Explicit Multi-soliton Solutions Research Articles

Solutions to Integrable Space-Time Shifted Nonlocal Equations

In this paper we present a reduction technique based on bilinearization and double Wronskians (or double Casoratians) to obtain explicit multi-soliton solutions for the integrable space-time shifted nonlocal equations introduced very recently by Ablowitz and Musslimani in [Phys. Lett. A, 2021]. Examples include the space-time shifted nonlocal nonlinear Schr\"odinger and modified Korteweg-de Vries hierarchies and the semi-discrete nonlinear Schr\"odinger equation. It is shown that these nonlocal integrable equations with or without space-time shift(s) reduction share same distributions of eigenvalues but the space-time shift(s) brings new constraints to phase terms in solutions.

Integrable nonlocal Hirota equations

We construct several new integrable systems corresponding to nonlocal versions of the Hirota equation, which is a particular example of higher order nonlinear Schrödinger equations. The integrability of the new models is established by providing their explicit forms of Lax pairs or zero curvature conditions. The two compatibility equations arising in this construction are found to be related to each other either by a parity transformation P, by a time reversal T or a P T-transformation possibly combined with a conjugation. We construct explicit multisoliton solutions for these models by employing Hirota’s direct method as well as Darboux-Crum transformations. The nonlocal nature of these models allows for a modification of these solution procedures as the new systems also possess new types of solutions with different parameter dependence and different qualitative behavior. The multisoliton solutions are of varied type, being, for instance, nonlocal in space, nonlocal in time of time-crystal type, regular with local structures either in time/space or of rogue wave type.

The canonical AB system: conservation laws and soliton solutions

Under investigation in this paper is the canonical AB system, which describes the marginally unstable baroclinic wave packets in the geophysical fluids and ultra-short pulses in nonlinear optics. Through symbolic computation, conservation laws are derived. Furthermore, by virtue of the Darboux transformation, the explicit multi-soliton solutions are generated. Figures are plotted to reveal the following dynamic features of the solitons: (1) Elastic interactions of two one-peak bright solitons, of two one-peak dark solitons and of two two-peak dark solitons; (2) Parallel propagations of two one-peak bright solitons, of two one-peak dark solitons and of two two-peak dark solitons; (3) Propagations of three types of bound solitons: periodic propagation of bound solitons taking on contrary trends, mutual attractions and repulsions of two bight bound solitons and of two dark bound solitons.

The Inverse Scattering Transform for the Defocusing Nonlinear Schrödinger Equations with Nonzero Boundary Conditions

A rigorous theory of the inverse scattering transform for the defocusing nonlinear Schrödinger equation with nonvanishing boundary values as is presented. The direct problem is shown to be well posed for potentials q such that , for which analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated and solved both via Marchenko integral equations, and as a Riemann‐Hilbert problem in terms of a suitable uniform variable. The asymptotic behavior of the scattering data is determined and shown to ensure the linear system solving the inverse problem is well defined. Finally, the triplet method is developed as a tool to obtain explicit multisoliton solutions by solving the Marchenko integral equation via separation of variables.

Integrability aspects and soliton solutions for an inhomogeneous nonlinear system with symbolic computation

Under investigation in this paper is an inhomogeneous nonlinear system, which describes the marginally unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics under inhomogeneous media. Through symbolic computation, the Painlevé integrable condition, Lax pair and conservation laws are derived for this system. Furthermore, by virtue of the Darboux transformation, the explicit multi-soliton solutions are generated. Figures are plotted to reveal the following dynamic features of the solitons: (1) Parallel propagation of solitons: separation distance of the two parallel solitons depends on the value of |Im(λ1)|-|Im(λ2)| (where λ1 and λ2 are the spectrum parameters); (2) Periodic propagation of bound solitons: periodic bound solitons taking on contrary trends, and mutual attractions and repulsions of two bright bound solitons; (3) Elastic interactions of two one-peak bright solitons and of two one-peak dark solitons.

Elastic–inelastic-interaction coexistence and double Wronskian solutions for the Whitham–Broer–Kaup shallow-water-wave model

By virtue of a variable transformation, Whitham–Broer–Kaup (WBK) model describing the propagation of the shallow water waves is transformed into the generalized Ablowitz–Kaup–Newell–Segur (AKNS) systems, the bilinear forms of which are derived with a rational transformation. Explicit multi-soliton solutions are obtained subsequently by means of the Wronskian technique and symbolic computation. Furthermore, interactions of the solitons are investigated graphically for the WBK model and a phenomenon is revealed that the elastic and inelastic interactions occur simultaneously without affecting each other, i.e., the elastic and fission interactions coexist at the same time, and they do not disturb mutually during the collision. Our results could be useful to explain certain physical phenomena in the shallow water models.

Conservation laws, soliton solutions and modulational instability for the higher-order dispersive nonlinear Schrödinger equation

In this paper, analytically investigated is a higher-order dispersive nonlinear Schrodinger equation. Based on the linear eigenvalue problem associated with this equation, the integrability is identified by admitting an infinite number of conservation laws. By using the Darboux transformation method, the explicit multi-soliton solutions are generated in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.

Vandermonde-like determinants’ representations of Darboux transformations and explicit solutions for the modified Kadomtsev–Petviashvili equation

Recently, the (2+1)-dimensional modified Kadomtsev–Petviashvili (mKP) equation was decomposed into two known (1+1)-dimensional soliton equations by Dai and Geng [H.H. Dai, X.G. Geng, J. Math. Phys. 41 (2000) 7501]. In the present paper, a systematic and simple method is proposed for constructing three kinds of explicit N-fold Darboux transformations and their Vandermonde-like determinants’ representations of the two known (1+1)-dimensional soliton equations based on their Lax pairs. As an application of the Darboux transformations, three explicit multi-soliton solutions of the two (1+1)-dimensional soliton equations are obtained; in particular six new explicit soliton solutions of the (2+1)-dimensional mKP equation are presented by using the decomposition. The explicit formulas of all the soliton solutions are also expressed by Vandermonde-like determinants which are remarkably compact and transparent.

REDUCTION PROBLEM FOR FOUR WAVE INTERACTION AND THE CLASSICAL r MATRIX

Integrable nonlinear partial differential equations, describing the process of four-wave interaction are derived by means of the reduction procedure. Explicit multisoliton solutions are constructed by formulating a degenerate Reimann–Hilbert problem; while the analytic structure of the Lax operators in the complex plane is used to deduce general Backlund transformations. Conservation laws are derived and the Hamiltonian structure is used to deduce a new form of the classical r matrix, which in turn is used to prove the integrability of our model in case of nontrivial boundary conditions. Lastly the corresponding Quantum R-matrix is derived in the lattice approximation and is seen to be related to an eight-vertex model.

SOLITON SOLUTIONS OF THE GENERALIZED NONLINEAR SCHR?DINGER EQUATION WITH HIGHER-ORDER CORRECTIONS BY THE DIRECT METHOD OF HIROTA

By using the direct method of Hirota, explicit multi-soliton solutions are found for the generalized nonlinear Schr?dinger equation with higherorder corrections which describes the ultra-short pulse propagation along monomode optical fibers.

N-soliton solutions for the sine-Hilbert equation

An exact method for constructing soliton solutions of the sine-Hilbert (sH) equation is developed. It is shown that the sH equation can be transformed into a system of nonlinear ordinary differential equations. By linearizing this system of equations, explicit multi-soliton solutions of the sH equation are presented.

A bogomolny bound for general relativity and solitons in N=2 supergravity

We derive a lower bound for the mass of a space-time, in terms of its electric and magnetic charges, which is the analogue of the Bogomolny bound for magnetic monopoles. We give an almost complete characterization of space-times for which the bound is attained, provide explicit multi-soliton solutions and describe how they fit into supermultiplets.

The Benjamin-Ono and related equations [1

Properties of the Benjamin-Ono equation are discussed. In particular a whole hierarchy of related equations are introduced. Explicit multi-soliton solutions are given both for the periodic and non-periodic cases.

Benjamin—Ono-related equations and their solutions

It is shown that the Benjamin-Ono equation is one of a hierarchy of nonlinear partial differential singular integral equations. Explicit multisoliton solutions are constructed for all equations of the hierarchy.