Long-time asymptotics for the Hirota equation on the half-line

Abstract

We consider the Hirota equation on the quarter plane with the initial and boundary values belonging to the Schwartz space. The goal of this paper is to study the long-time behavior of the solution of this initial-boundary value problem based on the asymptotic analysis of an associated matrix Riemann-Hilbert problem. 0 1 , J 2 (x, t, k) = (J 1 J −1 4 J 3 )(x, t, k) = 1 − r(k)e −tΦ(k) −r(k)e tΦ(k) 1 + r(k)r(k) ,

The Riemann–Hilbert problem to coupled nonlinear Schrödinger equation: Long-time dynamics on the half-line

“…2 (x,t, k) into an appropriate upper/lower triangular form. This may be a main difference compared with the long-time asymptotic analysis of integrable nonlinear evolution equations associated with 2 × 2 matrix spectral problems on the half-line [3,20,21].…”

“…Furthermore, the long-time asymptotics for the focusing NLS equation with t-periodic boundary condition on the half-line is analyzed in [5]. We also have done some work about determining the long-time asymptotics for integrable equations on the half-line, see [20,21]. However, there is only a little of literature [7] to consider the asymptotic behaviors for integrable nonlinear PDEs with Lax pairs involving 3 × 3 matrices on the half-line.…”

Long-time asymptotics for the initial-boundary value problem of coupled Hirota equation on the half-line

“…According to (2.10), the {S n (k)} 4 1 can be computed from initial and boundary values alone as well as the spectral functions s(k) and S(k). 27) wheres…”

“…2 (x, t; k) into an appropriate upper/lower triangular form. This is a main difference compared with the long-time asymptotic analysis of integrable nonlinear evolution equations associated with 2 × 2 matrix spectral problems on the half-line [3,26,27].…”

“…There also exist some meaningful papers [10,11,23] about the study of longtime asymptotics for the initial-value problems of integrable nonlinear evolution equations associated with 3 × 3 matrix spectral problems. For the large-time asymptotic analysis of the IBV problems of integrable nonlinear PDEs, we refer the readers to see [3,7,26,27,29,35]. More recently, by using the∂ generalization of the nonlinear steepest descent method recast by Dieng and McLaughlin [18] (where the error term was optimized to O(t −3/4 )), long-time behavior of solutions to the derivative NLS equation for soliton-free initial data was studied in [37].…”