Analytic multi-Baryonic solutions in the SU(N)-Skyrme model at finite density

We construct explicit analytic solutions of the SU(N)-Skyrme model (for generic N) suitable to describe different phases of nuclear pasta at finite volume in (3 + 1) dimensions. The first type are crystals of Baryonic tubes (nuclear spaghetti) while the second type are smooth Baryonic layers (nuclear lasagna). Both, the ansatz for the spaghetti and the ansatz for the lasagna phases, reduce the complete set of Skyrme field equations to just one integrable equation for the profile within sectors of arbitrary high topological charge. We compute explicitly the total energy of both configurations in terms of the flavor number, the density and the Baryonic charge. Remarkably, our analytic results allow to compare explicitly the physical properties of nuclear spaghetti and lasagna phases. Our construction shows explicitly that, at lower densities, configurations with N = 2 light flavors are favored while, at higher densities, configurations with N = 3 are favored. Our construction also proves that in the high density regime (but still well within the range of validity of the Skyrme model) the lasagna configurations are favored while at low density the spaghetti configurations are favored. Moreover, the integrability property of the present configurations is not spoiled by the inclusion of the subleading corrections to the Skyrme model arising in the ’t Hooft expansion. Finally, we briefly discuss the large N limit of our configurations.

Riemann-Hilbert Approach of The Complex Sharma-Tasso-Olver Equation and Its N-Soliton Solutions

In this paper, we use Riemann-Hilbert method to study the N-soliton solutions of the complex Sharma-Tasso-Olver(cSTO) equation. And then, based on analyzing the spectral problem of the Lax pair, the matrix Riemann-Hilbert problem for this integrable equation can be constructed, the N-soliton solutions about this system are given explicitly under the relationship of scattering matrix. At last, under the condition that some specifific parameter values are given, the three-dimensional diagram of the 2-soliton solution and the trajectory of the soliton solution will be simulated.

New (3+1)-dimensional integrable fourth-order nonlinear equation: lumps and multiple soliton solutions

Purpose This paper aims to introduce a new (3 + 1)-dimensional fourth-order integrable equation characterized by second-order derivative in time t. The new equation models both right- and left-going waves in a like manner to the Boussinesq equation. Design/methodology/approach This formally uses the simplified Hirota’s method and lump schemes for determining multiple soliton solutions and lump solutions, which are rationally localized in all directions in space. Findings This paper confirms the complete integrability of the newly developed (3 + 1)-dimensional model in the Painevé sense. Research limitations/implications This paper addresses the integrability features of this model via using the Painlevé analysis. Practical implications This paper presents a variety of lump solutions via using a variety of numerical values of the included parameters. Social implications This work formally furnishes useful algorithms for extending integrable equations and for the determination of lump solutions. Originality/value To the best of the author’s knowledge, this paper introduces an original work with newly developed integrable equation and shows useful findings of solitons and lump solutions.

The N-soliton solutions to the Hirota and Maxwell–Bloch equation via the Riemann–Hilbert approach

In this paper, we study the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation with physical meaning. From the Lax pair and Volterra integral equations, the Riemann–Hilbert problem of this integrable equation is constructed. By solving the matrix Riemann–Hilbert problem with the condition of no reflecting, the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation are obtained explicitly. Finally, we simulate the three-dimensional diagram of [Formula: see text] with 2-soliton solutions and the motion trajectory of [Formula: see text]-axis in the case of different [Formula: see text].

Conservation laws, classical symmetries and exact solutions of a (1 + 1)-dimensional fifth-order integrable equation

In this paper, we present a study of a fifth-order nonlinear partial differential equation, which was recently introduced in the literature. This equation can be used as a model for bidirectional water waves propagating in a shallow medium. Using elements of an optimal system of one-dimensional subalgebras, we perform similarity reductions culminating in analytic solutions. Rational, hyperbolic, power series and elliptic solutions are obtained. Furthermore, by using the multiple exponential function method we obtain one and two soliton solutions. Finally, local and low-order conserved quantities are derived by enlisting the multiplier approach.

On solitons: Propagation of shallow water waves for the fifth-order KdV hierarchy integrable equation

This article studies the fifth-order KdV (5KdV) hierarchy integrable equation, which arises naturally in the modeling of numerous wave phenomena such as the propagation of shallow water waves over a flat surface, gravity–capillary waves, and magneto-sound propagation in plasma. Two innovative integration norms, namely, the G ′ G 2 \left(\frac{{G}^{^{\prime} }}{{G}^{2}}\right) -expansion and ansatz approaches, are used to secure the exact soliton solutions of the 5KdV type equations in the shapes of hyperbolic, singular, singular periodic, shock, shock-singular, solitary wave, and rational solutions. The constraint conditions of the achieved solutions are also presented. Besides, by selecting appropriate criteria, the actual portrayal of certain obtained results is sorted out graphically in three-dimensional, two-dimensional, and contour graphs. The results suggest that the procedures used are concise, direct, and efficient, and that they can be applied to more complex nonlinear phenomena.

Stable novel and accurate solitary wave solutions of an integrable equation: Qiao model

This article investigates the dynamical and physical behavior of the second positive member in a new, utterly integrable hierarchy. This investigation depends on constructing novel analytical and approximate solutions to the Qiao model. The model’s name is after the researcher who derived the mathematical formula of it in 2007. This model possesses a Lax representation and bi-Hamiltonian structure. This study employs the unified and variational iteration (VI) method to obtain analytical and numerical solutions to the considered model. The obtained analytical solutions are used to calculate the necessary conditions for applying the suggested numerical method that makes checking the obtained solutions’ accuracy a valuable option. The obtained solutions are sketched in different techniques to explain more physical and dynamics details of the Qiao model and show the matching between obtained solutions.

An optimal system of group-invariant solutions and conserved quantities of a nonlinear fifth-order integrable equation

In this work, we perform Lie group analysis on a fifth-order integrable nonlinear partial differential equation, which was recently introduced in the literature and contains two dispersive terms. We determine a one-parameter group of transformations, an optimal system of group invariant solutions, and derive the corresponding analytic solutions. Topological kink, periodic and power series solutions are obtained. The existence of a variational principle for the underlying equation is proven using Helmholtz conditions and, thereafter, both local and nonlocal conserved quantities are obtained by utilising Noether’s theorem and a homotopy integral approach.