Narrow Results

This study seeks to explore the integrable dynamics of induced curves through the utilization of the complex-coupled Kuralay system. The importance of the coupled complex Kuralay equation lies in its role as a fundamental model that contributes to the understanding of intricate physical and mathematical concepts, making it a valuable tool in scientific research and applications. The soliton solutions originating from the Kuralay equations are believed to encapsulate cutting-edge research in various important domains such as optical fibers, nonlinear optics, and ferromagnetic materials. Analytical techniques are employed to derive traveling wave solutions for this model, given that the Cauchy problem cannot be resolved using the inverse scattering transform. In the quest for solitary wave solutions, the extended modified auxiliary equation mapping (EMAEM) method is employed. We derive several novel families of precise traveling wave solutions, encompassing trigonometric, hyperbolic, and exponential forms. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are incorporated alongside the analytical results to enhance comprehension of the dynamic characteristics of the solutions obtained. This study’s outcomes can offer valuable insights for addressing other nonlinear partial differential equations (NLPDEs). The soliton solutions obtained in this study offer important insights into the intricate nonlinear equation being examined.

In this paper, a fractional dual-core optical fiber model is considered, which is described as a coupled fractional nonlinear Schrödinger equation. The fractional ($e^{-\varphi (\xi )})$-expansion method and fractional rational sinh–cosh function method have been successfully employed to derive various new soliton solutions for the coupled fractional nonlinear Schrödinger equations. These obtained solutions are of great significance for a deeper understanding of the corresponding physical phenomena. Finally, some 3D and 2D graphs are provided to elaborate the physical explanation of the obtained solutions.

An important manifestation of the integrability of nonlinear mathematical and physical models is their solvability through inverse scattering transform (IST). The problem to be investigated in this paper is a system of fifth-order nonlinear evolution equations (NLEEs) with variable coefficients related to time-varying spectral problems. The expected result is to derive the system of fifth-order NLEEs, obtain its exact solutions, verify its integrability in the sense IST solvability, and reveal some novel local structures of the obtained solutions. First, the system of fifth-order NLEEs is derived. Then, by combining the IST with a time-varying spectral parameter, the associated scattering data are determined for the reconstruction of potentials. Using the determined scattering data, exact solutions are ultimately obtained. Meanwhile, some local structures with new features are analyzed for two pairs of special solutions, including kink solitons, wide/narrow top bell solitons, and small-scale peaks in the time variable direction, as well as sine/cosine-like fluctuations in the spatial variable direction. This paper not only demonstrates that equipping the Ablowitz–Kaup–Newell–Segur (AKNS) problems with appropriate time-varying spectra can be used to derive some other inverse scattering integrable systems of NLEEs with variable coefficients, but also graphically illustrates the regulatory effect of time-varying spectral parameter on local solution structures.

This paper investigates the dynamics of the nonlinear Landau–Ginzburg–Higgs equation within the domain of superfluids and Bose–Einstein condensates. The primary focus is on exploring novel exact solutions within this equation. To obtain soliton solutions of the proposed nonlinear model, a modified Sardar sub-equation method and an extended modified tanh function method are employed. From a significant physical perspective, a diverse range of solutions is presented, encompassing bright solitons, dark solitons, singular solitons, combo dark-bright solitons, dark-singular solitons, periodic solutions, exponential solutions, and rational solutions. The study utilizes 2D and 3D visualizations of the computed wave solutions to effectively illustrate the internal structure of the phenomenon. This study also presents a comparative analysis of wave speed parameters using 2D graphs. By visualizing these parameters, the research highlights the variations and intricate behavior of the soliton solutions, further enhancing the understanding of the nonlinear Landau–Ginzburg–Higgs equation. The study marks the first application of the modified Sardar subequation method and the extended modified tanh function method to the nonlinear Landau–Ginzburg–Higgs equation. This combination provides a comprehensive model for capturing complicated biological and physical events, with increased predictive power and adaptability.

We use loop group factorizations to construct Darboux transformations, Permutability formulas, Scaling Transformations, and explicit soliton solutions of the $1$-d Schrödinger flows on compact Hermitian symmetric spaces.

Firstly twenty-four types of doubly-periodic solutions of the reduction mKdV equation are given. Secondly based on the reduction mKdV equation and its solutions, a systemic transformation method (called the reduction mKdV method) is developed to construct new doubly-periodic solutions of nonlinear equations. Thirdly with the aid of symbolic computation, we choose the KdV equation, the coupled variant Boussinesq equation and the cubic nonlinear Schrödinger equation to illustrate our method. As a result many types of solutions are obtained. These show that this method is simple and powerful to obtain more exact solutions including doubly-periodic solutions, soliton solutions and singly-periodic solutions to a wide class of nonlinear wave equations. Finally we further extended the method to a general form.

By means of a more general ansatz and the computerized symbolic system Maple, a generalized algebraic method to uniformly construct solutions in terms of special function of nonlinear evolution equations (NLEEs) is presented. We not only successfully recover the previously-known traveling wave solutions found by Fan's method, but also obtain some general traveling wave solutions in terms of the special function for the asymmetric Nizhnik–Novikov–Vesselov equation.

The projective Riccati equations method is extended to find some novel exact solutions of a (2+1)-dimensional cubic nonlinear Schrödinger (NLS) equation. Applying the extended method and symbolic computation, six families of exact analytical solutions for this NLS equation are reported, which include some new and more general exact soliton-like solutions, trigonometric function forms solutions and rational forms solutions.

Based on the complex envelope ansatz method, the projective Riccati equation method and q-deformed hyperbolic functions, a method is developed for constructing a series of exact analytical solutions for higher-order nonlinear Schrödinger (HNLS) equation, which describes propagation of femtosecond light pulse in optical fiber under certain parametric conditions. With the help of symbolic computation, six families of new solitary wave solutions are obtained. The solitary wave solutions obtained by Li et al.¹⁸ are special cases of our solutions. The novel soliton solutions can describe W-shaped, bright and dark soliton properties in the same expression and their amplitude may approach nonzero when the time variable approaches infinity. Furthermore, the soliton propagation and solitons interaction scenario are discussed and simulated by computer.

We have studied the Maxwell–Higgs model on the surface of an infinite cylinder. In particular, we show that this model supports self-dual topological soliton solutions on the infinite tube. Finally, the Bogomol’nyi-type equations are studied from theoretical and numerical point of view.

The Hirota equation, a modified nonlinear Schrödinger (NLS) equation, takes into account higher-order dispersion and time-delay corrections to the cubic nonlinearity. Its wave propagation is like in the ocean and optical fibers can be viewed as an approximation which is more accurate than the NLS equation. By considering the potential application of two mode nonlinear waves in nonlinear fibers under a certain case, we use the algebraic reductions from the Lie algebra $gl(2,ℂ)$ to its commutative subalgebra $Z_2=ℂ[\mathrm{\Gamma }]/({\mathrm{\Gamma }}^2)$ and $\mathrm{\Gamma }={({\delta }_{i,j+1})}_{ij}$ to define a weakly coupled Hirota equation (called Frobenius Hirota equation) including its Lax pair, in this paper. Afterwards, Darboux transformation of the Frobenius Hirota equation is constructed. The Darboux transformation implies the new solutions of ($q^{[1]}$, $r^{[1]}$) generated from the known solution ($q$, $r$). The new solutions ($q^{[1]}$, $r^{[1]}$) provide soliton solutions, breather solutions of the Frobenius Hirota equation. Further, rogue waves of the Frobenius Hirota equation are given explicitly by a Taylor series expansion of the breather solutions. In particular, by choosing different parameter values for the rogue waves, we can get different images.

With the coupling of a scalar field, a generalization of the nonlinear Klein–Gordon equation which arises in the relativistic quantum mechanics and field theory, i.e., the coupled nonlinear Klein–Gordon equations, is investigated via the Hirota method. With the truncated Painlevé expansion at the constant level term with two singular manifolds, the coupled nonlinear Klein–Gordon equations are transformed to a bilinear form. Starting from the bilinear form, with symbolic computation, we obtain the N-soliton solutions for the coupled nonlinear Klein–Gordon equations.

A new generalized (2+1)-dimensional model, obtained from the Kadomtsev–Petviashvili (KP) equation considered as a system of nonlinear evolution partial differential equations (PDEs), is introduced. With the usage of the Hirota bilinear method and the KP-hierarchy reduction method, N-soliton solutions of the integrable system are constructed. Considering the case of s=−1 in the linear differential operators, L₁ and L₂, and a specific set of parameters, two dark solitons, mixed solutions consisting of soliton-type and periodic waves solution are obtained. Based on the particular definition of the matrix elements, one and two rogue waves solutions expressed in terms of rational functions are derived. It is shown that the fundamental rogue waves are line rogue waves, which is different from the property of the moving line solitons of the soliton equations.

The Kudryashov technique is employed to extract several classes of solitary wave solutions for the Joseph–Egri equation. The stability of the achieved solutions is tested. The numerical solution of this equation is also investigated. We also present the accuracy and the stability of the numerical schemes. Some two- and three-dimensional figures are shown to present the solutions on some specific domains. The used methods are found useful to be applied on other nonlinear evolution equations.

We investigate the solvability of an integrable nonlinear nonlocal reverse-time six-component fourth-order AKNS system generated from a reduced coupled AKNS hierarchy under a reverse-time reduction. Riemann–Hilbert problems will be formulated by using the associated matrix spectral problems, and exact soliton solutions will be derived from the reflectionless case corresponding to an identity jump matrix.

The paper aims to discuss nonlocal reverse-space multicomponent nonlinear Schrödinger equations and their inverse scattering transforms. The inverse scattering problems are analyzed by means of Riemann–Hilbert problems, and Gelfand–Levitan–Marchenko-type integral equations for generalized matrix Jost solutions are determined by the Sokhotski–Plemelj formula. Soliton solutions are generated from the reflectionless transforms associated with zeros of the Riemann–Hilbert problems.

In this paper, we solve nonlinear nonlocal reverse-time six-component sixth-order AKNS system. We used reverse-time reduction to reduce the coupled system to an integrable sixth-order nonlinear Schrödinger-type equation. Starting from the spectral problem of the AKNS system, a Riemann–Hilbert problem will be formulated. This formulation allows us to generate soliton solutions by using the vectors lying in the kernel of the matrix Jost solutions. When reflection coefficients are zeros, the jump matrix is identity and the corresponding Riemann–Hilbert problem yields soliton solutions, leading to explore their dynamics.

Based on a $2\times 2$ matrix $\bar{\partial }$ problem, we have obtained the Lax pair by constructing a spectral transformation matrix. The Hirota equation with self-consistent sources is derived by considering the nonanalytic part of the dispersion relation. Furthermore, explicit solutions including N-soliton solutions are computed by $\bar{\partial }$-dressing method, and N-soliton solutions of the Hirota equation with self-consistent sources are obtained by using the properties of Cauchy matrix. The dynamic behavior of soliton solutions is analyzed.

In this paper, abundant exact wave solutions of fourth-order Ablowitz–Kaup–Newell–Segur water wave (AKNS) equation have been investigated by using the innovative and efficient method called improved $tanh(\frac{\phi (\xi )}{2})$-expansion method (IThEM). The obtained solutions for AKNS equation are in the form of hyperbolic, trigonometric, exponential, logarithmic functions that are completely new and distant from previously derived solutions. To have the knowledge of dynamical physical characteristics of this equation, some important solutions have been discussed graphically in the form of two and three-dimensional along with contour plots by selecting suitable parameters with the aid of Maple program. The achieved outcomes exhibit that this new method is efficient, direct, and provides different classes of families. This technique can solve many nonlinear differential equations having importance in different field of sciences.

Under investigation in this paper is the long-wave-short-wave resonance system, which can describe a variety of nonlinear wave phenomena such as the two-dimensional packets of capillary-gravity waves in hydrodynamics and the optical-terahertz waves. The intended aim is carried out via considering a traveling wave reduction, adopting a modified version of the Jacobi elliptic expansion method and employing the Weierstrass elliptic function method to derive such analytic solutions as the bright and dark soliton solutions, periodic solutions, trigonometric-function and elliptic-function solutions in fluid mechanics.