The nonlinear steepest descent approach to the long-time asymptotics of the three-coupled Lakshmanan–Porsezian–Daniel model

In this paper, the long-time asymptotic behavior of the three-coupled Lakshmanan–Porsezian–Daniel (LPD) model with Schwartz initial data is investigated by the nonlinear steepest descent approach. Based on the Lax pair of the LPD model, a Riemann–Hilbert problem associated with the initial value problem is constructed. Further a sequence of transformations change the Riemann–Hilbert problem into a tractable form via the Deift–Zhou nonlinear steepest descent approach. Then the long-time asymptotics of the LPD model is obtained through reconstruction formula. What distinguishes the 2 × 2 spectral problem is that we treat the matrix in a block form, the advantage of which is that the matrix can be regarded as a block diagonal matrix without being strictly diagonal, necessitating the complexity of the function $\delta \left(k\right)$. The primary limitation of this approach is that it does not enable direct acquisition of the solution $\delta \left(k\right)$. To address this challenge, we employ a term involves $\delta \left(k\right)-I\cdot det\delta \left(k\right)$, then the term can be asymptotically estimated as $O\left(t^{-1/2}\right)$. The distinguishing feature of the long-time asymptotic analysis for our problem, when compared with the nonlinear Schrödinger equation and Hirota equation, lies in the presence of three critical points.