Ansatz Research Articles

New solitary wave solutions of the Sawada-Kotera equation and its bidirectional form

In this paper, the solitary wave ansatz method along with Hirota’s bilinear form are used to study the well-known Sawada-Kotera equation and its bidirectional form. Numerical simulations related to some of these solutions are given in the article. These figures provide us with a better understanding of visualizing the dynamic behavior of solutions. For the latter equation, we have found an indefinite quadratic bilinear Hirota solution with three homogenous terms and a constant. The techniques used in this paper can be adopted to solve other nonlinear models.

New exact solutions and numerical approximations of the generalized KdV equation

This paper is devoted to create new exact and numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with ansatz method and Galerkin finite element method based on cubic B splines over finite elements. Propagation of a single solitary wave is investigated to show the efficiency and applicability of the proposed methods. The performance of the numerical algorithm is proved by computing L2 and L∞ error norms. Also, three invariants I1, I2, and I3 have been calculated to determine the conservation properties of the presented algorithm. The obtained numerical solutions are compared with some earlier studies for similar parameters. This comparison clearly shows that the obtained results are better than some earlier results and they are found to be in good agreement with exact solutions. Additionally, a linear stability analysis based on Von Neumann's theory is surveyed and indicated that our method is unconditionally stable.

Solution of Radial Schrödinger Equation with Yukawa Potential Using Bethe Ansatz Method

Solution of Radial Schrödinger Equation with Yukawa Potential Using Bethe Ansatz Method

Exact solutions of the space time-fractional Klein-Gordon equation with cubic nonlinearities using some methods

Recently, finding exact solutions of nonlinear fractional differential equations has attracted great interest. In this work, the space time-fractional Klein-Gordon equation with cubic nonlinearities is examined. Firstly, suitable exact soliton solutions are formally extacted by using the solitary wave ansatz method. Some solutions are also illustrated by the computer simulations. Besides, the modified Kudryashov method is used to construct exact solutions of this equation.

Twisted rational r-matrices and algebraic Bethe ansatz: Application to generalized Gaudin and Richardson models

In the present paper we develop the algebraic Bethe ansatz approach to the case of non-skew-symmetric gl(2)⊗gl(2)-valued Cartan-non-invariant classical r-matrices with spectral parameters. We consider the two families of these r-matrices, namely, the two non-standard rational r-matrices twisted with the help of second order automorphisms and realize the algebraic Bethe ansatz method for them. We study physically important examples of the Gaudin-type and BCS-type systems associated with these r-matrices and obtain explicitly the Bethe vectors and the spectrum for the corresponding quantum hamiltonians in terms of solutions of Bethe equations.

Novel analytical cnoidal and solitary wave solutions of the Extended Kawahara equation

In this work, some novel analytic traveling wave solutions including the cnoidal and solitary wave solutions of the planar Extended Kawahara equation are deduced. Four different analytical methods (the Jacobian elliptic function, Weierrtrass elliptic function, the traditional tanh method and the sech-square) are devoted for solving this equation. By means of the Jacobian elliptic function ansatz, the cnoidal and soliary wave solutions are obtained. Also, new cnoidal wave solutions are derived via a new hypothesis in the form of the Weierrtrass elliptic function. Moreover, the standard tanh method is utilized to get a new set of solitary wave solutions for the evolution equation. Over and above, the hyperbolic ansatz method (a new ansatz in the form of squre-sech) is employed to get a new set of solitary wave solutions for the evolution equation. Furthermore, all obtained solutions of the planar Extended Kawahara equation cover the traveling wave solutions of the planar modified Kawahara equation. These solutions maybe useful to many researchers interested in studying the propagation of nonlinear waves in nonlinear dispersion mediums like plasma physics, optical fibers, fluid mechanics, and many different branches of science.

Solving the Liouvillian Gap with Artificial Neural Networks.

We propose a machine-learning inspired variational method to obtain the Liouvillian gap, which plays a crucial role in characterizing the relaxation time and dissipative phase transitions of open quantum systems. By using "spin bi-base mapping," we map the density matrix to a pure restricted-Boltzmann-machine (RBM) state and transform the Liouvillian superoperator to a rank-two non-Hermitian operator. The Liouvillian gap can be obtained by a variational real-time evolution algorithm under this non-Hermitian operator. We apply our method to the dissipative Heisenberg model in both one and two dimensions. For the isotropic case, we find that the Liouvillian gap can be analytically obtained and in one dimension even the whole Liouvillian spectrum can be exactly solved using the Bethe ansatz method. By comparing our numerical results with their analytical counterparts, we show that the Liouvillian gap could be accessed by the RBM approach efficiently to a desirable accuracy, regardless of the dimensionality and entanglement properties.

Dispersion and phase managed optical soliton solutions of a nonautonomous (3+1)-dimensional coupled nonlinear Schrödinger equation

The complex amplitude ansatz method and He's semi-inverse method have been used to investigate the optical soliton solutions of a nonautonomous (3+1)-dimensional coupled non-linear Schrödinger (NLS) equation. The effect of the time-dependent dispersion (TD) and the phase modulation (PM) coefficients in the equation has been analyzed. The intensity profiles of the soliton solutions have been obtained by assigning different test functions to the coefficients and distinct values to the parameters, which help to understand the physical characteristics of the solutions. The intensities of two circularly polarized waves have bright, dark, parabolic, dark-bright, parabolic-bright, bright-singular, homoclinic and breather solitons that depend significantly on the dispersion and phase modulation. Some of the obtained soliton solutions are novel and did not exist in the earlier studies. It is found that the optical soliton velocity and frequency are dependent on wave numbers and dispersion of the medium but are free from the phase modulation. Moreover, the amplitude of the solitons are managed by dispersion and phase modulation both but the soliton width remains constant. The results in this study may be used for experimental investigation of transmission of dispersion and phase managed optical soliton in non-linear optical fiber.

New vision for the soliton solutions to the complex Hirota-dynamical model

In this paper, the nonlinear complex Hirota-dynamical model NLCHM in which the third derivative term represents the self-interaction in the high-frequency subsystem is established. This model plays a vital role in plasma physics because there are agreements between the self-interaction in the high-frequency and the well- known self-focusing effect in plasma. Many soliton solutions to this equation model have been achieved perfectly using the solitary wave ansatz method (SWAM). Furthermore, in the same vein and related subject the extended simple equation method (ESEM) has been applied perfectly to achieve new perception of soliton solutions to this model. A good isomorphic between the achieved results and that achieved previous by other authors have been listed.

A New Derivation of Exact Solutions for Incompressible Magnetohydrodynamic Plasma Turbulence

The objective of this paper is to study the propagation of nonlinear, quasi-parallel, magnetohydrodynamic waves of small-amplitude in a cold Hall plasma of constant density. Magnetohydrodynamic equations, along with the cold plasma were expanded using the reductive perturbation method, which leads to a nonlinear partial differential equation that complies with a modified form of the derivative nonlinear evolution Schrödinger equation. Exact solutions of the turbulent magnetohydrodynamic model in plasma were formulated for the physical quantities that describe the problem completely by using the complex ansatz method. In addition, the complete set of equations was used taking into account the magnetic field, which can be considered to be constant in the x-direction to create stable vortex waves. Vortex solutions of the modified nonlinear Schrödinger equation (MNLS) were compared with the solutions of incompressible magnetohydrodynamic equations. The obtained equations differ from the standard NLS equation by one additional term that describes the interaction of the nonlinear waves with the constant density. The behavior of both the velocity profile and the waveform of pressure were studied. The results showed that there are clear disturbances in the identity of the velocity and thus pressure. The identity of both velocity and pressure results from that a magnetic field is formed.

Exact solutions of the Cn quantum spin chain

We study the exact solutions of quantum integrable model associated with the Cn Lie algebra, with either a periodic or an open one with off-diagonal boundary reflections, by generalizing the nested off-diagonal Bethe ansatz method. Taking the C3 as an example we demonstrate how the generalized method works. We give the fusion structures of the model and provide a way to close fusion processes. Based on the resulted operator product identities among fused transfer matrices and some necessary additional constraints such as asymptotic behaviors and relations at some special points, we obtain the eigenvalues of transfer matrices and parameterize them as homogeneous T−Q relations in the periodic case or inhomogeneous ones in the open case. We also give the exact solutions of the Cn model with an off-diagonal open boundary condition. The method and results in this paper can be generalized to other high rank integrable models associated with other Lie algebras.

Stability analysis, solitary wave and explicit power series solutions of a (2 + 1)-dimensional nonlinear Schrödinger equation in a multicomponent plasma

PurposeThe purpose of this paper is to study the stability analysis and optical solitary wave solutions of a (2 + 1)-dimensional nonlinear Schrödinger equation, which are derived from a multicomponent plasma with nonextensive distribution.Design Methodology ApproachBased on the ansatz and sub-equation theories, the authors use a direct method to find stability analysis and optical solitary wave solutions of the (2 + 1)-dimensional equation.FindingsBy considering the ansatz method, the authors successfully construct the bright and dark soliton solutions of the equation. The sub-equation method is also extended to find its complexitons solutions. Moreover, the explicit power series solution is also derived with its convergence analysis. Finally, the influences of each parameter on these solutions are discussed via graphical analysis.Originality ValueThe dynamics of these solutions are analyzed to enrich the diversity of the dynamics of high-dimensional nonlinear Schrödinger equation type nonlinear wave fields.

The action of the Virasoro algebra in quantum spin chains. Part I. The non-rational case

We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra (“Koo-Saleur generators” [1]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [2] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules Wj,1 for j ≠ 0 contains pairs of “conjugate states” with conformal weights (hr,s, hr,−s) and (hr,−s, hr,s) that give rise to dual structures: Verma or co-Verma modules. The limit of {W}_{0,{mathfrak{q}}^{pm 2}} contains diagonal fields (hr,1, hr,1) and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in {mathfrak{q}}^{pm 2} . In order to obtain matrix elements of Koo-Saleur generators at large system size N we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.

Dark-managed solitons in inhomogeneous cubic–quintic–septimal nonlinear media

We investigate the inhomogeneous higher-order nonlinear Schrodinger (INHLS) equation including cubic–quintic–septic (CQS) nonlinear terms and gain or loss with variable coefficients. The exact analytic solution that describes dark soliton-type pulse propagation is found for the model by employing the ansatz method. Unlike the traditional $$\text {tanh}$$ dark soliton in Kerr-type media, the functional form of this novel dark-managed soliton structure takes a $$\text {sech}^{2/3}$$ profile. Parametric conditions are presented in which these optical solitons exist. We also investigated the stability of these dark-managed solitons under some initial perturbations by employing the numerical simulation methods. Finally, the interaction dynamics of two and three dark-managed solitons has been numerically explored.

A variety of solitons to the sixth-order dispersive [formula omitted]-dimensional nonlinear time-fractional Schrödinger equation with cubic-quintic-septic nonlinearities

This study examines the exact soliton solutions under the effect of cubic-quintic-septic nonlinearities for a sixth-order (3+1)-dimensional nonlinear time-fractional Schrödinger equation with fourth-order and sixth-order dispersive terms. We apply the generalized Riccati equation mapping method and the modified Kudryashov method to obtain the dark soliton, singular, periodic, and other soliton-type solutions. To determine the bright solition, other singular and periodic solutions, ansatz method is employed. The results obtained can help to understand the dynamics of waves in a cubic-quintic-septic nonlinear material.

Transmission stability of chirped dark vector quasi-solitons in birefringent fiber system with nonlinear gain

In this article, we consider the coupled Ginzburg–Landau equation with variable coefficients including the nonlinear gain and obtain the exact solutions of chirped dark vector quasi-solitons via the ansatz method. Next, the propagation of chirped dark vector quasi-solitons is discussed to verify whether they can be transmitted stably in the birefringent optical fiber system. The numerical simulation shows that this can be achieved. We deeply add the small perturbation to the transmission of dark vector quasi-solitons to make the results above more general. The results further prove the correctness of our solutions.

Algebraic Bethe ansatz for $$\mathfrak o_{2n+1}$$-invariant integrable models

A class of $\mathfrak{o}_{2n+1}$-invariant quantum integrable models is investigated in the framework of algebraic Bethe ansatz method. A construction of the $\mathfrak{o}_{2n+1}$-invariant Bethe vector is proposed in terms of the Drinfeld currents for the double of Yangian $\mathcal{D}Y(\mathfrak{o}_{2n + 1})$. Action of the monodromy matrix entries onto off-shell Bethe vectors for these models is calculated. Recursion relations for these vectors were obtained. The action formulas can be used to investigate structure of the scalar products of Bethe vectors in $\mathfrak{o}_{2n+1}$-invariant models.

Exact solution of the Gaudin model with Dzyaloshinsky–Moriya and Kaplan–Shekhtman–Entin–Wohlman–Aharony interactions* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11847245 and 11874393).

We study the exact solution of the Gaudin model with Dzyaloshinsky–Moriya and Kaplan–Shekhtman–Entin–Wohlman–Aharony interactions. The energy and Bethe ansatz equations of the Gaudin model can be obtained via the off-diagonal Bethe ansatz method. Based on the off-diagonal Bethe ansatz solutions, we construct the Bethe states of the inhomogeneous XXX Heisenberg spin chain with the generic open boundaries. By taking a quasi-classical limit, we give explicit closed-form expression of the Bethe states of the Gaudin model. From the numerical simulations for the small-size system, it is shown that some Bethe roots go to infinity when the Gaudin model recovers the U(1) symmetry. Furthermore, it is found that the contribution of those Bethe roots to the Bethe states is a nonzero constant. This fact enables us to recover the Bethe states of the Gaudin model with the U(1) symmetry. These results provide a basis for the further study of the thermodynamic limit, correlation functions, and quantum dynamics of the Gaudin model.

Lump-type solutions, interaction solutions, and periodic lump solutions of the generalized (3+1)-dimensional Burgers equation

In this paper, the lump-type solutions, interaction solutions, and periodic lump solutions of the generalized ([Formula: see text])-dimensional Burgers equation were obtained by using the ansatz method. Based on a variable transformation, the generalized ([Formula: see text])-dimensional Burgers equation was transformed into a bilinear equation. And then, lump-type solutions, two kinds of interaction solutions, and periodic lump solutions were obtained via solutions of the bilinear equation. Fission and fusion phenomena are found in the process of interaction between lump-type soliton and one stripe soliton, which can derive the lumpoff wave solution. The dynamic characteristics of these solutions were vividly displayed by graphics.

Interactions solutions of various-type rogue with multi-stripe solitons and breather lump for the (2+1)-dimensional Maccari’s system

In this work, we consider the (2[Formula: see text]+[Formula: see text]1)-dimensional Maccari’s system, which is frequently introduced to describe the motion of the solitary waves. Abundant lump, line rogue wave, and dipole-type rogue wave are constructed by taking the ansatz method. Furthermore, the mixed interaction solutions between lump and multi-stripe solitons (such as fission and fusion) are obtained in combining rational function with exponential function. In particular, an interesting result is obtained: a rogue wave is excited from multi-stripe solitons.