Multiple rogue wave solutions and the linear superposition principle for a (3 + 1)‐dimensional Kadomtsev–Petviashvili–Boussinesq‐like equation arising in energy distributions

Abstract

The multiple exp-function method and multiple rogue wave solutions method are employed for searching the multiple soliton solutions to the (3 + 1)-dimensional Kadomtsev-Petviashvili-Boussinesq-like (KP-Boussinesq-like) equation. The obtained solutions contain the first-order, second-order, third-order, and fourth-order wave solutions. At the critical point, the second-order derivative and Hessian matrix for only one point are investigated, and the lump solution has one maximum value. Moreover, we employ the linear superposition principle to determine N-soliton wave solutions of the KP-Boussinesq-like equation. The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values.

Localized nonlinear waves on spatio-temporally controllable backgrounds for a (3+1)-dimensional Kadomtsev-Petviashvili-Boussinesq model in water waves

“…The previously reported solutions of (3+1)D KPB model in Refs. [46][47][48][49][50][51][52][53][54], especially the soliton solutions [46] and rogue waves [48,54] obtained through Hirota bilinear formalism does not have any varying background. However, such controllable background in the present work offered much freedom to manipulate the nonlinear waves accordingly and displayed diverse wave phenomena.…”

“…Also, the Painlevé integrability analysis [51], localized wave solutions using bilinear form [52] and the symmetry reductions along with conserved quantities are obtained [53]. Recently, Manafian has reported multi-rogue wave solutions using generalized polynomials and reduced bilinear form along with kink-soliton solutions using multiple exp-function method for the present KPB model [54]. For further detailed analysis and discussion, one can refer to these reports and references therein.…”

“…For further detailed analysis and discussion, one can refer to these reports and references therein. Note that the solutions reported in the above-mentioned works [46][47][48][49][50][51][52][53][54], especially the soliton and rogue wave solutions reported in [46,48,54], are on constant (zero/non-zero) background, while the solutions we intend to discuss here is on variable backgrounds. Considering the limited works for localized waves on variable backgrounds in higher-dimensional models, especially no such report is available for the present KPB model (1), we proceed to pursue our interest to construct localized nonlinear wave solutions on spatio-temporally controllable backgrounds and study their dynamics in detail.…”

Rational and semi-rational solution to the (3+1)-dimensional Kadomtsev-Petviashvili-Boussinesq-like equation

“…For equation (1), Yu et al have constructed a direct bilinear Bäcklund transformation [34]; Lu et al got diversity of interaction solutions by bilinear form [35]; Manafian obtained multiple rogue wave solutions for this equation and the linear superposition principle [36]; Liu et al researched the interaction phenomenon between lump solitons [37]; Han et al have described the three-wave solutions and periodic wave [38].…”

Reductions and Conservation Laws of a Generalized Third-Order PDE via Multi-Reduction Method

“…Nevertheless, several differential equations (DEs) with interesting physical and mathematical applications are not variational systems. This has motivated a great number of works over the last decades devoted to generalize Noether's theorem to non-variational DEs [5][6][7][8][9][10][11][12][13].…”