Bright N–solitons for the intermediate nonlinear Schrödinger equation

The integral sign with a backslash denotes the principal value. The constant σ is taken to be ±1. For positive (negative) σ, the equation is called the defocusing (focusing) INLS equation. Originally, the defocusing INLS equation was discovered by carrying out a reductive perturbation method for the ILW equation [8]. It describes the long–term evolution of quasi–harmonic wave packets whose wavelength is short compared to the fluid depth. The inverse scattering transform for the defocusing equation and its Hirota bilinear form are known [9, 4, 5]. The dark soliton solutions of this equation have been also investigated intensively. But as for the focusing type, which is also an integrable system in its own right, as far as the author knows, soliton solutions have never come up in the literature. In this paper, it will be shown that bright solitons exist for the focusing INLS equation and that they possess an interesting structure. An explicit formula for a bright N–soliton