mKdV Equation on Time Scales: Darboux Transformation and <i>N</i>-Soliton Solutions

Abstract

In this paper, the spectral problem of the mKdV equation satisfying the compatibility condition on time scales is directly constructed. By using the zero-curvature equation on time scales, the mKdV equation on time scales is obtained. When x∈R and t∈R, the equation degenerates to the classical mKdV equation. Then, the single-soliton, two-soliton, and N-soliton solutions of the mKdV equation under the zero boundary condition on time scales are presented via employing the Darboux transformation (DT). Particularly, we obtain the corresponding single-soliton solutions expressed using the Cayley exponential function on four different time scales (R, Z, q-discrete, C).

Keywords

• modified Korteweg–de Vries equation
• time scales
• Cayley exponential function
• Darboux transformation