Quasic-periodic Solutions Of Soliton Equations Associated With 2 × 2 Matrix Spectral Problems

One of the mathematical miracles of the 20 th century was the discovery of agroup of nonlinear wave equations being integrable. The beauty of the integrabletheory is reflected by the explicit formulas of nontrivial solutions of these nonlinearwave equations. Therefore, searching for the solutions of the nonlinear waveequations is a very important subject in this field. A great many of scholars areattracted to the study of this field and a large number of methods for obtainingthese explicit solution were propsed simultaneously. Such as, inverse scatteringtransform, B¨acklund transformation, Darboux transformation, Hirota’s bilinearmethods, algebro-geometric method.In this paper, under the help of the theory of algebraic curve and algebraicgeometry, we construct quasi-periodic solutions of four soliton hierarchies withimportant physical background. These hierarchies associated with 2 × 2 matrixspectral problems we discuss here are Konno-Oono equations, mixed AB equa-tions, fully local sine-Gordon hierarchy, generalized Jaulent-Miodek hierarchy,respectively.First, we introduce the Lenard recursion eauqtions to construct the hierar-chy of soliton equations which associated with 2 × 2 matrix spectral problemsin view of the zero-curvature equation. Using the characteristic polynomial ofthe Lax matrix, a hyperelliptic curve ????of arithmetic genus ?? is brought in.We introduce appropriate meromorphic function, Baker-Akhiezer function andelliptic variables on this algebric curve, from which soliton equations are decom-posed into solvable ordinary differential equations. Then, under the Abel-Jacobicoordinates, the flows of the soliton hierarchy are straightend. Furthermore,inaccordance with the properties of the zeros and poles of the meromorphic functionand Baker-Akhiezer function,we get their Riemann theta function representa-tions by means of the second and third Abel differentials,Riemann theorem andRiemann-Roch theorem.Combing the Riemann theta functionrepresentations ofthe meromorphic function and the Baker-Akhiezer function with their asymptoticproperties,we finally obtain the quasi-periodic solutions of soliton equations.