Complex Modified Korteweg De Vries Equation Research Articles

Преобразование Дарбу для т-симметричной нелокальной комплексной модифицированной системы уравнений Кортевега-де Фриза

onlinear partial differential equations are widely used as models to describe physical phenomena in various fields of sciences such as fluid mechanics, solid-state physics, plasma physics, chemical physics, condensed matter physics, optical fibers, biology, and geochemistry. One of the nonlinear partial differential equations is the complex modified Korteweg-de Vries equation. This equation has been proposed as a model for the nonlinear evolution of plasma waves and is the physical model that incorporates the propagation of transverse waves in a molecular chain model, and in a generalized elastic solid. In this paper, we study the T-symmetry nonlocal complex modified Korteweg-de Vries system of equations. This nonlocal system is obtained by Ablowitz-Musslimani type of reduction and is respectively T-symmetric nonlocal cmKdV system of equations. The Ablowitz - Musslimani type of reductions arises from remarkably simple symmetry reductions of general AKNS scattering problems. The method of the Darboux transformation is applied to obtain exact solutions.

A generalized complex mKdV equation: Darboux transformations and explicit solutions

A new generalized complex modified Korteweg–de Vries equation associated with a 3 × 3 matrix spectral problem is proposed by resorting to the zero-curvature equation. Based on the gauge transformations between the Lax pairs, a Darboux transformation for the generalized complex modified Korteweg–de Vries equation is constructed, from which the corresponding N-fold Darboux transformations are derived in terms of both iterative technique and determinants. As an application of the resulting Darboux transformations, explicit solutions, like one-soliton, two-soliton, and three-soliton solutions, first-order breather and second-order breather solutions, for the generalized complex modified Korteweg–de Vries equation are obtained.

Numerical solutions and stability analysis for solitary waves of complex modified Korteweg–de Vries equation using Chebyshev pseudospectral methods

AbstractIn this research article, the authors investigate the interaction of solitary waves for complex modified Korteweg–de Vries (CMKdV) equations using Chebyshev pseudospectral methods. The proposed method is established in both time and space to approximate the solutions and to prove the stability analysis for the equations. The derivative matrices are defined at Chebyshev–Gauss–Lobbato points and the problem is reduced to a diagonally block system of coupled nonlinear equations. For numerical experiments, the method is tested on a number of different examples to study the behavior of interaction of two and more than two solitary waves, single solitary wave at different amplitude parameters and different polarization angles. Numerical results support the theoretical results. A comprehensive comparison of numerical results with the exact solutions and other numerical methods are presented. The rate of convergence of the proposed method is obtained up to seventh‐order.

LINEARLY IMPLICIT ENERGY-PRESERVING FOURIER PSEUDOSPECTRAL SCHEMES FOR THE COMPLEX MODIFIED KORTEWEG–DE VRIES EQUATION

AbstractWe propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy in the discrete setting. Some numerical examples are also presented to validate the effectiveness of the proposed schemes.

Decomposition, Darboux transformation and soliton solutions for the (2 + 1)-dimensional integrable nonlocal “breaking soliton” equation

In this paper, we investigate an integrable nonlocal “breaking soliton” equation, which can be decomposed into the nonlocal nonlinear Schrödinger equation and the nonlocal complex modified Korteweg–de Vries equation. As an application, with the use of this decomposition and Darboux transformation, the dark solitons, antidark solitons, rational dark solitons and rational antidark solitons of the considered equation are given explicitly. In particular, the interaction mechanisms of these solutions are discussed and illustrated through some figures.

The bound-state soliton solutions of the complex modified KdV equation

The inverse scattering method is performed to the complex modified Korteweg–de Vries equation with zero boundary condition by developing an appropriate Riemann–Hilbert problem (RHP). We solve the RHP when reflection coefficient has multiple higher-order poles, and display the formulae of bound-state (BS) solitons and multiple BS solitons which correspond to one higher-order pole and multiple higher-order poles, respectively. The patterns of the BS solitons are shown and the interactions between solitons and BS solitons are considered.

Employing the dynamics of poles in the complex plane to describe properties of rogue waves: case studies using the Boussinesq and complex modified Korteweg–de Vries equations

The dynamics and properties of rogue waves of two classical evolution equations are studied in terms of trajectories of the poles of the exact solutions, by analytically continuing the spatial variable to be complex. The Boussinesq equation describes the motion of hydrodynamic waves in two opposite directions in the shallow water regime. The complex modified Korteweg–de Vries equation is relevant for wave packets in nonlinear media governed by a higher-order nonlinear Schrodinger equation, for the special case where second-order dispersion and cubic self-interaction are absent. On examining the movement of poles of the exact solutions for rogue waves, the real parts of the poles correlate well with the locations of maximum displacements in the physical space. This phenomenon holds for high precision numerically for the first-, second- and third-order rogue waves of the Boussinesq equation. A similar principle is also valid for the first- and second-order rogue waves of the complex modified Korteweg–de Vries equation. The imaginary parts of the poles can generate useful information too. For these two evolution models, a smaller imaginary part in the complex plane is associated with a larger amplitude of the rogue wave in physical space. An empirical formula is proposed which works well for the three lowest orders of rogue waves of the Boussinesq equation.

Deformed solitons of a typical set of (2+1)-dimensional complex modified Korteweg–de Vries equations

Deformed solitons of a typical set of (2+1)-dimensional complex modified Korteweg–de Vries equations

The soliton solutions for semidiscrete complex mKdV equation

The semidiscrete complex modified Korteweg–de Vries equation (semidiscrete cmKdV), which is the second member of the semidiscrete nonlinear Schrődinger hierarchy (Ablowitz–Ladik hierarchy), is solved using the Hirota bilinear formalism. Starting from the focusing case of semidiscrete form of cmKdV, proposed by Ablowitz and Ladik, we construct the bilinear form and build the multi-soliton solutions. The complete integrability of semidiscrete cmKdV, focusing case, is proven and results are discussed.

Deformed solitons of a typical set of (2+1)-dimensional complex modified Korteweg–de Vries equations - invalid DOI

Deformed solitons of a typical set of (2+1)-dimensional complex modified Korteweg–de Vries equations - invalid DOI

Nonlocal Complex Modified Korteweg-de Vries Equations: Reductions and Exact Solutions

We present three different two-dimensional nonlocal integrable cmKdV equations obtained by Ablowitz-Musslimani type of reductions which are respectively T-symmetric, S-symmetric, and ST-symmetric nonlocal cmKdV equations. Ablowiz - Musslimani type of reduction arises from remarkably simple symmetry reductions of general AKNS scattering problems where the nonlocality appears in both space and time or time alone. Moreover, we obtain exact solutions by Darboux transformation.

Vector rational and semi-rational rogue wave solutions in the coupled complex modified Korteweg–de Vries equations

With the aid of a generalized Darboux transformation (DT), we derive a hierarchy of rogue wave solutions to the coupled complex modified Korteweg–de Vries (mKdV) equations. Based on modulation instability (MI), two types of nth order rational rogue wave solutions in compact determinant forms are presented. Especially, the rational rogue wave solutions up to the second-order are performed explicitly and graphically. We find that there exist typical bright–dark composites and rogue wave doublets in first-order case and four or six fundamental rogue waves in second-order case. With appropriate choices of free parameters, the distribution shapes with four fundamental rogue waves admit triangular, quadrilateral, and line structures. Furthermore, triangular, ring, quadrilateral and line patterns can emerge with six fundamental rogue waves. Moreover, we exhibit the first-order semi-rational rogue wave solutions which can demonstrate the coexistence of one rational rogue wave and one breather. Our results can be applicable to the study of rogue wave manifestations in nonlinear optics.

Explicit solutions to a nonlocal 2-component complex modified Korteweg–de Vries equation

A nonlocal 2-component complex modified Korteweg–de Vries equation will be investigated in this paper. We derive the N-fold Darboux transformation in the form of determinants. Based on the plane wave solutions, explicit solutions can be generated via this N-fold Darboux transformation. Moreover, dynamical behaviours of nonsingular solutions are discussed.

An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation

The Hirota bilinear method has been studied in a lot of local equations, but there are few of works to solve nonlocal equations by Hirota bilinear method. In this letter, we show that the nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation admits multiple complex soliton solutions. A variety of exact solutions including the single bright soliton solutions and two bright soliton solutions are derived via constructing an improved Hirota bilinear method for nonlocal complex MKdV equation. From the gauge equivalence, we can see the difference between the solution of nonlocal integrable complex MKdV equation and the solution of local complex MKdV equation.

Do N-soliton breathers exist for the Hirota equation models?

We reveal the method for systematic construction of unexpected bound soliton states (localized breathers on the zero background) and formulate specific conditions for their appearance both in the framework of the canonical and the generalized nonautonomous Hirota models. These conditions show that both all parameters of solitons forming the breather and the dispersion and nonlinearity variations cannot be chosen independently, they are related by the exact integrability of the model. Novel N-soliton breathers derived for the nonautonomous Hirota equation with vanishing boundary conditions differ in that they generally move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain, or losses. We apply the specific conditions for the breather formation to derive novel soliton breather solutions of the nonlinear Schrödinger and the complex modified Korteweg–de Vries equations. Consequently, we demonstrate that the way is opened to generalize and apply the presented method for other evolution equations of the Ablowitz–Kaup–Newell–Segur hierarchy.

Novel conditions for soliton breathers of the complex modified Korteweg–de Vries equation with variable coefficients

We reveal novel unexpected conditions for soliton breathers of the generalized complex modified Korteweg–de Vries equation with variable coefficients and the loss (or gain) term (vc cmKdV). Novel relations between spectral parameters of two solitons giving rise to the breather substantially extend the well-known constraints for canonical soliton breathers. This finding allows us to systematically construct the variety of soliton breather solutions on a zero background of the considered model. These generalized breathers move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. Among other things, we establish that both the standing generalized breathers and envelope breathers exist as well.

Darboux transformation and soliton solution for the (2+1)-dimensional complex modified Korteweg-de Vries equations

In this paper, we consider the (2+1)-dimensional complex modified Korteweg-de Vries equations (CMKdVE). Using the Lax pair, we construct a Darboux transformation and namely one-fold and n-fold transformations. The soliton solutions are obtained from the different “seeds” by using this Darboux transformation.

Transformations between Nonlocal and Local Integrable Equations

AbstractRecently, a number of nonlocal integrable equations, such as the ‐symmetric nonlinear Schrödinger (NLS) equation and ‐symmetric Davey–Stewartson equations, were proposed and studied. Here, we show that many of such nonlocal integrable equations can be converted to local integrable equations through simple variable transformations. Examples include these nonlocal NLS and Davey–Stewartson equations, a nonlocal derivative NLS equation, the reverse space‐time complex‐modified Korteweg–de Vries (CMKdV) equation, and many others. These transformations not only establish immediately the integrability of these nonlocal equations, but also allow us to construct their Lax pairs and analytical solutions from those of the local equations. These transformations can also be used to derive new nonlocal integrable equations. As applications of these transformations, we use them to derive rogue wave solutions for the partially ‐symmetric Davey–Stewartson equations and the nonlocal derivative NLS equation. In addition, we use them to derive multisoliton and quasi‐periodic solutions in the reverse space‐time CMKdV equation. Furthermore, we use them to construct many new nonlocal integrable equations such as nonlocal short pulse equations, nonlocal nonlinear diffusion equations, and nonlocal Sasa–Satsuma equations.

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

Abstract The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg–de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg–de Vries equation.

Superregular breathers in a complex modified Korteweg-de Vries system.

We study superregular (SR) breathers (i.e., the quasi-Akhmediev breather collision with a certain phase shift) in a complex modified Korteweg-de Vries equation. We demonstrate that such SR waves can exhibit intriguing nonlinear structures, including the half-transition and full-suppression modes, which have no analogues in the standard nonlinear Schrödinger equation. In contrast to the standard SR breather formed by pairs of quasi-Akhmediev breathers, the half-transition mode describes a mix of quasi-Akhmediev and quasi-periodic waves, whereas the full-suppression mode shows a non-amplifying nonlinear dynamics of localized small perturbations associated with the vanishing growth rate of modulation instability. Interestingly, we show analytically and numerically that these different SR modes can be evolved from an identical localized small perturbation. In particular, our results demonstrate an excellent compatibility relation between SR modes and the linear stability analysis.