Abstract:

The nonlocal positive flow short-pules equation is first proposed based on the nonlinear transverse oscillation of elastic beam under tension in physics. By the nonlinear steepest descent method, the long-time asymptotics of the solution of the Cauchy problem for the equation is discussed. Starting from the WKI-type Lax pair it satisfies, the corresponding basic Riemann-Hilbert problem and reconstruction formula for the solution are established. Through a series of deformations such as reorientation, extending, cuting and rescaling, the basic Riemann-Hilbert problem is transformed into the model Riemann-Hilbert problem that can be solved using parabolic cylinder functions. Finally, the long-time asymptotics of the solution of the nonlocal positive flow short-pules equation is obtained.

Key words: integrable system, nonlinear steepest descent method, long-time asymptotics