Two-soliton Solutions Research Articles

The periodic soliton solutions for a nonlocal nonlinear Schrödinger equation with higher-order dispersion

A nonlocal nonlinear Schr¨odinger (NNLS) equation with fourth-order dispersion and cubic-quintic nonlinearities has been studied analytically and numeri- cally. Under the constraint conditions, auxiliary functions are introduced, and explicit one- and two-soliton solutions are obtained by the Hirota bilinear method. Accord- ing to the solutions, the propagation dynamics of soliton pulses are investigated. The influences of different parameters on the dynamics of one- and two-soliton solutions have been analyzed. The results show that the two-soliton solution exhibits diverse dy- namic characteristics under the suitable parameter selections. In addition, the stability of one- and two-soliton solutions against the constraint conditions deviations and under the initial perturbations are also studied numerically.

The mKdV equation under the Gaussian white noise and Wiener process: Darboux transformation and stochastic soliton solutions

In this paper, we propose a novel integrable system named the stochastic mKdV equation, along with its corresponding Lax pair. We aim to extend the methodology of deterministic integrable systems to construct and solve stochastic integrable systems. The Darboux transformation effectively obtains analytic solutions for the integrable stochastic mKdV equation. Using the Darboux transformation, soliton solutions incorporating stochastic terms are obtained as Wronskian determinants. Furthermore, we conduct an in-depth analysis of the dynamics exhibited by the stochastic one-soliton and the two-soliton solutions.

Integrability, breather, rogue wave, lump, lump-multi-stripe, and lump-multi-soliton solutions of a (3 + 1)-dimensional nonlinear evolution equation

In this article, we consider a new (3 + 1)-dimensional evolution equation, which can be used to interpret the propagation of nonlinear waves in the oceans and seas. We effectively investigate the integrable properties of the considered nonlinear evolution equation through several aspects. First of all, we present some elementary properties of multi-dimensional Bell polynomial theory and its relation with Hirota bilinear form. Utilizing those relations, we derive a Hirota bilinear form and a bilinear Bäcklund transformation. By employing the Cole–Hopf transformation in the bilinear Bäcklund transformation, we present a Lax pair. Additionally, using the Bell polynomial theory, we compute an infinite number of conservation laws. Moreover, we obtain one-, two-, and three-soliton solutions explicitly from Hirota bilinear form and illustrate them graphically. Breather solutions are also derived by employing appropriate complex conjugate parameters in the two-soliton solution. Choosing the generalized algorithm for rogue waves derived from the N-soliton solution, we directly obtain a first-order center-controllable rogue wave. Lump solutions are formulated by employing a well-established quadratic test function as a solution to the Hirota bilinear form. Further taking the test function in a combined form of quadratic and exponential functions, we obtain lump-multi-stripe solutions. Furthermore, a combined form of quadratic and hyperbolic cosine functions produces lump-multi-soliton solutions. The fission and fusion effects in the evolution of lump-multi-stripe solutions and lump-soliton-solutions are demonstrated pictorially.

Two soliton interaction of Zakharov–Mikhailov spinor models

We outline the spectral properties of their Lax operators for the Zakharov-Mikhailov spinor model (ZMm) and construct their fundamental analytic solutions (FAS). Using the dressing Zakahrov-Shabat method we derive the dressing factors for the one-soliton u 1(z 1, Φ1) and u 2(z 1 , z 2, Φ1, Φ2) for the two-soliton solutions. Next we calculate the asymptotics of the dressing factors of u 1(z 1, Φ1) for z 1 → ±∞. Inserting them into u 2(z 1 , z 2, Φ1, Φ2) we find that for z 1 → ±∞ u 2(z 1 , z 2, Φ1, Φ2) goes into one-soliton solutions but with shifted center of mass and shifted phase. Thus we conclude that the 2-soliton interactions of ZMm are purely elastic and are characterized by the above mentioned shifts.

Data-driven soliton solutions and parameters discovery of the coupled nonlinear wave equations via a deep learning method

In this paper, we extend the physics-informed neural networks (PINNs) to learn data-driven solutions and discover the coefficients of multi-component and high-dimensional coupled nonlinear partial differential equations (cNPDEs). In the forward problems, the data-driven solutions include “bell”-type (dark and bright) and “anti-kink”-type solitons of the generalized Hirota–Satsuma coupled KdV equation are investigated and reconstructed. Moreover, some special shape one-soliton and two-soliton solutions such as “Kink”-shaped, “U”-shaped, “Y”-shaped, and “X”-shaped of a (2+1)-dimensional cNPDE are learned and simulated as well. Particularly, some factors affecting the performance of PINNs are investigated, such as the structure of neural networks, the kinds of activation functions and optimizers, as well as the number of collocation points and iterations. For the inverse problems, the coefficients of the equations are successfully identified and driven by various kinds of solution data. The impact of the number and moment of observations on the model is discussed while the network’s robustness is validated by introducing mensurable initial noises to the training data. Numerical experiments not only show that the propagation direction and dynamical behaviors of the cNPDE can be well uncovered by utilizing the deep learning method, but also exhibit the advantage of finding equations with multiple parameters.

Localized wave solutions and interactions of the (2+1)-dimensional Hirota–Satsuma–Ito equation

This paper studies the (2+1)-dimensional Hirota–Satsuma–Ito equation. Based on an associated Hirota bilinear form, lump-type solution, two types of interaction solutions, and breather wave solution of the (2+1)-dimensional Hirota–Satsuma–Ito equation are obtained, which are all related to the seed solution of the equation. It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons, and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton. Furthermore, the breather wave solution is also obtained by reducing the two-soliton solutions. The trajectory and period of the one-order breather wave are analyzed. The corresponding dynamical characteristics are demonstrated by the graphs.

A ∂¯-Dressing Method for the Kundu-Nonlinear Schrödinger Equation

In this paper, we employed the ∂¯-dressing method to investigate the Kundu-nonlinear Schrödinger equation based on the local 2 × 2 matrix ∂¯ problem. The Lax spectrum problem is used to derive a singular spectral problem of time and space associated with a Kundu-NLS equation. The N-solitions of the Kundu-NLS equation were obtained based on the ∂¯ equation by choosing a special spectral transformation matrix, and a gradual analysis of the long-duration behavior of the equation was acquired. Subsequently, the one- and two-soliton solutions of Kundu-NLS equations were obtained explicitly. In optical fiber, due to the wide application of telecommunication and flow control routing systems, people are very interested in the propagation of femtosecond optical pulses, and a high-order, nonlinear Schrödinger equation is needed to build a model. In plasma physics, the soliton equation can predict the modulation instability of light waves in different media.

Optical nondegenerate solitons in a birefringent fiber with a 35 degree elliptical angle.

In this paper, we investigate the optical nondegenerate solitons in a birefringent fiber with a 35 degree elliptical angle. We derive the nondegenerate bright one- and two-soliton solutions by solving the coupled Schrödinger equation. The formation of nondegenerate solitons is related to the wave numbers of the solitons, and we further demonstrate that it is caused by the incoherent addition of different components. We note that the interaction between two degenerate solitons or a nondegenerate soliton and a degenerate soliton is usually inelastic. This is led to the incoherent interaction between solitons of different components and the coherent interaction between solitons of the same component. Through the asymptotic analysis, we find that the two degenerate solitons are elastic interactions under certain conditions, and analyzed the influence of the Kerr nonlinear intensity coefficient γ and the second-order group velocity dispersion β2 in this system on solitons: the velocity and amplitude of the solitons are proportional to |β2|, while the amplitude of the solitons is inversely proportional to γ. Two nondegenerate solitons are elastic interactions, but the phase of the soliton can be adjusted to make it inelastic. Furthermore, regardless of the situation mentioned above, total intensities of the solitons before the interaction are equal to that after the soliton interaction.

Degenerate behaviors and decompositions of the two-soliton solution in the (1+1)-dimensional Ito equation

In this paper, we investigate the degenerate behaviors and decompositions of two-soliton solution on the non-vanishing constant background for the (1+1)-dimensional Ito equation. By tuning the background constant, interaction phenomena of various types such as the elastic interaction of solitons, fission or fusion phenomena are presented. The degenerate two-soliton solutions which lead to a higher amplitude soliton in collisions of two solitons or describe the decay of a soliton into two new solitons are obtained. To better understand the nature of these interaction phenomena, we explore different decompositions to express the two-soliton solution as a sum of two functions. One of the function exhibits a special soliton excited by the collisions of two solitons, and the other describes the energy exchange of two solitons. Finally, under the module resonance of wave numbers, other types of the degenerate solutions of two solitons are derived and analyzed, such as the breather and rogue wave solutions.

Solitons, multi-solitons and multi-periodic solutions of the generalized Lax equation by Darboux transformation and its quasiperiodic motions

Using the Darboux transformation method, the general Lax equation is solved and a collection of new exact solutions together with one-soliton solutions, singular one-soliton solutions, periodic solutions, singular periodic solution, two-soliton solutions, singular two-soliton solutions, two-periodic solutions and singular two-periodic solutions is obtained. Using traveling wave transformation, the Lax equation is transfigured to a conservative dynamical system (CDS) of dimension four with three equilibrium points involving two parameters [Formula: see text] and v. The CDS has various quasi-periodic motions for fixed values of the parameters [Formula: see text] and v at different initial conditions. Furthermore, effects of the parameters [Formula: see text] and v are shown on the quasiperiodic motions of the CDS by means of phase sections and time series plots. This approach can be applied to a heterogeneity of nonlinear model equations or partial differential equations for describing their inherent nonlinear phenomena.

Various types of discrete exact localized wave solutions and dynamical analysis for the discrete complex modified Korteweg-de Vries equation

Under consideration is the second-order integrable discretization of complex modified Korteweg-de Vries (mKdV) equation which is regarded as the discrete counterpart of the mKdV equation having an essential role in describing the propagation behavior of water waves and acoustic waves in nonlinear media. First of all, based on the known linear spectral problem, the discrete generalized (n,N−n)-fold Darboux transformation is constructed to derive various types of discrete exact localized wave solutions, including soliton and semi-rational soliton solutions on vanishing background, breather, rogue wave and hybrid interaction solutions on plane wave background, and rational soliton solution on constant background, and the relevant evolution structures are studied graphically. Secondly, the asymptotic analysis is used to discuss the elastic interaction for two-soliton solutions and limit states for rational soliton solutions. Finally, the numerical simulation is utilized to investigate the dynamical behavior and propagation stability of some exact solutions. The findings presented in this paper may contribute to explaining physical phenomena described by the mKdV equation.

Painlevé integrable property, Bäcklund transformations, Lax pair, and soliton solutions of a (3 + 1)-dimensional variable-coefficient Hirota bilinear system in a fluid

In this paper, we focus our attention on a (3 + 1)-dimensional variable-coefficient Hirota bilinear system in a fluid with symbolic computation. The Painlevé integrable property is derived. Via the Ablowitz–Kaup–Newell–Segur procedure, we obtain a Lax pair under the coefficient constraints. Based on the Hirota method, we obtain a bilinear form and a bilinear Bäcklund transformation under the coefficient constraints. We derive the auto-Bäcklund transformations based on the truncated Painlevé expansions. According to the bilinear form, we give the two-soliton solutions under the coefficient constraints. We also discuss the relation between the variable coefficients and soliton solutions, i.e., how the two solitons present different types with the different forms of the variable coefficients.

Exact multi-soliton solutions of the KdV equation with a source: Riemann–Hilbert formulation

The soliton solutions with non-zero background of KdV equation with a source are constructed by Riemann–Hilbert approach. The irregular Riemann–Hilbert problem is constructed by direct and inverse scattering transform firstly, and then it can be regularized by introducing a novel transformation. The Residue theorem is utilized to derive the multi-soliton solutions at the simple poles of the Riemann–Hilbert problem. In particular, the interaction dynamics of the two-soliton solution are illustrated by considering their evolutions at different time.

Cauchy matrix structure and solutions of the spin-1 Gross–Pitaevskii equations

In this paper, we investigate the Cauchy matrix structure of the spin-1 Gross–Pitaevskii (GP) equations. By utilizing the Cauchy matrix approach, we derive a 2 × 2 matrix nonlinear Schrödinger (NLS) equation, which serves as an unreduced model for the spin-1 BEC system and allows solutions with explicit formulae. Then we provide suitable constraints which lead to reductions for obtaining the classical and nonlocal spin-1 GP equations and their solutions. Some obtained solutions, including one-soliton solution, two-soliton solution and double-pole solution, are analyzed and illustrated.

Conservation laws, bound-state solitons and modulation instability for a variable-coefficient higher-order nonlinear Schrödinger equation in an optical fiber

Under investigation in this paper is a variable-coefficient sixth-order nonlinear Schrödinger equation, which describes the propagation of attosecond pulses in an optical fiber. Based on Lax pair, infinitely-many conservation laws are constructed. With the aid of auxiliary functions, bilinear forms are derived. In addition, the one- and two-soliton solutions are obtained via the Hirota method. The influences of variable coefficients for soliton velocity and profile are discussed. Particularly, the interaction periods and soliton separation factor of bound-state solitons are analyzed. Finally, modulation instability is investigated. The reported results could be used to understand related soliton molecule and optical instability phenomena in nonlinear optics.

-dressing method for the complex modified KdV equation

The dressing method based on the 2 × 2 matrix -problem is generalized to study the complex modified KdV equation (cmKdV). Through two linear constraint equations, the spatial and time spectral problems related to the cmKdV equation are derived. The gauge equivalence between the cmKdV equation and the Heisenberg chain equation is obtained. Using a recursive operator, a hierarchy of cmKdV with source is proposed. On the basis of the -equation, the N-solition solutions of the cmKdV equation are obtained by selecting the appropriate spectral transformation matrix. Furthermore, we get explicit one-soliton and two-soliton solutions.

Soliton interactions and Mach reflection in gas bubbles–liquid mixtures

In this study, we assume that blood is assumed to be a viscoelastic and incompressible homogeneous media in which several uniform sized oxygen bubbles are uniformly distributed. Based on this, we establish a (3 + 1)-dimensional modified Kadomtsev–Petviashvili (mKP) equation to describe the long nonlinear pressure waves in the gas bubbles–liquid mixtures. Using bell polynomials, a new bilinear form of the mKP equation is constructed, and then the one- and two-soliton solutions of the equation are obtained by the Hirota method. Via the one-soliton solutions, parametric conditions of the existence of shock wave, elevation and depression solitons, and the Mach reflection characters in the mixtures are discussed. Soliton interactions have been discussed on the basis of the two-soliton solutions. We find that the (i) parallel elastic interactions can exist between the shock and elevation solitons; (ii) oblique elastic interactions can exist between the (a) shock and depression solitons and (b) the elevation and depression solitons; and (iii) oblique inelastic interactions can exist between the two depression solitons.

A non-autonomous Gardner equation and its integrability: Solitons, positons and breathers

The work studies some integrable properties and soliton type solutions of a non-autonomous Gardner equation with damping and forcing terms. A bilinear form, a bilinear Bäcklund transformation and a Lax pair are derived for the considered Gardner equation explicitly. K-Soliton solution with proper existence condition, smooth positons, breathers and their interaction solutions are presented via the bilinear form. Moreover, the amplitude as well as velocity of the soliton solutions are derived, and a first-order breather solution and a second-order smooth positon are generated from the two-soliton solution. The interaction between a single-breather solution and the single-soliton solution and the interaction of a second-order smooth positon and the single-soliton solution are studied analytically, based on the three-soliton solution. Profiles of various types of the obtained solutions and their interactions are illustrated graphically.

Nondegenerate solitons of the (2+1)-dimensional coupled nonlinear Schrödinger equations with variable coefficients in nonlinear optical fibers

We derive the multi-hump nondegenerate solitons for the (2+1)-dimensional coupled nonlinear Schrödinger equations with propagation distance dependent diffraction, nonlinearity and gain (loss) using the developing Hirota bilinear method, and analyze the dynamical behaviors of these nondegenerate solitons. The results show that the shapes of the nondegenerate solitons are controllable by selecting different wave numbers, varying diffraction and nonlinearity parameters. In addition, when all the variable coefficients are chosen to be constant, the solutions obtained in this study reduce to the shape-preserving nondegenerate solitons. Finally, it is found that the nondegenerate two-soliton solutions can be bounded to form a double-hump two-soliton molecule after making the velocity of one double-hump soliton resonate with that of the other one.

Study on the (2+1)-dimensional extension of Hietarinta equation: soliton solutions and Bäcklund transformation

The (1+1)-dimensional bilinear Hietarinta equation was firstly proposed when searching for integrable nonlinear evolution equations by the three-soliton method. In this paper, we focus on the (2+1)-dimensional extension of Hietarinta equation, which enjoys potential application in environmental engineering. Based on the bilinear form, one-soliotn and two-soliton solutions are derived. Bilinear Bäcklund transformation and Bell-polynomial-typed Bäcklund transformation are derived through the Hirota bilinear method and Bell polynomials, respectively. The three-dimensional plots of soliton solutions have been given by selecting appropriate parameters.