Asymptotic solutions of the one-dimensional linearized Korteweg–de Vries equation with localized initial data

2017 ◽  
Vol 102 (3-4) ◽  
pp. 403-416 ◽  
Author(s):  
S. A. Sergeev
Fractals ◽  
2019 ◽  
Vol 27 (01) ◽  
pp. 1940010 ◽  
Author(s):  
FENG GAO ◽  
XIAO-JUN YANG ◽  
YANG JU

The one-dimensional modified Korteweg–de Vries equation defined on a Cantor set involving the local fractional derivative is investigated in this paper. With the aid of the fractal traveling-wave transformation technology, the nondifferentiable traveling-wave solutions for the problem are discussed in detail. The obtained results are accurate and efficient for describing the fractal water wave in mathematical physics.


2016 ◽  
Vol 30 (35) ◽  
pp. 1650318 ◽  
Author(s):  
Jun Chai ◽  
Bo Tian ◽  
Xi-Yang Xie ◽  
Han-Peng Chai

Investigation is given to a forced generalized variable-coefficient Korteweg–de Vries equation for the atmospheric blocking phenomenon. Applying the double-logarithmic and rational transformations, respectively, under certain variable-coefficient constraints, we get two different types of bilinear forms: (a) Based on the first type, the bilinear Bäcklund transformation (BT) is derived, the [Formula: see text]-soliton solutions in the Wronskian form are constructed, and the [Formula: see text]- and [Formula: see text]-soliton solutions are proved to satisfy the bilinear BT; (b) Based on the second type, via the Hirota method, the one- and two-soliton solutions are obtained. Those two types of solutions are different. Graphic analysis on the two types shows that the soliton velocity depends on [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the soliton amplitude is merely related to [Formula: see text], and the background depends on [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are the dissipative, dispersive, nonuniform and line-damping coefficients, respectively, and [Formula: see text] is the external-force term. We present some types of interactions between the two solitons, including the head-on and overtaking interactions, interactions between the velocity- and amplitude-unvarying two solitons, between the velocity-varying while amplitude-unvarying two solitons and between the velocity- and amplitude-varying two solitons, as well as the interactions occurring on the constant and varying backgrounds.


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