Through symbolic computation with Maple, a class of lump solutions, rationally localized in all directions in the space, to the (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation is presented, ...making use of its Hirota bilinear form. The resulting lump solutions contain six free parameters, two of which are due to the translation invariance of the KP equation and the other four of which satisfy a non-zero determinant condition guaranteeing analyticity and rational localization of the solutions. Three contour plots with different determinant values are sequentially made to show that the corresponding lump solution tends to zero when the determinant approaches zero. Two particular lump solutions with specific values of the involved parameters are plotted, as illustrative examples.
•Positive quadratic function solutions.•Solitons rationally-localized in all directions in the space.•Solving systems of nonlinear algebraic equations by symbolic computation with Maple.
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.
The paper aims to present the inverse scattering transforms and soliton solutions for nonlocal reverse-time nonlinear Schrödinger equations. The inverse scattering problems are formulated via ...Riemann–Hilbert problems, and their solutions are determined by the Sokhotski–Plemelj formula, which close the systems for the Jost solutions. Soliton solutions, corresponding to the reflectionless transforms, are generated from zeros and kernel vectors of the Riemann–Hilbert problems with the identity jump matrix.
We would like to propose a kind of nonlocal real reverse-spacetime integrable hierarchies of PT-symmetric matrix AKNS equations through nonlocal symmetry reductions on the potential matrix, and ...formulate their associated Riemann–Hilbert problems to determine generalized Jost solutions of arbitrary-order matrix spectral problems. The Sokhotski–Plemelj formula is applied in transforming the associated Riemann–Hilbert problems into Gelfand–Levitan–Marchenko type integral equations. The Riemann–Hilbert problems corresponding to the reflectionless case are solved explicitly, where eigenvalues could equal adjoint eigenvalues, and thus, soliton solutions are presented for the resulting nonlocal real reverse-spacetime integrable PT-symmetric matrix AKNS equations.
•Matrix integrable hierarchies.•Matrix nonlocal integrable equations.•Innovative group reductions, which keep the existence of infinitely many symmetries and conservation laws.
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 3 × 3 matrix spectral problem is introduced and its associated AKNS integrable hierarchy with four components is generated. From this spectral problem, a kind of Riemann–Hilbert problems is ...formulated for a system of coupled mKdV equations in the resulting AKNS integrable hierarchy. N-soliton solutions to the coupled mKdV system are presented through a specific Riemann–Hilbert problem with an identity jump matrix.
We consider a linear combination of the potential KP equation and the BKP equation, and call it a combined pKP–BKP equation. We prove that the combined pKP–BKP equation satisfies the Hirota N-soliton ...condition and thus it possesses an N-soliton solution. The proof is an application of an algorithm to compare degrees of homogeneous polynomials associated with the Hirota function in N wave vectors.
•A new integrable PDE model of sixth-order.•An innovative algorithm to verify the Hirota N-soliton condition.•Existence of N-solion solutions under general dispersion relations.
The paper aims to explore the existence of diverse lump and interaction solutions to linear partial differential equations in (3+1)-dimensions. The remarkable richness of exact solutions to a class ...of linear partial differential equations in (3+1)-dimensions will be exhibited through Maple symbolic computations, which yields exact lump, lump-periodic and lump–soliton solutions. The results expand the understanding of lump, freak wave and breather solutions and their interaction solutions in soliton theory.
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