Two-parameter Fractional Derivative And Some Generalizations Of Integrable Systems And Their Nonlinear Dynamical Properties

Fractional calculus and integrable systems have very important research value both in theory and practical applications.There is a close relationship between fractional calculus and soliton theory,and polymer is one of them.The combination of fractional calculus with soliton and integrable system theory has attracted the attention of researchers,but the progress of related research is not ideal.What is the physical meaning of fractional derivative?Can fractional models meet new challenges,new results and new understandings of integrable systems?Can rogue wave solutions and soliton solutions with fractional order produce novel local spatial structures and dynamic evolution behaviors?The answers to these key questions are worth exploring.In this dissertation,several important problems are studied:defining the two-parameter fractional derivative;giving a physical explanation of the conformable fractional derivative;generalizing some integrable systems to the fractional cases;generating a fractional Yang hierarchy and its Hamiltonian structure and bi-integrable coupling system;constructing the rogue wave solutions,lump solutions,soliton solutions,fractal solutions of the fractional models and the long-time asymptotic solution of the variable-coefficient nonlinear Schrodinger(NLS)equation and the fractional NLS equation;simulating the spatial structures,dynamic evolution and nonlinear phenomena of some novel solutions;generalizing the bilinear method,the inverse scattering transform,the Riemann-Hilbert approach and several other methods for nonlinear waves;exploring the influences of fractional order on the convergence and divergence of approximate solution sequences and the evolution behavior of the obtained solutions of fractional differential equations.This dissertation consists of eight chapters,which can be divided into six parts.In the first part,Chapter 1,the relationship between fractional calculus and soliton theory is established by using polymers,and some hot issues in the research field and the research background and development of this dissertation are briefly introduced.The second part consists of chapters 2 and 3.In Chapter 2,we first define a twoparameter fractional derivative,which takes the conformable fractional derivative and the local fractional derivative as special cases,and prove some of its basic properties;secondly,we design a functional expression of the relationship between cosmic time and individual time;and we then give a physical explanation of the conformable fractional derivative from the perspective of variable acceleration;finally,we construct the operator solution and analytical solutions of two new fractional generalized forms of the Kolmogorov-Petrovskii-Piskunov(KPP)equation.In Chapter 3,we generate the two-parameter fractional integrable Yang hierarchy based on a Lie algebra,and construct its Hamiltonian structure and bi-integrable coupling systems.The third part is composed of chapters 4 and 5.In Chapter 4,a comfortable fractional NLS equation with Lax Integrability and its first-order and second-order fractional rogue wave solutions with translational coordinates are derived by using Hirota’s bilinear method in fractional form.Such translational coordinates provide the degree of freedom to adjust the position of rogue waves on the coordinate plane to a certain extent.Compared with the rogue wave solutions of the integer order NLS equation,the peaks of the fractional rogue wave solutions are steeper,and the time from the occurrence to the end of the rogue wave is shorter.Aiming at the construction of the asymmetric fractional rogue wave of the NLS equation with different backgrounds and amplitudes,this chapter attempts to modify the first-order and second-order fractional rogue wave solutions of the NLS equation.Chapter 5 derives a conformable fractionalorder Kadomtsev-Petviashvili(KP)equation with Lax integrability and its fractional N-soliton solutions and rational solutions,and use the obtained fractional 1-soliton solution to simulate the swallow shaped shallow water waves,solitons with multiple crests and peakons;at the same time,the fractional 2-and 3-soliton solutions are used to simulate the X-type,Y-type and 3-in-2-out interaction of shallow water waves;the fractional 2-soliton solution is also used to simulate the falling and scattering phenomenon of a long columnar wave structure in deep water,the columnar waves formed by many of these processes may be the merging effect(which is considered to cause the tsunami caused by earthquake)concerned by Ablowitz and Baldwin.The dynamic processes from annular structure and conical structure to bump structures in the coordinate plane are simulated by using the special case of rational solutions(fractional bump solutions),the selectivity of fractional order value will lead to a series of new features,such as the change of propagation velocity,the nonlinearity of motion trajectory,the slope and steepness of external profile,the variability of amplitude and so on.The fourth part,Chapter 6,derives the space-time fractional isospectral AKNS hierarchy and the time-fractional non-isospectral AKNS hierarchy which are not reported in the literature by introducing local partial derivatives to AKNS spectral problem and its adjoint equations,in which three reductions of the space-time fractional isospectral AKNS hierarchy generate the fractional and nonlocal NLS,modified KdV and reverset NLS hierarchies.By extending Hirota’s bilinear method to transform the space-time fractional isospectral AKNS hierarchy into two fractional bilinear forms and the inverse scattering transform to reconstruct potentials from the fractional scattering data which correspond to the time-fractional nonisospectral AKNS hierarchy,three new pairs of N-fractal solutions with Mittag-Leffler functions are obtained in this chapter.Some obtained solutions are restrained to Cantor set,we depict the continuous but nondifferentiable 1-fractal solutions and 2-fractal solutions,which are obviously different from smooth soliton solutions.However,we cannot induce the essential characteristics of 1-fractal solutions and 2-fractal solutions.On the one hand,these fractal solutions have been restrained to Cantor set,on the other hand,this is related to the properties of these two fractional AKNS hierarchies and the parameters chosen in simulating figures.The fifth part,Chapter 7,generalizes and uses analytical methods and numerical algorithms to solve some partial differential equations,lattice equations,fractional differential equations,fractional integral-differential equation,and initial value and initial boundary value problems,the methods involved includes the Riemann-Hilbert approach,the negative power expansion method proposed in this dissertation,and the variational iteration method,the homotopy perturbation method and the F-expansion method in the literature.Its main purpose is to develop some effective and constructive methods for nonlinear waves and extend the Riemann-Hilbert approach to solve the initial value problems of the variable-coefficient NLS equation and the fractional NLS equation,and find the interesting correlation that the convergence of the approximate solution sequence of fractional equations sometimes depends on the fractional value of the equation when solving fractional order initial value problems by numerical algorithms.The sixth part,Chapter 8,summarizes the content of this dissertation and prospects the future research work.This dissertation contains 51 figures,4 tables and 190 references.