Envelope soliton resonances and broer-kaup-type non-madelung fluids

Multiple soliton solutions for the Whitham–Broer–Kaup model in the shallow water small-amplitude regime

“…Solitons can be observed in the atmosphere, forming cloud rolls, known as morning glory, that appear in northern Australia, and can also be observed in tidal bores, known as bore solitons. Solitons possess many useful and unique properties in solitary waves theory [1][2][3][4][5][6][7][8][9][10][11][12][13].…”

“…Toward this goal, a variety of powerful methods to study the integrability of nonlinear partial differential equations and to construct multiple soliton solutions have been established by many physicists and mathematicians. Examples of the methods that have been used are: the inverse scattering method, Hirota's bilinear method and its simplified form, the Bäcklund transformation method [11][12][13][14][15][16][17][18][19][20][21][22][23][24], Darboux transformation, Pfaffian technique, the Painlevé analysis, the Lax pair, the Bell polynomial approach, the generalized symmetry method, the mapping and the deformation approach and many other methods.…”

“…The simplified Hirota method does not depend on the construction of the bilinear forms, instead it assumes the multi-soliton solutions can be expressed as polynomials of exponential functions. Hirota's bilinear method and the simplified Hirota approach are rather heuristic and significant for handling nonlinear equations [1][2][3][4][5][6][7][8][9][10][11][12][13].…”

“…In [12], the resonant nonlinear Schrodinger equation (RNLS) was investigated. For the RNLS equation, in addition to the Madelung fluid representation, an alternative non-Madelung fluid system in the form of a Broer-Kaup (BK) system was obtained in [12], and using the bilinear form for the RNLS equation, the soliton resonances for the BK system were constructed in [12]. Moreover, the coupled heat equation, the BK hydrodynamic representation and the classical Boussinesq equation were also investigated, and soliton solutions were derived.…”

Relativistic dissipatons in integrable nonlinear Majorana type spinor model

“…In addition, the mapping of the RNLS hierarchy, the second and the third flow to KP-II equation [16], established link between dissipaton and envelope soliton resonances with planar solitons of KP-II, creating the web type structure in shallow water [27], [28]. Several modifications of RNLS models, as the derivative RNLS [5], [6], [29], modified RNLS [30], generic RNLS [31] and related generalized equations [32], [33], [34], [35], [36], were studied. However, all these developments are related with non-relativistic dissipatons and envelope solitons, so it is not clear if exist relativistic nonlinear equations with dissipaton solutions and resonant property of their mutual interaction.…”