Inverse Scattering Transform Method Research Articles

Travelling wave solution for the Landau-Ginburg-Higgs model via the inverse scattering transformation method

The Landau-Ginzburg-Higgs (LGH) equation explains the ocean engineering models, superconductivity and drift cyclotron waves in radially inhomogeneous plasma for coherent ion-cyclotron waves. In this paper, with a simple modification of the Ablowitz-Kaup-Newell-Segur (AKNS) formalism, the integrability of LGH equation is proved by deriving the Lax pair. Hence for that, the inverse scattering transformation (IST) is applied, and the travelling wave solutions are obtained and graphically represented in 2d and 3d profiles.

Optical soliton solutions for the integrable Lakshmanan-Porsezian-Daniel equation via the inverse scattering transformation method with applications

The Lakshmanan-Porsezian-Daniel equation (LPD) is a nonlinear Schrödinger type (NLS) equation which describe the effect on the integrable properties of Heisenberg bilinear spin chains under the classical limit by the biquadratic interactions. In this paper, the integrability of LPD equation is illustrated by obtaining the Lax pair via the AKNS scheme. The exact soliton solutions via the inverse scattering transformation (IST) method are obtained and illustrated in 3D graphical representation.

Simple Equations Method (SEsM): An Effective Algorithm for Obtaining Exact Solutions of Nonlinear Differential Equations.

Exact solutions of nonlinear differential equations are of great importance to the theory and practice of complex systems. The main point of this review article is to discuss a specific methodology for obtaining such exact solutions. The methodology is called the SEsM, or the Simple Equations Method. The article begins with a short overview of the literature connected to the methodology for obtaining exact solutions of nonlinear differential equations. This overview includes research on nonlinear waves, research on the methodology of the Inverse Scattering Transform method, and the method of Hirota, as well as some of the nonlinear equations studied by these methods. The overview continues with articles devoted to the phenomena described by the exact solutions of the nonlinear differential equations and articles about mathematical results connected to the methodology for obtaining such exact solutions. Several articles devoted to the numerical study of nonlinear waves are mentioned. Then, the approach to the SEsM is described starting from the Hopf-Cole transformation, the research of Kudryashov on the Method of the Simplest Equation, the approach to the Modified Method of the Simplest Equation, and the development of this methodology towards the SEsM. The description of the algorithm of the SEsM begins with the transformations that convert the nonlinearity of the solved complicated equation into a treatable kind of nonlinearity. Next, we discuss the use of composite functions in the steps of the algorithms. Special attention is given to the role of the simple equation in the SEsM. The connection of the methodology with other methods for obtaining exact multisoliton solutions of nonlinear differential equations is discussed. These methods are the Inverse Scattering Transform method and the Hirota method. Numerous examples of the application of the SEsM for obtaining exact solutions of nonlinear differential equations are demonstrated. One of the examples is connected to the exact solution of an equation that occurs in the SIR model of epidemic spreading. The solution of this equation can be used for modeling epidemic waves, for example, COVID-19 epidemic waves. Other examples of the application of the SEsM methodology are connected to the use of the differential equation of Bernoulli and Riccati as simple equations for obtaining exact solutions of more complicated nonlinear differential equations. The SEsM leads to a definition of a specific special function through a simple equation containing polynomial nonlinearities. The special function contains specific cases of numerous well-known functions such as the trigonometric and hyperbolic functions and the elliptic functions of Jacobi, Weierstrass, etc. Among the examples are the solutions of the differential equations of Fisher, equation of Burgers-Huxley, generalized equation of Camassa-Holm, generalized equation of Swift-Hohenberg, generalized Rayleigh equation, etc. Finally, we discuss the connection between the SEsM and the other methods for obtaining exact solutions of nonintegrable nonlinear differential equations. We present a conjecture about the relationship of the SEsM with these methods.

Integrable local and non-local vector Non-linear Schrödinger Equation with balanced loss and gain

The local and non-local vector Non-linear Schrödinger Equation (NLSE) with a general cubic non-linearity are considered in presence of a linear term characterized, in general, by a non-hermitian matrix which under certain condition incorporates balanced loss and gain and a linear coupling between the complex fields of the governing non-linear equations. It is shown that the systems posses a Lax pair and an infinite number of conserved quantities and hence integrable. Apart from the particular form of the local and non-local reductions, the systems are integrable when the matrix representing the linear term is pseudo hermitian with respect to the hermitian matrix comprising the generic cubic non-linearity. The inverse scattering transformation method is employed to find exact soliton solutions for both the local and non-local cases. Further, it is shown that the presence of the linear term restricts the possible form of the norming constants and hence the polarization vector.

Nonautonomous solitons of the generalized complex mKdV equation in external potentials

The generalized nonautonomous complex modified Korteweg–de Vries (mKdV) equation with external potentials is introduced in the framework of the nonisospectral generalization of the Inverse Scattering Transform method with associated spectral parameter varying in accordance with the Riccati equation. Nonautonomous accelerating and self-compressing mKdV solitons exist only if the varying dispersion, nonlinearity, and the external linear, parabolic, and cubic potentials satisfy to the conditions of the exact integrability of the introduced model.

Nonautonomous solitons of the novel nonlinear Schrödinger equation: Self-compression, amplification, and the bound state decay in external potentials

The generalized nonautonomous nonlinear Schrödinger equation is introduced in the framework of the nonisospectral generalization of the Inverse Scattering Transform method with associated spectral parameter varying in accordance with the Riccati equation. Nonautonomous solitons of the introduced model conserve the soliton main feature to interact elastically both in the linear, parabolic, and cubic external potentials, and are controlled only if the varying dispersion and nonlinearity satisfy to the conditions of the exact integrability. Novel features of nonautonomous solitons arising due to the dependence between the soliton amplitudes and their velocities consist in the decay of the soliton bound states.

Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods.

The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a “small” parameter ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of to solution for arbitrary finite values of . Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.

On the double-pole and two-soliton solutions of the Harry Dym equation

The double-pole solution of the initial value problem for the Harry Dym equation (HDE) was obtained by using the inverse scattering transform (IST) method, i.e., solving the associated Gelfand–Levitan–Marchenko (GLM) equation. Further, we show that the double-pole solution can be viewed as some proper limits of the two-soliton solution obtained by Riemann–Hilbert problem (RHP) and IST method, respectively.

Nonlinear Dynamics of Extremely Short Solitons of the Generalized Reduced Maxwell–Bloch System

The effect of remote quantum states on two allocated atomic levels during the propagation of extremely short electromagnetic pulses in a medium is considered without using the slowly varying envelope approximation. A generalization of the reduced Maxwell–Bloch system is obtained that is integrable via the inverse scattering transformation method. Its soliton and breather solutions are found and studied.

Few-cycle solitons of an integrable generalization of the reduced Maxwell-Bloch equations

The physical generalization of the system of the reduced Maxwell-Bloch equations describing the nonlinear interaction of the laser pulses with the multilevel quantum medium is obtained without using the slowly varying envelope approximation. It is shown that this system is integrable in the frameworks of the inverse scattering transformation method. The soliton and breather solutions in the form of the unipolar and few-cycle pulses are constructed. Their distinctive features caused solely by the multilevel structure of the medium are discussed. It is revealed that the collision of the solitons can lead to an appearance of the large-amplitude short-living pulse, whose dynamics resembles that of rogue waves.

Few-cycle solitons in a quasiequilibrium nanodispersed medium of asymmetric molecules

The nonlinear dynamics of extremely short electromagnetic pulses in a nanodispersed birefringent medium is investigated. The matrix of the medium and chaotically mixed nanogranules form an amorphous photon crystal containing asymmetric molecules of identical chemical composition. The concentrations of molecules in the nanogranules and in the matrix are different. This nanodispersed (discrete) structure of the medium leads to spatial dispersion. The wave equation for the ordinary component of pulses propagating in such a medium is derived under the conditions of the sudden perturbation approximation. This equation generalizes the sine-Gordon and Rabelo–Fokas equations and appears to be integrable in the frameworks of the inverse scattering transformation method if an additional restriction on the parameters characterizing the spatial dispersion and anisotropy of the medium is imposed. This restriction implies that the medium is prepared in the quasiequilibrium state before the pulse effect, when the molecules in the matrix or the ones in the nanogranules are in the excited state. The soliton and breather solutions of the integrable wave equation are investigated.

The massive Thirring system in the quarter plane

AbstractThe unified transform method (UTM) or Fokas method for analyzing initial-boundary value (IBV) problems provides an important generalization of the inverse scattering transform (IST) method for analyzing initial value problems. In comparison with the IST, a major difficulty of the implementation of the UTM, in general, is the involvement of unknown boundary values. In this paper we analyze the IBV problem for the massive Thirring model in the quarter plane, assuming that the initial and boundary data belong to the Schwartz class. We show that for this integrable model, the UTM is as effective as the IST method: Riemann-Hilbert problems we formulated for such a problem have explicit (x, t)-dependence and depend only on the given initial and boundary values; they do not involve additional unknown boundary values.

Integrable generalizations of the sine-Gordon, short pulse, and reduced Maxwell–Bloch equations

The multi-parameter generalizations of the sine-Gordon (SG), short pulse (SP), and reduced Maxwell–Bloch (RMB) equations are considered. These generalizations are integrable by the inverse scattering transformation method and connected with the modified SG equation, its limiting case, and the modified RMB equations by the changes of the dependent and independent variables expressed in terms of the conserved densities and fluxes of the latter. The particular cases of the generalizations considered include the Rabelo–Fokas (RF) equation, the modified SP equation, and some other equations. The properties of the soliton solutions of some of the generalizations are studied. It is revealed that the interaction of the well-defined solitons of the RF equation and the corresponding version of the RMB equations leads to an appearance of the intervals in some regions of the soliton parameters, where the solution becomes multi-valued, and to the blow-up of the solution. In addition, the compacton-like soliton solutions can exist in some cases.

Propagation of few-cycle pulses in a nonlinear medium and an integrable generalization of the sine-Gordon equation

The generalized sine-Gordon equation is obtained under the theoretical investigation of interaction of few-cycle pulses in a nonlinear medium modeled by a set of four-level atoms. This equation is derived without the use of the slowly varying envelope approximation and is shown to be integrable in the frameworks of the inverse scattering transformation method. Its solutions describing the propagation of the solitons and breathers and their interaction are investigated. In the case of different signs of the parameters of the equation considered, it is revealed, in particular, that the collision of solitons with opposite polarities can lead to an appearance of the short-living pulse having extraordinarily large amplitude, whose dynamics is similar to that of rogue waves. Also, the solitons of ``rectangular'' form and the breathers with rectangular oscillations exist in the case of the same signs of the parameters.

Ablowitz-Kaup-Newel-Segur Formalism and N-Soliton Solutions of Generalized Shallow Water Wave Equation

We apply Ablowitz-Kaup-Newel-Segur hierarchy to derive the generalized shallow waterwave equation and we also investigate N-soliton solutions of the derived equation using InverseScattering Transform method and Hirota’s bilinear method.

KdV-type equations linked via Bäcklund transformations: Remarks and perspectives

Third order nonlinear evolution equations, that is the Korteweg–de Vries (KdV), modified Korteweg–de Vries (mKdV) equation and other ones are considered: they all are connected via Bäcklund transformations. These links can be depicted in a wide Bäcklund Chart which further extends the previous one constructed in [22]. In particular, the Bäcklund transformation which links the mKdV equation to the KdV singularity manifold equation is reconsidered and the nonlinear equation for the KdV eigenfunction is shown to be linked to all the equations in the previously constructed Bäcklund Chart. That is, such a Bäcklund Chart is expanded to encompass the nonlinear equation for the KdV eigenfunctions [30], which finds its origin in the early days of the study of Inverse scattering Transform method, when the Lax pair for the KdV equation was constructed. The nonlinear equation for the KdV eigenfunctions is proved to enjoy a nontrivial invariance property. Furthermore, the hereditary recursion operator it admits [30] is recovered via a different method. Then, the results are extended to the whole hierarchy of nonlinear evolution equations it generates. Notably, the established links allow to show that also the nonlinear equation for the KdV eigenfunction is connected to the Dym equation since both such equations appear in the same Bäcklund chart.

Soliton solutions and eigenfunctions of linearized operator for a higher-order nonlinear Schrödinger equation

The Inverse Scattering Transform (IST) method is applied to find soliton solutions for a higher-order nonlinear Schrödinger (NLS) equation. Eigenfunctions of linearized operator which have a central role in soliton perturbation theory are explicitly found.

New integrable model of propagation of the few-cycle pulses in an anisotropic microdispersed medium

We investigate the propagation of the few-cycle electromagnetic pulses in the anisotropic microdispersed medium. The effects of the anisotropy and spatial dispersion of the medium are created by the two sorts of the two-level atoms. The system of the material equations describing an evolution of the states of the atoms and the wave equations for the ordinary and extraordinary components of the pulses is derived. By applying the approximation of the sudden excitation to exclude the material variables, we reduce this system to the single nonlinear wave equation that generalizes the modified sine–Gordon equation and the Rabelo–Fokas equation. It is shown that this equation is integrable by means of the inverse scattering transformation method if an additional restriction on the parameters is imposed. The multisoliton solutions of this integrable generalization are constructed and investigated.

Existence of Global Solutions to the Derivative NLS Equation with the Inverse Scattering Transform Method

We address existence of global solutions to the derivative nonlinear Schr\"{o}dinger (DNLS) equation without the small-norm assumption. By using the inverse scattering transform method without eigenvalues and resonances, we construct a unique global solution in $H^2(\mathbb{R}) \cap H^{1,1}(\mathbb{R})$ which is also Lipschitz continuous with respect to the initial data. Compared to the existing literature on the spectral problem for the DNLS equation, the corresponding Riemann--Hilbert problem is defined in the complex plane with the jump on the real line.

Soliton solutions and dynamical evolutions of a generalized AKNS system in the framework of inverse scattering transform

It is important and interesting to derive integrable systems or construct soliton solutions of nonlinear partial differential equations (PDEs). In this paper, a new and more general Ablowitz–Kaup–Newell–Segur (AKNS) system with Lax integrability is derived and solved in the framework of inverse scattering transform (IST). To be specific, the known AKNS’s linear spectral problem and its time evolution equation are first generalized by embedding a nonisospectral parameter whose varying with time obeys the rational function of spectral parameter. Based on the generalized AKNS spectral problem and time evolution equation, we then derive a generalized AKNS system with infinite number of terms. Finally, in the case of reflectionless potentials, explicit n-soliton solutions of the generalized AKNS system are formulated through the IST method. It is shown that the dynamical evolutions of the obtained one-soliton solutions and two-soliton solutions possess time-varying speeds and amplitudes in the process of propagations.