| Today,systems science and systems engineering,in the convergence and intersection of many disciplines,are devoted to the study of a.wide variety of the nonlinear systems,because most systems in science and engineering are nonlinear.That’s why the nonlinear problems are of interest to engineers,physicists,mathematicians,astronomers,chemists,biologists,financiers,etc.Outline of the interdisciplinary innovation in this dissertation:In the optical fiber communications,magneto-optical materials,cosmic plasmas,oceans,bubble liquids,atmospheres,incompressible fluids,biomedicine,acoustics,hydrodynamics,lattice dynamics,etc.,analytic studies are performed on certain nonlinear systems,resulting in the scaling transformations,auto-B(?)cklund transformations,hetero-B(?)cklund transformations.bilinear forms,similarity reductions,Painleve analysis,multi-soliton solutions,physical-mechanism analysis,consistency with the relevant experimental results SCIreported by others,etc.During the analytic investigations in this dissertation,symbolic computation,singular manifolds,Painleve expansions.Bell polynomials,Hirota method,Clarkson-Kruskal direct method,etc.,are involved.Entirety/systematicness of this dissertation is reflected in the interdisciplinary mainline that runs through this dissertation,i.e.,analytic explorations on certain nonlinear systems in the aforementioned fields.Interdisciplinary achievements in this dissertation include the following aspects:On the first aspect,related to optical communication and optical network,for the picosecond-pulse attenuation/amplification in a nulti-component inhomogeneous optical fiber with diverse polarizations/frequencies,innovation of this dissertation:Analytic investigation and symbolic computation are carried out on a three-coupled variablecoefficient nonlinear Schrodinger system.For the slowly-varying envelopes of optical modes,this dissertation obtains a similarity reduction,an auto-B(?)cklund transformation and some analytic solutions,which rely on the optical-fiber variable coefficients,i.e.,the fiber loss/gain,nonlinearity and group velocity dispersion.Relevant variablecoefficient constraints are presented.Possible applications:Construction of logic gates,optical computing,soliton switching,design of the fiber directional couplers,quantum information processing,soliton amplification in the wavelength division multiplexing systems,solitonic studies in the all-optical devices and birefringence fiber systems,etc.On the second aspect,related to plasma astrophysics,magneto-optics and ferromagnetism,for certain electromagnetic waves in an isotropic charge-free infinite ferromagnetic thin film,or Alfven/ion-acoustic/solitonic waves in a cosmic/laboratory plasma,innovation of this dissertation:Analytic investigation and symbolic computation are carried out on a variable-coefficient modified Kadomtsev-Petviashvili system.With respect to,e.g.,introducing an oblique propagation of the electromagnetic wave in that ferromagnetic thin film,the first derivative of the angle made between the direction of propagation of the electromagnetic wave and the uniform magnetization of the medium,this dissertation works out(1)one set of the variable-coefficient-dependent bilinear forms,(2)two sets of the variable-coefficient-dependent multi-solitonic solutions and(3)two sets of the variable-coefficient-dependent auto-B(?)cklund transformations via the generalized Laurent series,along with some solitonic features.Relevant constraints on the variable coefficients are presented.Possible applications:Studies on the sky,magnetooptical recording,computer data-storage,waveguides,etc.On the third aspect,related to astrophysics and oceanography,for the shallow water waves in a lake or near an ocean beach(while the Earth,Enceladus,Europa and Titan all have oceans),innovation of this dissertation:Analytic investigation and symbolic computation are carried out on a Boussinesq-Burgers system.For the water-wave horizontal velocity and height of the water surface,this dissertation constructs a scaling transformation,two sets of the bilinear forms through the binary Bell polynomials,two sets of the multi-soliton solutions,and two auto-B(?)cklund transformations together with the solitonic solutions.Our results are dependent on the water-wave dispersive power.Possible applications:Studies on the energy development,marine/offshore engineering,hydraulic engineering,mechanical engineering,etc.On the fourth aspect,as for certain wave processes in acoustics,hydrodynamics or oceanography,innovation of this dissertation:Analytic investigation and symbolic computation are carried out on the an extended coupled(2+1)-dimensional Burgers system.This dissertation finds out that the system passes the Painlevé test and works out two sets of the similarity reductions through symbolic computation.Each set relies on the coefficients in the system,and can develop into a known ordinary differential equation.Possible applications:Wave studies in acoustics,hydrodynamics,oceanography and related fields.On the fifth aspect,related to magneto-optics,ferromagnetism,fluid mechanics and plasma physics,for the electromagnetic waves in a ferromagnetic material,or water waves,or dust-acoustic/ion-acoustic/dust-ion-acoustic waves in a plasma,innovation of this dissertation:Analytic investigation and symbolic computation are carried out on a generalized(3+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili system.Special cases of the system in those fields are listed out(such as one special case in magneto-optics,which describes the electromagnetic waves in an isotropic charge-free ferromagnetic thin film with the potential application in magneto-optical recording).For that system,this dissertation finds out.(1)two sets of the variable-coefficient-dependent auto-B(?)cklund transformations along with some solitonic features,(2)the variablecoefficient-dependent bilinear forms and(3)two branches of the variable-coefficientdependent multi-soliton solutions.Relevant constraints on the variable coefficients are presented.Possible applications:Plasmas,fluids,computer data-storage,waveguides,magneto-optical recording,etc.On the sixth aspect,related to astrophysics and oceanography,taking into account the nonlinear and dispersive long gravity waves in two horizontal directions on the shallow water of an open sea or a wide channel of finite depth,innovation of this dissertation:Analytic investigation and symbolic computation are carried out on a generalized(2+1)-dimensional dispersive long-wave system.With respect to the horizontal velocity and wave elevation above the undisturbed water surface,this dissertation leads to a scaling transformation,two hetero-B(?)cklund transformations and two auto-B(?)cklund transformations with certain solitons.All of our results are dependent on the constant coefficients in the original system.Possible applications:Oceanic and related water-wave studies,etc.,in the Solar System.On the seventh aspect,related to biomedicine and plasma astrophysics,innovation of this dissertation:Analytic investigation and symbolic computation are carried out on ① for a(3+1)-dimensional generalized variable-coefficient Kadomtsev-PetviashviliBurgers-type equation for different types of cosmic dusty plasmas plus observational/experimental supports and ② a variable-coefficient generalized forced-perturbed Korteweg-de Vries-Burgers model for a dilated artery,blood vessel or circulatory system.This dissertation presents some auto-B(?)cklund transformations,solitons and similarity reductions.Features:Consistency with the relevant experimental results SCI-reported by others.Possible applications:Studies on the cosmic dusty plasmas,dilated arteries,blood vessels and circulatory systems.On the eighth aspect.Appendix 1 presents the SCI citation information.On the ninth aspect,Appendix 2 lists out the author’s other contributions.Conclusion:In order to express the entirety/systematicness,this dissertation has an interdisciplinary mainline,i.e.,analytic explorations on certain nonlinear systems in the optical communications,magneto-optical materials,cosmic plasmas,oceans,bubble liquids,atmospheres,incompressible fluids,biomedicine,acoustics,hydrodynamics,lattice dynamics,etc. |
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