Riemann–Hilbert approach of the Newell-type long-wave–short-wave equation via the temporal-part spectral analysis

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In this paper, the Riemann–Hilbert approach is systematically established for the Newell-type long-wave–short-wave equation. Firstly, we start spectral analysis from the t-part of the Lax pair other than the x-part to obtain enough analytic spectral functions formulating the desired Riemann–Hilbert problems. The associated Riemann–Hilbert problems are determined by the t-part of the Lax pair with the x-part playing only an auxiliary role. Secondly, the zero structures of the Riemann–Hilbert problems are obtained, from which Riemann–Hilbert problems corresponding to the reflectionless cases are explicitly solved. Then, based on the solutions of the Riemann–Hilbert problems, a general form of multi-soliton solutions is derived for the long-wave–short-wave equation. Thirdly, some interesting soliton dynamics of the obtained multi-soliton solutions are explored and then demonstrated via the symbolic computation system Mathematica. The Riemann–Hilbert approach proposed in this paper allows us to investigate the complicated spectral structure of the long-wave–short-wave equation and is very efficient to derive the general multi-soliton solutions of the equation.