<i>N</i>-Fold Darboux Transformation of the Jaulent-Miodek Equation

Abstract

In this paper, based on the Lax pair of the Jaulent-Miodek spectral problem, we construct the Darboux transformation of the Jaulent-Miodek Equation. Then from a trivial solution, we get the exact solutions of the Jaulent-Miodek Equation. We obtain a kink-type soliton and a bell-kink-type soliton. Particularly, we obtain the exact solutions which describe the elastic-inelastic-interaction coexistence phenomenon.

New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis

“…The Equations ( 1) and ( 2) also have a relation with Euler-Darboux equation, which has been presented by Matsuno [19]. The Darboux transformation of the JM spectral problem has been studied by Xu [20]. By using hereditary symmetries, Ruan and Lou [21] have presented the symmetries of Jaulent-Miodek hierarchy.…”

The New Solitary Solutions to the Time-Fractional Coupled Jaulent–Miodek Equation

“…Owing to the importance of studying this class of equations, several distinct techniques have been proposed such as the fractional Davey-Stewartson equation with power law nonlinearity [9], the Calogero-Bogoyavlenskii-Schiff equation in (2 + 1) dimensions with time-fractional conformable derivative [10], the fractional mitigating Internet bottleneck with quadratic cubic nonlinearity [11], the density-dependent conformable fractional diffusion-reaction equation [12], the space-time fractional Klein-Gordon equation with symmetry analysis [13], and the space-time fractional advection-diffusion equation with convergence analysis [14]. e further analytical methods for the (2 + 1)-dimensional Jaulent-Miodek (JM) evolution equation have been well scrutinized by a lot of researchers containing the results such as the Hirota's bilinear method [15], the expfunction method [16], the symmetry reductions method [17], the homotopy perturbation method [18], the optimal hidden symmetries [19], and the related topics of the Jaulent-Miodek equation with various topics, e.g., the (G′/G)expansion method [20], the bifurcation and exact traveling wave solutions [21], the integrating factors method in an unbounded domain [22], the Adomian's decomposition method [23], the finite-band solution method [24], N-fold Darboux transformation method [25], the Hermite wavelets method [26], and the modified Riemann-Liouville derivative and exterior derivatives [27]. In the following, we take the nonlinear time-fractional coupled Jaulent-Miodek (FCJM) equation, firstly introduced by Jaulent and Miodek [27][28][29], that is,…”

Wave motions in discontinuous initial-value problem of the inviscid shallow water wave Jaulent-Miodek model

“…The quasi-periodic solutions, also named algebro-geometric solutions, related to the JM system are discussed in [9,24]. Besides, the exact kink-type and bell-kink-type of soliton solutions of the JM system are derived by the method of Darboux transformation [57]. The tanh-coth and the sech methods [54], the homotopy analysis method [50],…”