On The Structures And Propagation Properties Of Solitons In Several Nonlocal Nonlinear Systems

Soliton is an important concept of nonlinear science created in the 20th century, and its origin can be traced back to the observations of the long, shallow, water waves by British naval engineer Scott Russell in 1834. After 10 years, Scott Russell reported this phenomenon in the meetings of the British Association for the Advancement of Science. Since then, a large number of nonlinear systems existing in natural world have been studied. However, until the second half of the 20th century, the soliton and related nonlinearity began to be studied systematically. In recent decades, the rapid development of the soliton physics has been extended to a number of areas, such as nonlinear optics, semiconductor electronics, photonics, plasma physics, Bose-Einstein condensation, biology, thermal conductivity, liquid crystal and so on. At present, the solitary phenomena are found in different physical systems, such as shallow-water wave, deep-water wave, plasma density wave, matter-wave in BECs, DNA chain etc. However, the area where the soliton can best embody the diversity is the nonlinear optics. Due to the potential demand of the modern communication industry for the optical technology, such as high-capacity optical transmission, all-optical information processing and all-optical control, optical solitons become the most cutting-edge and popular subject in the soliton field. On the other hand, the governing equations controlling dynamics and evolution of solitons in different nonlinear fields have the same or similar form, so different nonlinear areas can inspire each other, complement each other and jointly promote the development of nonlinear science and technology.Soliton physics is a branch of the nonlinear science on the formation, evolution, control and interaction of solitons in nonlinear medium, which covers the formation mechanism, stability analysis, and transmission characteristics of all types of solitons, as well as communication and control applications. In recent years, there have been some new trends in soliton researchs, such as solitons in the nonlocal nonlinear media, spinning soliton, and Laguerre and Hermite soliton clusters, and so on. All these have greatly enriched the contents of the study on soliton and provide a broad prospect for achieving all-optical control and atomic interferometry, as well as the development of new materials. As such, we would like to choose nonlocal solitons as our research subject. This paper will mainly focus on the work of nonlocal solitons, and the work is summarized as follows:1. Influnence of Nonlocality on the Modulational Stable Property of the Plane Wave and the Propogation Stability of the SolitonsTo begin, we briefly review the status quo related to the studies of spatial soliton stability, and describled the suppressed effect of nonlocality on spatial soliton, and then derive nonlocal nonlinear models. Based on this, the stable property of the continous wave (CW) modulation is discussed for the weakly nonlocal nonlinear system with the quintic nonlinearity. The results show that the stable property of CW modulation depends on the signs of the cubic-quintic nonlinearities, the degree of nonlocality and the light intensity. The stable properties of CW in noncompetiting nonlinear optical media are the same as those in Kerr media. However, competiting nonlinearity changes the stable property of CW and the structure and character of gain spectrum for modulational instability.By considering the weakly nonlocal competiting nonlinear system with the nonlinear gain, we found that a relatively strong nonlocality reduces the stability of the soliton propagation under the certain condition. We carefully analyse the physical mechanism implied in this phenomenon, and show that the nonlocality not only restricts the nonlinearity of medium and the diffraction of beam, but also lead to the mutual association of perturations at the different positions in the medium. Therefore, solitons are influenced by the perturbations in the whole region of the medium and the excessively strong nonlocality inevitably results in the instability of the soliton propagation. The newly discovered phenomenon will help us better understand the nature of nonlocality. 2. Theory Analysis of the Impact of Rectangular Boundary Conditions on the Structure and Symmetry of Strongly Nonlocal Soliton ClusterNonlocality relies on boundary conditions and physical properties of media, and then impacts the structure and propagation properties of solitons by the boundary conditions. After introducting the experiment on the observation of coherent elliptic solitons and of vortex-ring solitons carried out by C. Rotschild et al, we solve the model which corresponds to the experimental system adopting the self-similarity technique, and theorelically and numerically analyse the structures of the symmetries of the spatial soliton clusters to obtain a number of the results.First of all, the Hermite-Gaussian soliton cluster, single-soliton or multisoliton, can exist and propagate in a thermal lead glass with the rectangular boundaries, and their structures are determined by two integer parameters in the given initial conditions and system parameter.Secondly, for certain medium, the soliton, single-soliton or soliton cluster, is elliptic, and its symmetry depends not only on the boundary conditions of the mediun, but also on the power and symmetry of input beam as well as the propagation distance. By the parallel analysis, we predict that the difference between theoretical analysis and the experimental conclusion results from the choice of a limited number of samples in experiment.The particular interest lies in that the soliton cluster shows several unique features. For instance, the Hermite-Gaussian solitons form a rectangular matrix cluster with certain rows and columns where the symmetry of the solitons in the cluster is the same as the single-soliton. Moreover, the intensities of the solitons in the edges of the rectangular cluster are bigger than those on the center, the most intense solitons occurring at four angles. These characteristics provide us the probability remotely controlling solitons by the boundary conditions.3. Three-Dimensional Strongly Nonlocal Cold Atomic Gas Model and Connection between Two- and Three-Dimensional Strongly Nonlocal Solitons The nonlinearity in the system of Bose-Einstein condensates (BEC) is composed of both local and nonlocal nonlinearities, where the local nonlinearity can be tuned by Feshbach resonance technique. In the case that the local nonlinearity can be tuned to a very small value, the local nonlinear effect is negligible, but the nonlocal term is dominant, related to a long-range interaction such as the dipole-dipole interaction in a condensate. For such a special case, we have proposed and exactly solved an extended nonlocal nonlinear model. The results showed that for the strongly nonlocal case, the three-dimensional matter wave soliton, forming Hermite-Laguerre-Gaussian soliton cluster, is a new type of solitons. The soliton cluster with comlex structures and new propagation properties displays the necklace, ring, speckle or rectangle in geometry, different from Hermite-Gaussian or Laguerre-Gaussian soliton family in two-dimension. Such soliton solution is mathematically the product of Hermite, Laguerre and Gassian functions and physically behaves as the modulations of three functions each other. Moreover, the central spacing in the soliton matrix depends on the azimuthal quantum number. The width of soliton cluster in the evolution changes depending on the coefficient of resonance in the corresponding direction and the initial width of atomic density distribution.At the same time, we comparatively analyse features of the two- and three-dimensional strongly nonlocal soliton to find out the connection and difference between them and to interpret the internal reasons from mathematics and physics. We believe that the Hermite-Gaussian soliton, Laguerre-Gaussian soliton, Hermite-Laguerre-Gaussian soliton existing on different planes, or other similar solitons (e.g., accessible soliton, bear-shaped spinning soliton) are a class of soliton clusters, being classified as the Whittaker-Gaussian soliton cluster.4. Variable Parameter Method for Solving Nonlinear Schrodinger EquationNonlinear Schrodinger (NLS) equation is a mathematical description of much nonlinear physical phenomena existing in nature. To search for analytical solution to nonlinear equations is a key to grasp the essence of physical phenomena and to reveal characteristics of soliton. So far, there exist many methods for exactly solving NLS equation, such as inverse scattering transform, Darboux-Backlun transform, Painleve expansion method and self-similarity technique etc. These methods play different roles in solving nonlinear problems. On the basis of the current method, to find new methods and skills as well as technique is still one of the main contents in the soliton field.The NLS with the cubic-quintic nonlinearities as well as the self-steeping considered in Chap.5 can not pass the Painleve PDE test and can not be solved by employing inverse scattering transform or Darboux-Backlun transform transform. In this connection, we especially introduce a new method, so-called Variable parameter method, referring to variable coefficient method in the theory of linear differential equations. The main idea follows as. First, we choice the soliton solution to the basic NLS equation as a trial solution, where part parameters are the functions of the longitudinal and transverse coordinates. Second, we substitute the trial solution into the equation considered and find these parametric functions. Variable parameter method not only is of great significance in studying of the properties of solitons for ultra-short pulses, but also provides a new thinking of way for solving the other NLS equations.In this article, for the purpose of studying the structures and dynamical behaviors of soliton solutions in nonlinear systems, on the basis of searching for the analytical solutions to the nonlinear models we altogether consider four kinds of models, this is, one-dimensional weakly nonlocal nonlinear optical systems with including quintic nonlinearity and / or nonlinear gain, two-dimensional strongly nonlocal thermal lead glass system with rectangular boundaries, three-dimensional strongly nonlocal cold atomic gase as well as the generalized nonlinear Schrodinger model.