Abstract
General soliton and (semi-)rational solutions to the y-non-local Mel’nikov equation with non-zero boundary conditions are derived by the Kadomtsev–Petviashvili (KP) hierarchy reduction method. The solutions are expressed in N × N Gram-type determinants with an arbitrary positive integer N. A possible new feature of our results compared to previous studies of non-local equations using the KP reduction method is that there are two families of constraints among the parameters appearing in the solutions, which display significant discrepancies. For even N, one of them only generates pairs of solitons or lumps while the other one can give rise to odd numbers of solitons or lumps; the interactions between lumps and solitons are always inelastic for one family whereas the other family may lead to semi-rational solutions with elastic collisions between lumps and solitons. These differences are illustrated by a thorough study of the solution dynamics for N = 1, 2, 3. Besides, regularities of solutions are discussed under proper choices of parameters.
Highlights
In the past two decades, the studies on parity-time (PT )-symmetric systems have grown significantly
It is noted that for even N = 2J, in Case I, solitons or lumps always appear in pairs, whereas Case II can give rise to odd numbers of solitons or lumps
For the dynamics of semi-rational solutions with N = 2J, the collisions between soliton and lump in Case I are always inelastic, while elastic collisions between them may appear for Case II
Summary
In the past two decades, the studies on parity-time (PT )-symmetric systems have grown significantly. The collisions between lumps and solitons that correspond to semi-rational solutions of the non-local DS I equation are inelastic. Studies of the partial reverse space-time (x,t)- 3 non-local Mel’nikov equation have been carried out in [40,41], where solutions containing even numbers of solitons or lumps were derived. 2. General soliton and (semi-)rational solutions of the y-non-local Mel’nikov equation. Take the variable transformations x1 1⁄4 x, x2 1⁄4 Àyi, x3 1⁄4 À4t, xÀ1 1⁄4 kt: With these conditions, it will be shown that the functions in (2.1) satisfy equation (1.3) in theorem 2.1 as long as proper parameter constraints are imposed.
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