
Brief introduction on optical solitons 1 Nonlinear Schrödinger equation
So it is perfectly acceptable for a particular optical field to satisfy an equation of the same form as . where p is momentum and the terms inside the square root stand for kinetic energy or the difference between the total energy E and the potential energy V of the particle, we get the Schrödinger equation [9]. where m is the particle mass and h the Planck constant. Brief introduction to optical solitons3.1 Non-linear Schrödinger equation. we arrive at an important relationship by using Eq.
Nonlinear and quantum optical sensor principles
- Optical systems
- Mapping optical systems
The method developed in this work can be implemented to identify patterns of nonlinear modes that contribute to the distortion of the optical signal in the transmission system [17]. We can explore the optical dynamics of a beam and waves propagating through an optical lattice that can be described by PT symmetric complex potentials.
New non-Hermitian optical systems
- New models and sensors
The application of non-Hermitian optical systems in the modeling of the optical gratings and waveguides [18] can be performed by the variational method proposed by [19]. Non-linear and non-Hermitian optical systems can be modeled now by the method based on the Lagrangian.
Conclusions
As can be seen from expression (2), the same value of OAM can correspond to a different composition of vortex modes with squared amplitude C2m. This changes both the magnitude and the spectrum of the vortex modes including their initial phases.
Preliminary remarks
For example, in a weakly turbulent medium [37] or optical fibers [38], there is an intense energy exchange between the vortex modes of the single beam. In addition, knowing the digital spectrum, the beam can be restored again, and by adjusting the spectral vortex amplitudes, we can improve the structure of the emitted field.
Theoretical background of the digital vortex sorting and experimental results
- Nondegenerate case
- Degenerate case and vortex avalanche
We then write the complex amplitude of the perturbed beam in the form ΨðR,φ, zÞ ¼ XN. Small variations in the holographic lattice structure lead to a cardinal reconstruction of the vortex spectrum (Figure 4(а)).
OAM, informational entropy and topological charge of truncated vortex beams
- Sector aperture
- Circular and annular apertures
We investigated TC changes under beam sector perturbations [52] and holographic grid perturbations [47]. In the perturbed vortex beams shown in Fig. 9(Ibc), a broadening of the vortex spectrum and a reduction of the tail amplitudes are observed.
Conclusions
Abramochkin (Samara branch of the Lebedev Institute of Physics, Russian Academy of Sciences, Samara, Russia) for helpful discussion on the mathematical approach. Measurement of the vortex spectrum in a vortex beam array without cut and paste.
Factors affecting the efficiency of switches
In general, most molecular switches can be divided into two categories, namely stereoisomerism and structural isomerism. Finally, we illustrate the influence of the environment on NLO switching properties with examples.
Molecular switches based on stimuli
- Light-induced optical switches/photoswitches
- Thermal optical switches
- Chemically activated optical switches
The photochemical isomerization of an olefinic bond is one of the fundamental processes in vision. The switching ability of the NLO chromophores was studied by monitoring the SHG at the opening/closing states.
Molecular switches based on materials 1 Optical switches based on organic chromophores
The fall time and rise time of the switching device measured with a 200 Hz square wave voltage were found to be 370.2 and 287.5 μs, respectively. Schematic representation of the reversible optical switching between the SP and MC states of a spirobenzopyran associated with a cysteine residue in a protein (reused from [38]; American Chemical Society Publication, 2005).
