The Modified Korteweg-de Vries Equation

Abstract. We present a systematic way to construct solutions of the (n = 5)-reduction of the BKP and CKP hierarchies from the general τ function τ (n+k) of the KP hierarchy. We obtain the one-soliton, two-soliton, and periodic solution for the bi-directional Sawada-Kotera (bSK), the bi-directional Kaup-Kupershmidt (bKK) and also the bi-directional Satsuma-Hirota (bSH) equation. Different solutions such as left-and right-going solitons are classified according to the symmetries of the 5th roots of e iε . Furthermore, we show that the soliton solutions of the nreduction of the BKP and CKP hierarchies with n = 2j + 1, j = 1, 2, 3, . . ., can propagate along j directions in the 1 + 1 space-time domain. Each such direction corresponds to one symmetric distribution of the nth roots of e iε . Based on this classification, we detail the existence of two-peak solitons of the n-reduction from the Grammian τ function of the sub-hierarchies BKP and CKP. If n is even, we again find two-peak solitons. Last, we obtain the "stationary" soliton for the higher-order KP hierarchy.

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“…The 5-reduction of the CKP hierarchy yields the bKK equation equation (15). Let the initial value be u = 0 in Eqs.…”

Section: Periodic and Soliton Solutions Of Bkk Equationmentioning

confidence: 99%

Solving bi-directional soliton equations in the KP hierarchy by gauge transformation

Cheng

Römer

2006

J. High Energy Phys.

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“…Among these methods, the IST [1] is a systematic method. Since being put forward in 1967, the IST has achieved considerable development and received a wide range of applications [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. As a famous method in mathematical physics, the IST is a milestone in the process of developing analytical methods for solving nonlinear PDEs.…”

Section: Introductionmentioning

confidence: 99%

Mixed spectral AKNS hierarchy from linear isospectral problem and its exact solutions

Zhang

Gao

2015

Open Physics

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Abstract:In this paper, the AKNS isospectral problem and its corresponding time evolution are generalized by embedding three coe cient functions. Starting from the generalized AKNS isospectral problem, a mixed spectral AKNS hierarchy with variable coe cients is derived. Thanks to the selectivity of these coe cient functions, the mixed spectral AKNS hierarchy contains not only isospectral equations but also nonisospectral equations. Based on a systematic analysis of the related direct and inverse scattering problems, exact solutions of the mixed spectral AKNS hierarchy are obtained through the inverse scattering transformation. In the case of re ectionless potentials, the obtained exact solutions are reduced to n-soliton solutions. This paper shows that the AKNS spectral problem being nonisospectral is not a necessary condition to construct a nonisospectral AKNS hierarchy and that the inverse scattering transformation can be used for solving some other variable-coe cient mixed hierarchies of isospectral equations and nonisospectral equations.

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“…(1) reflect in a simple general form the dispersion relation k(ω) of the medium and its cubic nonlinear susceptibility χ (3) . It should be emphasized that the mKdV equation is completely integrable by means of the inverse scattering transform (IST) method [13,14].…”

Section: Modelsmentioning

confidence: 99%

Few-cycle solitons in supercontinuum generation dynamics

Leblond

Grelu

Mihalache

et al. 2016

Eur. Phys. J. Spec. Top.

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Abstract. We review several propagation models that do not rely on the slowly-varying-envelope approximation (SVEA), and can thus be considered as fundamental models addressing the formation and propagation of few-cycle pulsed field structures and solitary waves arising in the course of intense ultrashort optical pulse evolution in nonlinear media and beyond octave-bandwidth optical spectrum broadening. These generic models are: the modified-Korteweg-de-Vries (mKdV), the sineGordon (sG), and the mixed mKdV-sG equations. To include wave polarization dynamics, the vector extensions of both mKdV and sG equations are introduced. Multi-octave-spanning supercontinuum generation and few-cycle soliton structures are highlighted from numerical simulations.

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The Modified Korteweg-de Vries Equation

Cited by 501 publications

References 7 publications

Solving bi-directional soliton equations in the KP hierarchy by gauge transformation

Solving bi-directional soliton equations in the KP hierarchy by gauge transformation

Mixed spectral AKNS hierarchy from linear isospectral problem and its exact solutions

Few-cycle solitons in supercontinuum generation dynamics

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