Study On The Key Techniques Of Retrieving Surface Mass Transport Based On GRACE/GFO Satellite Gravimetry Data

This thesis is mainly divided into two parts.In the first part(Chapter 2),we extend the construction of soliton solutions for integrable equations to higher-order spectral problems with decaying initial values and finite density initial values,and extend the application of Riemann-Hilbert(RH)theory.In the second part(Chapters 3 to 6),we investigate the long-time asymptotic behavior of several integrable equations regarding the soliton resolution conjecture of nonlinear dispersion equations.This conjecture is positively answered and extended to negative order equations with finite density initial values,and the asymptotic stability of soliton solutions is given.In Chapter 1,we mainly review the development of soliton theory,the development and application of inverse scattering theory,the working motivation,and a brief introduction to the content of this thesis.In Chapter 2,the RH method is extended to study the initial value problem of multi-component nonlinear integrable equations with high-order Lax matrices by constructing adjoint spectrum problems,matrix partitioning,and introducing Joukowsky transformation techniques.This overcomes the lack of analyticity and solves the problem of constructing soliton solutions for multi-component integrable equations under Schwartz initial values and finite density initial values.Three types of soliton solutions and some new phenomena are obtained by classifying discrete spectrums.In Chapter 3,the (?)-steepest descent method is extended to study long-time asymptotic behavior of solutions to the coupled Kundu-nonlinear Schrodinger(KN-NLS)equation based on the special reduction.The mapping property from the initial value to the reflection coefficient is strictly proved to ensure the controllability of the interpolation matrix.The research results demonstrate that the KN-NLS equation with a steady-state phase point satisfies the soliton resolution conjecture,and a conical region can be found to approximate the N-soliton solution of the coupled integrable KN-NLS equationIn Chapter 4,combining the Beals-Coifman operator theory,we extend the (?)descent method to study the long-time asymptotic behavior of the Cauchy problem for the extended mKdV equation with discrete spectrums and multiple phase points.Along the orbit ξ=x/t∈(-9α2/20β+ε,-ε),we obtain the asymptotic solution of the extended mKdV equation.With the initial value q0(x)∈H4,1(R),the research results obtain an interesting conclusion:the asymptotic expansion of the solution to the extended mKdV equation can be characterized by finite soliton solutions on the discrete spectrum,leading-terms on the continuous spectrum,and error terms from the (?)-equation,and the error can be improved from O(ln t/t)to O(t-3/4).In Chapter 5,we investigate the long-time behavior of the Fokas-Lenells(FL)equation in two regions of x/t+α<0 and x/t+α>0 using the generalized (?)-steepest descent method based on singularity characteristics.We overcome the problem of only constructing partial derivative solution qx(x,t)in a previous work and obtain the longtime asymptotic behavior of the original solution q(x,t)in two regions.In addition,we prove that the soliton resolution conjecture holds,indicating that at t→∞,the solution of the focusing FL equation can be represented as soliton solutions and the leading-terms and error terms.In Chapter 6,the initial value problem of the modified Camassa-Holm(mCH)equation on a straight line is studied based on the development of the (?)-descent method using multi-gauge transformations,where the initial value u0 tends to a constant at infinity and satisfies u0(x)-1 ∈ H4,1(R).We obtain the long-time behavior of the mCH equation under finite density initial value conditions in four different asymptotic regions,where regions ξ∈(-∞,-1/4)and ξ∈(2,∞)do not have phase points,and the solution of the mCH equation can be represented as soliton solutions and error terms.The regions ξ∈(-1/4,0)and ξ∈(0,2)have eight and four phase points,respectively.This proves the validity of the soliton resolution conjecture and the asymptotic stability of the N-soliton solution.In Chapter 7,there is a summary of the work of this thesis and further prospects for related research.