A Number Of Limited Parameters Of The (2 +1)-dimensional Soliton Equations Solution

It is difficulty to get the explicit solution for the (2+1)-dimensional soliton equation. Quite a few progress has been made during the recent decades. Nevertheless, each method has its limitation. The purpose of the present paper is using the nonlinearization approach to study the explicit finite-parameter solution to the difficult (2+1) dimensional soliton equation.Four (2+1)-dimensional soliton models are investigated: the (2+1)-dimensional sine-Gordon equation (SG), two kinds of modified Kadomtsev-Petviashvili equation (mKP), a (2+1)-dimensional integrable equation associated with Boiti-Pempinelli- Tu spectral problem (BPT).First, starting from the spectral problem associated with the soliton model, we construct the fundamental identity and find out the Lenart operator pair and the explicit expression of the Lenart's gradients by using the commutative condition, from which the hierarchy of (1+1)-dimensional soliton equations is obtained and the interesting (2+1)-dimensional soliton equation is constructed by the compatible condition of two (1+1)-dimensional soliton equations. Then, through the nonlinearization procedure, an integrable finite dimensional Hamiltonian system is acquired, whose complete integrability in Liouville sense is proved by Lax matrix and generating function of the conserved integrals. Thus a hierarchy of integrable finite dimensional Hamiltonian systems is obtained, by which the integrable decomposition of the (2+1)-dimensional soliton equation is acquired.As an application of algebraic curve theory, the elliptic coordinates and the Abel-Jacobi coordinates are introduced suitably, from which the straightening out of the Hamiltonian flows are made and the quadrature is obtained directly. Finally, by virtue of the Riemann-Jacobi inversion, the solution of the soliton equation is represented in the original variable explicitly.Additionally, the relationship between the (2+1)-dimensional sine-Gordon equation (SG) and the modified Korteweg-de Vries equation (mKdV) is also considered.