Evolution of the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in fluid mechanics in three spatial dimensions can be described by a generalized ( 3 + 1 )-dimensional variable-coefficient Kadomtsev-Petviashvili equation, which is studied in this paper with symbolic computation. Via the truncated Painleve expansion, an auto-Backlund transformation is derived, based on which, under certain variable-coefficient constraints, one-soliton, two-soliton, homoclinic breather-wave and rouge-wave solutions are respectively obtained via the Hirota method. Graphic analysis shows that the soliton propagates with the varying soliton direction. Change of the value of any one of g ( t ) , m ( t ) , n ( t ) , h ( t ) , q ( t ) and l ( t ) in the equation can cause the change of the soliton shape, while the soliton amplitude cannot be affected by that change, where g ( t ) represents the dispersion, m ( t ) and n ( t ) respectively stand for the disturbed wave velocities along the y and z directions, h ( t ) , q ( t ) and l ( t ) are the perturbed effects, y and z are the scaled spatial coordinates, and t is the temporal coordinate. Soliton direction and type of the interaction between the two solitons can vary with the change of the value of g ( t ) , while they cannot be affected by m ( t ) , n ( t ) , h ( t ) , q ( t ) and l ( t ) . Homoclinic breather wave and rouge wave are respectively displayed, where the rouge wave comes from the extreme behaviour of the homoclinic breather wave.
['Jun Chai', 'Bo Tian', 'Wen-Rong Sun', 'Xi-Yang Xie']
Solitons and rouge waves for a generalized ( 3 + 1 )-dimensional variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics