Painlevé analysis, restricted bright-dark N-solitons, and N-rogue waves of a (4+1)-dimensional variable-coefficient generalized KP equation in nonlinear sciences

This research work studies the complete integrability, bright-dark solitons, and rogue waves of a recently formed variable coefficient generalized (4+1)-dimensional Kadomtsev-Petviashvili equation. It analyses the integrability of the investigated generalized equation by applying the Painlevé test with arbitrary choices and fulfilling the condition for compatibility for the resonances. It generates the bilinear equation with the Cole-Hopf transformation in the auxiliary function and, by using the bilinear differential operator, constructs Hirota’s bilinear form of this equation. Utilizing Hirota’s bilinear technique for N-soliton solutions, we obtain soliton solutions and their X-type and Y-type interactions for 1-, 2-, and 3-soliton solutions under the obtained restrictions and showcase their analytic dynamics. Also, it obtains the N-rogue wave solutions up to second order with center-controlled parameters with appropriate parameters and the variable coefficients and displays the dynamical structures. We form the bright-dark solitons and rogue waves with appropriate choices of parameters in the third and second order, respectively. By applying the computer algebra system software Mathematica, it displays the dynamical structures for the generated solutions with several chosen parameter values. Solitons appear in different fields of nonlinear sciences, such as fluid mechanics, nonlinear optics, oceanography, plasma physics, water waves, and other sciences.