We compute
N
-soliton solutions and analyze the Hirota
N
-soliton conditions, in (2+1)-dimensions, based on the Hirota bilinear formulation. An algorithm to check the Hirota conditions is proposed by ...comparing degrees of the polynomials generated from the Hirota function in
N
wave vectors. A weight number is introduced while transforming the Hirota function to achieve homogeneity of the resulting polynomial. Applications to three integrable equations: the (2+1)-dimensional KdV equation, the Kadomtsev–Petviashvili equation, the (2+1)-dimensional Hirota–Satsuma–Ito equation, are made, thereby providing proofs of the existence of
N
-soliton solutions in the three model equations.
Lump solutions are analytical rational function solutions localized in all directions in space. We analyze a class of lump solutions, generated from quadratic functions, to nonlinear partial ...differential equations. The basis of success is the Hirota bilinear formulation and the primary object is the class of positive multivariate quadratic functions. A complete determination of quadratic functions positive in space and time is given, and positive quadratic functions are characterized as sums of squares of linear functions. Necessary and sufficient conditions for positive quadratic functions to solve Hirota bilinear equations are presented, and such polynomial solutions yield lump solutions to nonlinear partial differential equations under the dependent variable transformations u=2(lnf)x and u=2(lnf)xx, where x is one spatial variable. Applications are made for a few generalized KP and BKP equations.
A class of Riemann–Hilbert problems on the real axis is formulated for solving the multicomponent AKNS integrable hierarchies associated with a kind of bock matrix spectral problems. Through special ...Riemann–Hilbert problems where a jump matrix is taken to be the identity matrix, soliton solutions to all integrable equations in each hierarchy are explicitly computed. A class of specific reductions of the presented integrable hierarchies is also made, together with its reduced Lax pairs and soliton solutions.
A class of higher-order matrix spectral problems is formulated and the associated integrable hierarchies are generated via the zero-curvature formulation. The trace identity is used to furnish ...Hamiltonian structures and thus explore the Liouville integrability of the obtained hierarchies. Illuminating examples are given in terms of coupled nonlinear Schrödinger equations and coupled modified Korteweg–de Vries equations with four components.
With symbolic computation, two classes of lump solutions to the dimensionally reduced equations in (2+1)-dimensions are derived, respectively, by searching for positive quadratic function solutions ...to the associated bilinear equations. To guarantee analyticity and rational localization of the lumps, two sets of sufficient and necessary conditions are presented on the parameters involved in the solutions. Localized characteristics and energy distribution of the lump solutions are also analyzed and illustrated.
The aim of the paper is to construct nonlocal reverse‐space nonlinear Schrödinger (NLS) hierarchies through nonlocal group reductions of eigenvalue problems and generate their inverse scattering ...transforms and soliton solutions. The inverse scattering problems are formulated by Riemann‐Hilbert problems which determine generalized matrix Jost eigenfunctions. The Sokhotski‐Plemelj formula is used to transform the Riemann‐Hilbert problems into Gelfand‐Levitan‐Marchenko type integral equations. A solution formulation to special Riemann‐Hilbert problems with the identity jump matrix, corresponding to the reflectionless transforms, is presented and applied to N‐soliton solutions of the nonlocal NLS hierarchies.
This paper aims to explore a kind of lump solutions in nonlinear dispersive waves with higher-order rational dispersion relations. We show that the second member in the commuting ...Kadomtsev–Petviashvili hierarchy is such an example, and construct its lump solutions, based on a Hirota trilinear form. The presented lump solutions have one peak and two valleys, where the global maximum and minimum values are achieved. A few three-dimensional plots and contour plots are made for a specific example of the lumps.
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