$N$-double poles solutions for nonlocal Hirota equation with nonzero boundary conditions using Riemann-Hilbert method and PINN algorithm

Abstract

We systematically investigate the nonlocal Hirota equation with nonzero boundary conditions via Riemann-Hilbert method and multi-layer physics-informed neural networks algorithm. Starting from the Lax pair of nonzero nonlocal Hirota equation, we first give out the Jost function, scattering matrix, their symmetry and asymptotic behavior. Then, the Riemann-Hilbert problem with nonzero boundary conditions are constructed and the precise formulaes of $N$-double poles solutions and $N$-simple poles solutions are written by determinants. Different from the local Hirota equation, the symmetry of scattering data for nonlocal Hirota equation is completely different, which results in disparate discrete spectral distribution. In particular, it could be more complicated and difficult to obtain the symmetry of scattering data under the circumstance of double poles. Besides, we also analyse the asymptotic state of one-double poles solution as $t\rightarrow \infty$. Whereafter, the PINN algorithm is applied to research the data-driven soliton solutions of the nonzero nonlocal Hirota equation by using the training data obtained from the Riemann-Hilbert method. Most strikingly, the integrable nonlocal equation is firstly solved via PINN algorithm. As we all know, the nonlocal equations contain the $\mathcal{PT}$ symmetry $\mathcal{P}:x\rightarrow -x,$ or $\mathcal{T}:t\rightarrow -t,$ which are different with local ones. Adding the nonlocal term into the NN, we can successfully solve the integrable nonlocal Hirota equation by PINN algorithm. The numerical results indicate the algorithm can well recover the data-driven soliton solutions of the integrable nonlocal equation. Noteworthily, the inverse problems of the integrable nonlocal equation are discussed for the first time through applying the PINN algorithm to discover the parameters of the equation in terms of its soliton solution.