The bi-Hamiltonian structure is established for the perturbation equations of KdV hierarchy and thus the perturbation equations themselves provide also examples among typical soliton equations. Besides, a more general bi-Hamiltonian integrable hierarchy is proposed and a remark is given for a generalization of the resulting perturbation equations to 1 + 2 dimensions. Perturbation theory is the ...

Recent concept of integrable nonholonomic deformation found for the KdV equation is extended to the mKdV equation and generalized to the AKNS system. For the deformed mKdV equation we find a matrix Lax pair, a novel two-fold integrable hierarchy and exact N-soliton solutions exhibiting unusual accelerating motion. We show that both the deformed KdV and mKdV systems possess infinitely many gener...

We study the traveling waves of the Nonlinear Schrödinger Equation in dimension one. Through various model cases, we show that for nonlinearities having the same qualitative behaviour as the standard Gross-Pitaevkii one, the traveling waves may have rather different properties. In particular, our examples exhibit multiplicity or nonexistence results, cusps (as for the Jones-Roberts curve in the...

of solutions of KdV whose role is similar to that of e ( kx−ω(k)t ) is discussed. Theory of these solutions, referred to here as harmonic breathers, is developed and it is shown that these solutions may be used to construct more general solutions of KdV similarly to how the functions e ( kx−ω(t) ) are used to perform the same task in the theory of Fourier transform. Nonlinear superposition form...

The conservation laws for the integrable coupled KDV type system, complexly coupled kdv system, coupled system arising from complex-valued KDV in magnetized plasma, Ito integrable system, and Navier stokes equations of gas dynamics are computed by multipliers approach. First of all, we calculate the multipliers depending on dependent variables, independent variables, and derivatives of dependen...

Darboux transformation is reconsidered for the supersymmetric KdV system. By iterating the Darboux transformation, a supersymmetric extension of the Crum transformation is obtained for the Manin-Radul SKdV equation, in doing so one gets Wronskian superdeterminant representations for the solutions. Particular examples provide us explicit supersymmetric extensions, super solitons, of the standard...

1 KdV equation and Schrödinger operator 2 1.1 Integrability of Korteweg – de Vries equation . . . . . . . . . . . . . . . . . . 2 1.2 Elements of scattering theory for the Schrödinger operator . . . . . . . . . . . 5 1.3 Inverse scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Dressing operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

We study solitary wave interactions in the Euler–Poisson equations modeling ion acoustic plasmas and their approximation by KdV n-solitons. Numerical experiments are performed and solutions compared to appropriately scaled KdV n-solitons. While largely correct qualitatively the soliton solutions did not accurately capture the scattering shifts experienced by the solitary waves. We propose corre...

In this paper, we study the propagation of bores over a long distance. We employ experimental data as input for numerical simulations using COULWAVE. The flume is extended numerically to an effective relative length x/h=3000, which allows all far-field solitons emerge from undular bore in simulation data. apply periodic KdV-based nonlinear Fourier transform (KdV-NFT) time series taken at differ...

Numerical simulations of the interaction internal solitary waves (ISWs) opposite polarity are conducted by solving incompressible Euler equations under Boussinesq approximation. A double-pycnocline stratification is used. method to determine when ISWs both polarities exist also presented. The confirm previous work that same soliton-like; however, here it shown a fast ISW with as Korteweg–de Vri...