Auto-Bäcklund transformations and soliton solutions on the nonzero background for a (3+1)-dimensional Korteweg-de Vries-Calogero-Bogoyavlenskii-Schif equation in a fluid

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

“…For system (1), certain similarity reductions (Gao et al, 2022b(Gao et al, , 2023a(Gao et al, , 2021b, hetero-Bäcklund transformations (Gao et al, 2023a(Gao et al, , 2021c(Gao et al, , 2020, bilinear forms (Gao et al, 2022b(Gao et al, , 2021c, scaling transformations (Gao et al, 2021c), solitons (Gao et al, 2021c(Gao et al, , 2020 and auto-Bäcklund transformations (Gao et al, 2020) have been reported. By the way, with the view of investigating the oceanic shallow water (Cheng et al, 2023;Shen et al, 2023aShen et al, , 2023bShen et al, , 2023cNOAA, 2023;Gao, , 2023bGao, , 2023cWu et al, 2023c;Zhou et al, 2023aZhou et al, , 2023bZhou et al, , 2023cFeng et al, 2023;Liu et al, 2019Liu et al, , 2021Zayed, 2014;Liu et al, 2018;Gao et al, 2023bGao et al, , 2023c, people have presented some other dispersive-type systems, e.g., a (2þ1)-dimensional generalized modified dispersive water-wave system modeling some nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth (Gao et al, 2023b), a variable-coefficient generalized dispersive water-wave system modeling certain long-weakly nonlinear and weakly dispersive surface waves of variable depth in the shallow water (Zayed, 2014;Liu et al, 2018) and a variable-coefficient dispersive-wave system modeling certain long gravity waves in a shallow ocean (Gao et al, 2023c).…”

“…By the way, with the view of investigating the oceanic shallow water (Cheng et al , 2023; Shen et al , 2023a, 2023b, 2023c; NOAA, 2023; Gao, 2023a, 2023b, 2023c; Wu et al ., 2023c; Zhou et al ., 2023a, 2023b, 2023c; Feng et al ., 2023; Liu et al ., 2019, 2021; Zayed, 2014; Liu et al , 2018; Gao et al , 2023b, 2023c), people have presented some other dispersive-type systems, e.g., a (2+1)-dimensional generalized modified dispersive water-wave system modeling some nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth (Gao et al , 2023b), a variable-coefficient generalized dispersive water-wave system modeling certain long-weakly nonlinear and weakly dispersive surface waves of variable depth in the shallow water (Zayed, 2014; Liu et al , 2018) and a variable-coefficient dispersive-wave system modeling certain long gravity waves in a shallow ocean (Gao et al , 2023c).…”

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:…”