A modified complex short pulse equation of defocusing type

In this paper, we are concerned with a modified complex short pulse (mCSP) equation of defocusing type. Firstly, we show that the mCSP equation is linked to a complex coupled dispersionless equation of defocusing type via a hodograph transformation, thus, its Lax pair can be deduced. Then the bilinearization of the defocusing mCSP equation is formulated via dependent variable and hodograph transformations. One- and two-dark soliton solutions are found by Hirota’s bilinear method and their properties are analyzed. It is shown that, depending on the parameters, the dark soliton solution can be either smoothed, cusponed or looped one. More specifically, the dark soliton tends to be evolved into a singular (cusponed or looped) one due to the increase of the spatial wave number in background plane waves and the increase of the depth of the trough. In the last part of the paper, we derive the defocusing mCSP equation from the single-component extended KP hierarchy by the reduction method. As a by-product, the N-dark soliton solution in the form of determinants for the defocusing mCSP is provided.