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In this paper, we present an integrable semi-discretization of the modified Korteweg-deVries (mKdV) equation. We discretize the “time” variable of the mKdV equation and get an integrable differential-difference system. Under a standard limit, the differential-difference system converges to the continuous mKdV equation. By Hirota’s bilinear method, we find some explicit solutions including solitons and breather solutions. From the semi-discrete system, we design a numerical scheme to the mKdV equation and carry out several numerical experiments with the 3-soliton solution and breather solution. [ABSTRACT FROM AUTHOR]

In this paper, we give a proof of the existence of stationary dark soliton solutions or heteroclinic orbits of nonlinear equations of Schrödinger type with periodic inhomogeneous nonlinearity. The result is illustrated with examples of dark solitons for cubic and photorefractive nonlinearities. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the modified Camassa–Holm equation is Painlevé integrable. We also study the orbital stability problem of negative solitary waves for this integrable equation. It is shown that the negative solitary waves are stable for arbitrary wave speed of propagation. [ABSTRACT FROM AUTHOR]

In this paper, we numerically investigate the dynamics of the solitary waves in the presence of damping and external stochastic forces for the improved Boussinesq equation. The hydrodynamical and Stokes damping will be considered respectively. Since the analytic solutions for the considered problems are not available, Monte Carlo method will be used in random space and finite volume element method will be used in physical space. Numerical results demonstrate that solitary wave profile is not strongly affected by the Gaussian white noise in our study. The average of the velocity of the solitary waves is very close to that for the deterministic case. [ABSTRACT FROM AUTHOR]