Analytical and numerical studies for integrable and non-integrable fractional discrete modified Korteweg-de Vries hierarchies

Abstract

Under investigation in this paper is the fractional integrable and non-integrable discrete modified Korteweg-de Vries hierarchies. The linear dispersion relations, completeness relations, inverse scattering transform, and fractional soliton solutions of the fractional integrable discrete modified Korteweg-de Vries hierarchy will be explored. The inverse scattering problem will be solved accurately by using Gel'fand-Levitan-Marchenko (GLM) equations and Riemann-Hilbert (RH) problem. The peak velocity of fractional soliton solutions will be analyzed. The numerical solutions of the non-integrable fractional averaged discrete modified Korteweg-de Vries equation which has a simpler form than the integrable one will be obtained by a split-step fourier method.

Figures (10)

• Figure 1: Fractional one-soliton solution of Eq. (2.10) with $z_1=2,\ \bar{z}_1=0.5,\ C_1\left(0\right)=\bar{C}_1\left(0\right)=1,\ \epsilon=0.25$. (a) GLM equations; (b) RH problem.
• Figure 2: Fractional one-soliton solution of Eq. (2.11) with $z_1=2,\ \bar{z}_1=0.5,\ C_1\left(0\right)=\bar{C}_1\left(0\right)=1,\ \epsilon=0.25$. (a) GLM equations; (b) RH problem.
• Figure 3: Fractional two-soliton solution of Eq. (2.10) with $z_1=2.5,\ \bar{z}_1=0.4,\ z_2=2,\ \bar{z}_2=0.5, \ C_1\left(0\right)=\bar{C}_1\left(0\right)=C_2\left(0\right)=\bar{C}_2\left(0\right)=1,\ \epsilon=0.1$.
• Figure 4: Fractional two-soliton solution of Eq. (2.11) with $z_1=2.5,\ \bar{z}_1=0.4,\ z_2=2,\ \bar{z}_2=0.5, \ C_1\left(0\right)=\bar{C}_1\left(0\right)=C_2\left(0\right)=\bar{C}_2\left(0\right)=1,\ \epsilon=0.1$.
• Figure 5: The solitary wave solutions of the fAdmKdV equation at different simulation times with $h=1,\ \eta=0.5,\ \phi_0=0$. (a) $\epsilon=0.1$; (b) $\epsilon=0.2$.