Narrow Results

Computers have a great potential in the analytical investigations on various physics problems. In this paper, we make use of computerized symbolic computation to obtain two similarity reductions as well as a rational solution for the variable-coefficient cylindrical Korteweg–de Vries equation, which was originally introduced in the studies of plasma physics. One of the reductions is to the second Painlevé equation, while the other to either the first Painlevé equation or the Weierstrass elliptic function equation. Our results are in agreement with the Painlevé conjecture.

In this paper, via the limit technique of long wave, the N-order rational solution of the (2+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation is derived from the N-soliton solution. In particular, the bright–dark M-lump solution can be obtained from resulting N-order rational solution. This kind of lump wave in BKP equation exhibits the one-peak-one-valley structure which is different from that in KPI equation. In addition, graphical illustration presents the collision dynamics of multi-bright–dark lump waves.

Two Sawada–Kotera-like equations are introduced by the generalized bilinear operators $D_p$ associated with two prime numbers $p=3$ and $p=5$, respectively. Rational solutions of the two presented Sawada–Kotera-like equations are generated by searching polynomial solutions of the corresponding two generalized bilinear equations.

In this paper, we investigate some symmetries and Lie-group transformations of an integrable system by using the symmetry analysis method. It follows that the resulting similarity solutions are obtained by applying the characteristic equations of the symmetries. By applying the software Maple,we work out some exact solutions of the generalized KdV system, including the rational solutions, the periodic solutions, the dark soliton solutions, and so on. Finally, we make use of the self-adjoint operators to investigate the nonlinear self-adjointness and the conservation laws of the generalized KdV integrable system.

In this paper, a variable-coefficient KdV equation in a fluid, plasma, anharmonic crystal, blood vessel, circulatory system, shallow-water tunnel, lake or relaxation inhomogeneous medium is discussed. We construct the reduction from the original equation to another variable-coefficient KdV equation, and then get the rational, periodic and mixed solutions of the original equation under certain constraint. For the original equation, we obtain that (i) the dispersive coefficient affects the solitonic background, velocity and amplitude; (ii) the perturbed coefficient affects the solitonic velocity, amplitude and background; (iii) the dissipative coefficient affects the solitonic background, and there are different mixed solutions under the same constraint with the dispersive, perturbed and dissipative coefficients changing.

This paper continues the investigation of the theory of rational solutions of the CYBE for o(n) from the point of view of orders in the corresponding loop algebra, as it was developed in [8]. As suggested by [8], in the case of "singular vertices", we use the list of connected irreducible subgroups of SO(n) locally transitive on the Grassmann manifolds of isotropic k-dimensional subspaces in ℂⁿ obtained in [11]. New arguments based on the analysis of the structure of the stationary subalgebra of a generic point allow us to construct several rational solutions in o(7), o(8) and o(12).