Beam Properties In Fractional Competing Nonlocal Schr(?)dinger Equation

Spatial soliton refers to the optical pulse that keeps the spatial shape unchanged during propagation in nonlinear media.It can form and propagate stably in a variety of nonlinear materials because the optical pulse induces refractive index waveguides when it propagates in the material.The self-focusing properties of the waveguide can balance the diffraction broadening of the beam.In addition,the interaction between spatial solitons has the properties of elastic collision similar to particles,which can realize the all-optical control and complete optical logic operation.So far,the research on spatial solitons has been focused on local,non-local and competitive non-local materials.In different materials,spatial solitons have unique properties.Recently,the spatial soliton in competitive nonlocal materials have attracted much attention.However,there are few reports on the competitive non-local model proposed by Jung et al.Based on the model proposed by Jung,this paper studies the beam properties in the competitive nonlocal and fractional competitive nonlocal Schr(?)dinger equation.The main research contents are as follows:Firstly,the modulation instability of plane waves in nematic liquid crystals is studied based on fractional nonlinear Schr(?)dinger equation.It is found that the competition between the molecular orientation effect and the thermal effect in the fractional competitive non-local model leads to the special modulation instability of plane waves.The fractional Lévy index only affects the bandwidth of modulation instability,but does not affect its gain peak.The nonlocality of molecular orientation effect and nonlocality of thermal effect will suppress the modulation instability,and the increase of thermal nonlinear coefficient will also suppress the modulation instability of plane wave.The most important finding is that the competitive nonlinear effect leads to the existence of a critical value of amplitude.When the amplitude of plane wave is less than the critical intensity,the plane wave appears modulation instability.Modulation instability does not occur when the plane wave amplitude is greater than the critical intensity value.Then,the effects of fractional effect and competitive nonlinear effect on the power and width of bright solitons are studied.When the nonlocality of molecular orientation increases,the power and width of bright solitons increase,while the increase of Lévy index decreases the power and width of bright solitons.The power and width of bright solitons increase with the increase of thermal nonlinearity coefficient,and so does the fractional Lévy index.It is found that the power and width of bright solitons increase monotonically with the increase of Lévy exponent when the nonlocality of molecular orientation is1,while the power and width of bright solitons decrease monotonically with the increase of Lévy exponent when the non-local degree of molecular orientation is 3.Secondly,based on the model proposed by Jung,the stable solution of modulation instability in competitive non-local materials is studied by using analytical and numerical methods.The exponential response function is used to simulate the nonlinear evolution of modulation instability.When the plane wave is the superposition of the zero harmonic and the first harmonic,the approximate analytical solution of the stable solution under the induced modulation instability of the competitive nonlocal nonlinear equation is obtained.The exact numerical solution of the competing nonlocal equation is obtained by Newton iteration method.The solution is composed of multiple harmonics.Finally,the discrete Fourier algorithm is used to prove that the solution can stabilize the propagation.Finally,soliton interactions in competing nonlocal materials are studied based on Jung’s model.Bright soliton solutions are obtained by using square operator method.Taking the numerical solution as the initial input of the fractional Fourier algorithm,the interaction between oblique incidence solitons,parallel incidence solitons,symmetric incidence solitons and three-beam solitons is simulated numerically.On this basis,we simulate the modulation of weak signal optical transmission by soliton-induced waveguides,and find that weak signal light is restricted to stable transmission in soliton-induced refractive index waveguides.At the point where the soliton-induced waveguides meet,the signal light splits into two beams that travel along each waveguide.The principle can be used to design a beam splitting device based on soliton interaction.Secondly,the interlocking refractive index waveguides are formed during the interaction of the three solitons,and the signal light will propagate from one waveguide to the adjacent waveguide at the place(node)where the distance between the refractive index waveguides is minimum.This finding provides theoretical support for all-optical interconnected devices based on the interaction of solitons.