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Within the (2 + 1)-dimensional Korteweg-de Vries equation framework, new bilinear Backlund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation. By introducing an arbitrary function, a family of deformed soliton and deformed breather solutions are presented with the improved Hirotas bilinear method. Choosing the appropriate parameters, their interesting dynamic behaviors are shown in three-dimensional plots. Furthermore, novel rational solutions are generated by taking the limit of obtained solitons. Additionally, two dimensional [2D] rogue waves (localized in both space and time) on the soliton plane are presented, we refer to it as deformed 2D rogue waves. The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane, and its evolution process is analyzed in detail. The deformed 2D rogue wave solutions are constructed successfully, which are closely related to the arbitrary function. This new idea is also applicable to other nonlinear systems.
With the assistance of one fold Darboux transformation formula, we derive rogue wave solutions of the complex modified Korteweg-de Vries equation on an elliptic function background. We employ an algebraic method to find the necessary squared eigenfun
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation with steplike initial data leading to a rarefaction wave. In addition to the leading asymptotic we also compute the next term in
We show that the long-time behavior of solutions to the Korteweg-de Vries shock problem can be described as a slowly modulated one-gap solution in the dispersive shock region. The modulus of the elliptic function (i.e., the spectrum of the underlying
Based on integrable Hamiltonian systems related to the derivative Schwarzian Korteweg-de Vries (SKdV) equation, a novel discrete Lax pair for the lattice SKdV (lSKdV) equation is given by two copies of a Darboux transformation which can be used to de
The $n$-fold Darboux transformation $T_{n}$ of the focusing real mo-di-fied Kor-te-weg-de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the $n$-soliton solutions of the mKdV equation are als