Starting from the bilinear form of the (2+1)-dimensional Sawada–Kotera (SK) equation, the y-periodic soliton, the algebraic soliton and the solution to describe the interaction between the y-periodic soliton and the algebraic soliton are obtained. The behaviours of interaction between the y-periodic soliton and the algebraic soliton are analysed in detail. Some representational graphs are also given to make the problem much clearer.
A set of generalized symmetries with
arbitrary functions of t for the Konopelchenko–Dubrovsky (KD)
equation in 2 + 1 space dimensions is given by using a direct method
called formal function series method presented by Lou. These
symmetries constitute an infinite-dimensional generalized w∞ algebra.
Starting from the variable separation approach, the algebraic soliton solution and the solution describing the interaction between line soliton and algebraic soliton are obtained by selecting appropriate seed solution for (2+1)dimensional ANNV equation. The behaviors of interactions are discussed in detail both analytically and graphically. It is shown that there are two kinds of singular interactions between line soliton and algebraic soliton: 1) the resonant interaction where the algebraic soliton propagates together with the line soliton and persists infinitely; 2) the extremely repulsive interaction where the algebraic soliton affects the motion of the line soliton infinitely apart.
A two-dimensional monatomic lattice with nearest-neighbor interaction is investigated by the method of multiple scales combined with a quasidiscreteness approximation. The Davey-Stewartson II equation (DS-II) is obtained from the original two-dimensional (2D) differential-difference system. By solving the DS-II, explicit periodic solutions, soliton solutions and rational function solutions are obtained, and the leading order approximated solutions of the 2D monatomic lattice are constructed by explicit solutions of the DS-II.
The (3+1)-dimensional Jimbo–Miwa (JM)
equation is solved approximately by using the conformal invariant
asymptotic expansion approach presented by Ruan. By solving the new
(3+1)-dimensional integrable models, which are conformal invariant
and possess Painlevé property, the approximate solutions are
obtained for the JM equation, containing not only one-soliton
solutions but also periodic solutions and multi-soliton solutions.
Some approximate solutions happen to be exact and some approximate
solutions can become exact by choosing relations between the parameters properly.
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