Auto-Bäcklund transformations, Lax pair, bilinear forms and bright solitons for an extended (3+1)-dimensional nonlinear Schrödinger equation in an optical fiber

Painlevé analysis, auto-Bäcklund transformations, bilinear forms and soliton solutions for a (2+1)-dimensional variable-coefficient modified dispersive water-wave system in fluid mechanics

“…Fluid mechanics has been regarded as the study of the fundamental mechanisms and force of liquids, plasmas and gases, with applications in astrophysics, meteorology, oceanography and biomedical engineering [1][2][3][4]. Methods have been introduced to solve the nonlinear evolution equations, including the Bäcklund transformation, Darboux transformation and the Lie symmetry approach [5][6][7][8][9][10].…”

Letter to the Editor: Thinking of the oceanic shallow water in the light of a (2+1)-dimensional generalized dispersive long-wave system related to HFF 33, 3272; 33, 965 and 32, 2282

“…To our knowledge, for system (1), hetero-Bäcklund transformations different from those in Gao et al (2023aGao et al ( , 2021cGao et al ( , 2020 and similarity reductions different from those in Gao et al (2022bGao et al ( , 2023aGao et al ( , 2021b have not been worked out. Employing symbolic computation (Zhou and Tian, 2022;Wu et al, 2023b;Shen et al, 2023dChen et al, 2021;He and Lü, 2022) [1], we purpose to build up two sets of the hetero-Bäcklund transformations which differ from those in Gao et al (2023aGao et al ( , 2021cGao et al ( , 2020, as well as a set of the similarity reductions which differs from those in Gao et al (2022bGao et al ( , 2023aGao et al ( , 2021b.…”

Letter to the Editor: Singular-manifold view on a (3+1)-dimensional fourth-order nonlinear equation in a fluid via HFF 32, 1664 (2022)

“…In equation (1) let us put the truncated Painlevé expansion, in a generalized Laurent series (Zhou and Tian, 2022; Zhou et al ., 2023; Gao, 2023a, 2023b, 2023c), around a noncharacteristic movable singular manifold conferred by an analytic function ψ ( x , y , z , t ) = 0, as: where v k ( x , y , z , t ) ’s also represent the analytic functions, with v 0 ( x , y , z , t ) ≠ 0, ψ x ( x , y , z , t ) ≠ 0 and ψ t ( x , y , z , t ) ≠ 0, and if the powers of ψ at the lowest orders cancel out, the positive integer: …”

“…For equation (1), Wazwaz (2022) has investigated the Painlev e integrability, lump and multiple soliton solutions, while Meng et al (2023) has presented the special cases in fluid dynamics, bilinear auto-Bäcklund transformations, breather and mixed lump-kink solutions. This Letter, based on the work in Wazwaz (2022) and Meng et al (2023), aims to seek an auto-Bäcklund transformation for equation (1), which is different from those in Meng et al (2023).In equation (1) let us put the truncated Painlev e expansion, in a generalized Laurent series (Zhou and Tian, 2022;Zhou et al, 2023;Gao, 2023aGao, , 2023bGao, , 2023c, around a noncharacteristic movable singular manifold conferred by an analytic function c(x, y, z, t) ¼ 0, as:Letter to the Editor 3561…”

“…In equation (1) let us put the truncated Painlev e expansion, in a generalized Laurent series (Zhou and Tian, 2022;Zhou et al, 2023;Gao, 2023aGao, , 2023bGao, , 2023c, around a noncharacteristic movable singular manifold conferred by an analytic function c(x, y, z, t) ¼ 0, as:…”