Long Time Asymptotic Behavior of the Focusing Nonlinear Schrodinger Equation

Long-time behavior of solutions to the derivative nonlinear Schrödinger equation for soliton-free initial data

“…we obtain a RHP for N (1) with a new jump matrix e 2itθ ad σ3 V (1) . We give expressions for V (1) for each of the four cases ±t > 0, ±x > 0 in (B.2), (B.6), (B.10), and (B.14) respectively.…”

“…we obtain a RHP for N (1) with a new jump matrix e 2itθ ad σ3 V (1) . We give expressions for V (1) for each of the four cases ±t > 0, ±x > 0 in (B.2), (B.6), (B.10), and (B.14) respectively. The new RHP's are 'prepared' for the steepest descent method in the sense that contours can be deformed so that the exponential functions e ±itθ have maximum decay in |z − ξ|.…”

“…These ideas were implemented by Miller and McLaughlin [18] to the study of asymptotic stability of orthogonal polynomials. In the context of NLS with soliton solutions, they were successfully applied to prove asymptotic stability of N -soliton solutions to defocusing NLS [2] and address the soliton resolution problem for focusing NLS [1].…”

“…As discussed earlier, the proof of Theorem 1.4, following [1,8], consists of several steps corresponding to transformations of the initial RHP 1.1 implemented successively. For sake of clarity, we present in Section 2 a summary of the analysis of the various steps in each of the four cases, x ≶ 0, t → ±∞ and we show how the RHPs and the respective factorizations are modified to take into account the signs of x and t. In the next Sections (Sections 3 to 7), we provide the details of each step in one case x > 0, t → ∞ as follows.…”

Asymptotic stability of N-solitons in the cubic NLS equation

“…Finally in Section 7 the results will be collected in order to prove Theorem 1.1. The results in [BJM16] (and also the proofs) are basically the same but developed independently of each other.…”