New analytical positon, negaton and complexiton solutions of a coupled KdV–mKdV system

A Fourth Order Finite Difference Method for the Good Boussinesq Equation

“…By making use of the differential system (7) and (8), all terms inside the brackets in (22) are zero. Similar analysis can be done for (20).…”

“…which provides a balance between dispersion and nonlinearity, that leads to the existence of soliton solutions, similar to the Korteweg-de Vries (KdV) and cubic nonlinear Schrödinger equation [1,8,22]. The initial displacement associated with the partial differential equation given in (1) is assumed to take the form [23][24][25] ( , 0 ) = ( ) ; 0 ≤ ≤ 1 ,…”

Nth-order smooth positon and breather-positon solutions of a generalized nonlinear Schrödinger equation

“…By considering the importance of positon solutions, efforts have been made to identify singular positon solutions in many nonlinear integrable evolution equations including modified KdV equation [16], Toda chain [17], sine-Gordon equation [18], KdV and modified KdV hierarchies [19], fifth-order KdV equation [20], extended KdV equation [21] and KdV equation with self-consistent sources [22]. Non-singular positon solutions on vanishing background are named as smooth positons or degenerate soliton solutions [23][24][25][26][27][28][29]. This kind of solution has also been constructed for several nonlinear partial differential equations including nonlinear Schrödinger (NLS) equation [23,30], Bogoyavlensky-Konopelchenko equation [24], coupled KdV [25] and mKdV equations [26], derivative NLS equation [27], nonlocal Kundu-NLS equation [28], complex mKdV equation [29], Wadati-Konno-Ichikawa equation [31] and higher-order Chen-Lee-Liu equation [32].…”

Nth order smooth positon and breather-positon solutions of a generalized nonlinear Schrödinger equation

“…Non-singular positon solutions on vanishing background are named as smooth positons or degenerate soliton solutions [26][27][28][29][30][31][32]. This kind of solution has also been constructed for several nonlinear partial differential equations including nonlinear Schrödinger (NLS) equation [25,26], Bogoyavlensky-Konoplechenko equation [27], coupled KdV [28] and mKdV equations [29], derivative NLS equation [30], nonlocal Kundu-NLS equation [31], complex mKdV equation [32], Wadati-Konno-Ichikawa equation [33] and higher-order Chen-Lee-Liu equation [34]. Recently positons on nonvanishing background for the NLS equation have also been constructed and they have been coined as breather-positon (B-P) solutions.…”