Long-time Asymptotics for mKdV Type Reduced Equations of the AKNS Hierarchy in Weighted \u3cem\u3eL\u3c/em\u3e\u3csup\u3e2\u3c/sup\u3e Sobolev Spaces

The long-time asymptotics of nonlinear integrable partial differential equations is one of the important research areas in the field of integrable systems. The main tool to analyze the long-time behaviors is the so-called nonlinear steepest descent method, or Deift-Zhou\u27s method, which was born in 1993. To apply Deift-Zhou\u27s method, one first uses the inverse scattering transform to formulate the nonlinear PDEs in terms of an oscillatory 2 by 2 matrix Riemann-Hilbert problem (RHP). After about 15 years of development, a generalized version of Deift-Zhou\u27s method, the ∂—steepest method, came out. The ∂—steepest descent method is a useful method for analyzing RHP with rough scattering data, or in the view of inverse scattering, with rough initial data. Recently, the ∂—steepest descent method has been applied to study long-time behaviors of NLS, DNLS, mKdV and sine-Gordon equations as well as their corresponding soliton resolutions. In this dissertation, we use the ∂—steepest descent method to fully study the asymptotic behavior of a class of oscillatory Riemann-Hilbert problems. We restrict ourselves to the case of defocusing mKdV type of reductions of the AKNS hierarchy and consider the initial data in Hn-1,1 for the nth member of the hierarchy, n is an odd integer. The formulas for the long-time asymptotics in three regions are presented. The three regions are defined in the main results. In the oscillatory region (i.e. region I), we find q(x,t) = qas(x,t) + 퓞(t -¾), t→ ∞, where qas(x,t) = -2i ∑l j=1 ︳η(zj)︳½ / √2tθ (zj) 푒풾ℓ(t), where θ, ƞ, ℓ(t) and zj \u27s are defined in the introduction. In the case of mKdV hierarchy, we derive some interesting result for the other two regions. In this case, θ = xz + ctzn, n is odd. In the Painlevé region (i.e. region II), we find q(x,t) = (nt) -1/nun(x(nt)-1/n) + O(t-3/2n), t→∞, where un(x) belongs to Painlevé II hierarchy. In the fast decay region (i.e. region III), we find q(x,t) = 퓞(t-1), t→∞, In the second part, we study exact solutions to the focusing 5th-order mKdV and formulate multi-poles soliton solutions, i.e., solitons associated with the reflection coefficients having arbitrary orde rpoles. We use the generalized Vandermonde-like determinant to present the resulting solitons, which reduces the complexity of the involved computation. In conclusion, in the first part, we develop a generic scheme of applying the ∂-steepest descent method to an oscillatory RHP with arbitrary stationary phase points. Our results can be directly applied to any nonlinear PDEs generated from defocusing reductions of the AKNS hierarchy. In the second part, we show that the generalized Vandermonde-like determinant is a more efficient way to present higher order pole soliton solutions, based on generalized Darboux transformations