Exact Analytic N -Soliton-Like Solution in Wronskian Form for a Generalized Variable-Coefficient Korteweg–de Vries Model from Plasmas and Fluid Dynamics

Abstract

Applicable in fluid dynamics and plasmas, a generalized variable-coefficient Korteweg-de Vries (vcKdV) model is investigated. The bilinear form and analytic N-soliton-like solution for such a model are derived by the Hirota method and Wronskian technique. Additionally, the bilinear auto-Bkklund transformation between (N-1)soliton-like and N-soliton-like solutions is verified.

Investigation on a nonisospectral fifth-order Korteweg-de Vries equation generalized from fluids

“…[9][10][11][12][13][14][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] Generally speaking, it is easier to solve analytically the NLEEs with the scaled "temporal" variable coefficients than those with the scaled "spatial" ones from the viewpoint of integrability, just as those shown in Refs. [42][43][44][45][46]. Whereas, if the dissipative coefficient with respect to the scaled "spatial" coordinate x is linear, the NLEE could still be expected to be integrable, which forms the nonisospectral problem.…”

Theoretical investigations on a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers model for a dilated artery, blood vessel or circulatory system with experimental support

“…( ) ˆ( ), a variable-coefficient generalized Korteweg-de Vries model with dissipative, perturbed and external-force terms for the pulse waves in a blood vessel or dynamics in a circulatory system [17] (and references therein) r r h r h By the bye, more nonlinear evolution equations might be found, for instance, in [24][25][26][27][28].…”

“…3 Reference [16] has discussed the situation with s = 0 2 ˜. 4 Reference [17] has also, with references therein, listed out other applications of equation (3) in the fluid-filled tubes, Bose-Einstein condensates, circular rods with variable cross-sections and material densities, varied-depth shallow-water channels, lakes, and so on. applications of equation (4) in the dusty plasmas, interactionless plasmas, two-layer liquids, atmospheric flows, shallow seas and deep oceans.…”

Symmetries and solutions of a variable coefficient KdV equation via gauge transformation

“…the bilinear KdV equation (4.2) is then transformed to the bilinear vcKdV equation We note that both (4.6) and (4.7) have been given in [12]. Now that the bilinear equations admit the coordinate relation (1.3b), it is easy to give Wronskian solutions for the bilinear vcKdV equation.…”

“…where a and b are real constants and a = 0. Under the above relation the vcKdV equation (1.1) can pass the Painlevé test [10], can be bilinearized with multi-soliton-like solutions [11,12] and can have infinitely many forms of conservation laws [11,13]. In fact, behind these results there is a gauge transformation [14] between the vcKdV equation (1.1) and the standard KdV equation when f (t) and g(t) satisfy (1.2).…”