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