Complex Modified Korteweg De Vries Equation Research Articles

Multiple solitons and breathers on periodic backgrounds in the complex modified Korteweg–de Vries equation

This study explores multiple soliton and breather solutions on periodic backgrounds in the complex modified Korteweg–de Vries equation. The compact determinant formulas and their detailed derivation process for these solutions are provided via the bilinear method. We confirm that on periodic backgrounds, soliton amplitudes exhibit regular periodic behaviors, while breather amplitudes display quasi-periodic behaviors, as is expected for a breather with one period propagating over a periodic wave with another period. The asymptotic expressions for the solitons and breathers, which establish the high accuracy of the derived solutions, are provided to reveal the soliton and breather dynamics on the periodic backgrounds.

The Cauchy Problem for the Nonlinear Complex Modified Korteweg-de Vries Equation with Additional Terms in the Class of Periodic Infinite-Gap Functions

The Cauchy Problem for the Nonlinear Complex Modified Korteweg-de Vries Equation with Additional Terms in the Class of Periodic Infinite-Gap Functions

Multi-soliton solutions, breather-like and bound-state solitons for complex modified Korteweg–de Vries equation in optical fibers

Under investigation in this paper is a complex modified Korteweg–de Vries (KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.

Evolution of dispersive shock waves to the complex modified Korteweg–de Vries equation with higher-order effects

In this paper, new dispersive shock waves (DSWs) in step-like initial value problems to the complex modified Korteweg–de Vries (cmKdV) equation with higher-order effects are found via Whitham modulation theory. For the aforementioned equation, the 1-genus and 2-genus periodic solutions and the associated Whitham equations which are used to describe DSWs are firstly given by the finite-gap integration method, and we also analyze nine types of rarefaction waves appearing before DSWs under the 0-genus Whitham equations. Subsequently, the DSW solutions with step-like initial data are discussed, where we acquire some DSW structures that have not been previously proposed. These notable new results include 1-genus DSW satisfying that one Riemann invariant is constant and the other three are variables and 2-genus DSW in the DSW solutions with one step-like initial data, as well as 3-genus DSW resulting from the collision to 1-genus and 2-genus or two 2-genus DSWs propagating toward each other in the possible DSW solutions with two step-like initial data. Ultimately, the dam break problem is explored to demonstrate the significant physical application of the theoretical findings.

Soliton solutions of a novel nonlocal Hirota system and a nonlocal complex modified Korteweg–de Vries equation

This paper proposes a novel nonlocal Hirota system by adding constraints to the coupled Hirota equation, which can be transformed into a nonlocal complex modified Korteweg–de Vries (mKdV) equation with a Galilean transformation. Since the nonlocal Hirota equation and nonlocal complex mKdV equation can be converted into each other, we only need to solve one of them. Starting from the eigenfunctions of the Lax pair and the adjoint Lax pair of the nonlocal complex mKdV equation, the N-fold Darboux transformation (DT) of this equation is constructed. Subsequently, bright soliton solutions, double-hump soliton solutions, breather-like solutions under zero background, and multi-breather solutions under nonzero constant background are obtained for the nonlocal complex mKdV equation. Besides that, the interaction behaviors of the multi-soliton waves are also studied. There are some interesting phenomenons. The breather-II soliton wave, that can be seen as a combination of bright wave and dark wave, will appear. It does not appear alone but in the multi-breather solution.

Riemann–Hilbert approach to the focusing and defocusing nonlocal complex modified Korteweg–de Vries equation with step-like initial data

Recently, research about nonlocal integrable systems has become a popular topic. Here, we mainly use the Riemann–Hilbert (RH) approach to discuss the nonlocal complex modified Korteweg–de Vries (cmKdV) equation with step-like initial value. That is the Cauchy problem, i.e., we establish the analytical relation between the solutions r(z, t), r(−z, −t) of the nonlocal cmKdV equation and the solution of a matrix RH problem. First, we analyze the eigenfunctions of the linear spectral problem of the nonlocal cmKdV equation. Second, we discuss the scattering matrix T(ɛ) and its spectral functions α1(ɛ), β(ɛ) and α2(ɛ) depending on the prescribed step-like initial value. Finally, we find that the solution of the Cauchy problem of the nonlocal cmKdV equation can be represented by the solution of the corresponding matrix RH problem.

Solitons and dynamics for the shifted reverse space–time complex modified Korteweg–de Vries equation

Solitons and dynamics for the shifted reverse space–time complex modified Korteweg–de Vries equation

Two-component complex modified Korteweg–de Vries equations: New soliton solutions from novel binary Darboux transformation

Two-component complex modified Korteweg–de Vries equations: New soliton solutions from novel binary Darboux transformation

Rogue waves, modulation instability of the (2+1)-dimensional complex modified Korteweg-de Vries equation on the periodic background

In this paper, we construct rogue periodic wave solutions (rogue wave solutions under the Jacobian elliptic dn- and cn-periodic waves background) and the modulation instability of a (2+1)-dimensional complex modified Korteweg–de Vries (CMKdV) equation. Under the linear stability analysis, we reveal that modulation instability is the condition for the existence of rogue waves. Next, we derive the expression of eigenvalues and corresponding squared periodic eigenfunctions by the method of nonlinearization of the Lax pair. Then, we construct the periodic and non-periodic solution of the Lax pair. At last, we present the rogue periodic wave solutions of the (2+1)-dimensional CMKdV equation by the Darboux transformation (DT) method, and analyze the non-linear dynamics in some figures.

The orbital stability of the periodic traveling wave solutions to the defocusing complex modified Korteweg–de Vries equation

The stability of the elliptic solutions to the defocusing complex modified Korteweg–de Vries (cmKdV) equation is studied. Using the integrability of the defocusing cmKdV equation, we prove the spectral stability of the elliptic solutions. We show that one special linear combination of the first seven conserved quantities produces a Lyapunov functional, which implies that the elliptic solutions are orbitally stable with respect to the subharmonic perturbations.

Positon and hybrid solutions for the (2+1)-dimensional complex modified Korteweg–de Vries equations

Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method is used to derive several positon and hybrid solutions for the (2+1)-dimensional complex modified Korteweg–de Vries equations. Based on the zero seed solution, the positon solution and the hybrid solutions of positon and soliton are constructed. The composition of positons is studied, showing that multi-positons of (2+1)-dimensional equations are decomposed into multi-solitons as well as the (1+1)-dimensions. Moreover, the interactions between positon and soliton are analyzed. In addition, the hybrid solutions of b-positon and breather are obtained using the plane wave seed solution, and their evolutions with time are discussed.

Rational soliton solutions in the nonlocal coupled complex modified Korteweg–de Vries equations

Abstract In this article, our work oversees with the nonlocal coupled complex modified Korteweg–de Vries equations (cmKdV), which is a nonlocal generalization for coupled cmKdV equations. The n-fold Darboux transformation (DT) is constructed in the form of determinants for the nonlocal coupled cmKdV equations. Via generalized DT method, we obtain the rational soliton solutions describing M-shaped soliton, W-shaped soliton, and the interactions on the plane wave and periodic background. The results can be useful to study the dynamical behaviors of soliton solutions in nonlocal wave models.

Soliton solution to the complex modified Korteweg–de Vries equation on both zero and nonzero background via the robust inverse scattering method

In this paper, based on the robust inverse scattering method, we construct two kinds of solutions to the focusing modified Korteweg–de Vries equation. One is the classical soliton solution under the zero background condition and the other one is given through the nonzero background. Especially, for the nonzero background case, we choose a special spectral parameter such that the nonzero background solution is changed into the rational travelling waves. Finally, we also give a simple analysis of the soliton as the time is large, then we give the comparison between the exact solution and the asymptotic solution.

Interaction Behaviours between Soliton and Cnoidal Periodic Waves for Nonlocal Complex Modified Korteweg–de Vries Equation

The reverse space-time nonlocal complex modified Kortewewg–de Vries (mKdV) equation is investigated by using the consistent tanh expansion (CTE) method. According to the CTE method, a nonauto-Bäcklund transformation theorem of nonlocal complex mKdV is obtained. The interactions between one kink soliton and other different nonlinear excitations are constructed via the nonauto-Bäcklund transformation theorem. By selecting cnoidal periodic waves, the interaction between one kink soliton and the cnoidal periodic waves is derived. The specific Jacobi function-type solution and graphs of its analysis are provided in this paper.

Rational solutions of multi‐component nonlinear Schrödinger equation and complex modified KdV equation

In this paper, the crucial condition to achieve rational solutions of the multi‐component nonlinear Schrödinger equation is proposed by introducing two nilpotent Lax matrices. Taking the series multisections of the vector eigenfunction as a set of fundamental eigenfunctions, an explicit formula of the nth‐order rational solution is obtained by the degenerate Darboux transformation, which is used to generate some new patterns of rogue waves. A conjecture about the degree of the nth‐order rogue waves is summarized. This conjecture also holds for rogue waves of the multi‐component complex modified Korteweg–de Vries equation. Finally, the semi‐rational solutions of the Manakov system are discussed.

Breather-soliton molecules and breather-positons for the extended complex modified KdV equation

In this paper, we investigate an extended complex modified Korteweg–de Vries equation. On zero background, based on Darboux transformation, breather solutions, breather molecules, breather-soliton molecules and breather-positons for extended complex modified Korteweg–de Vries equation are obtained by module resonance, velocity resonance and degenerate Darboux transformation. On nonzero background, breather solutions and breather-positons for extended complex modified Korteweg–de Vries equation are generated through Darboux transformation and degenerate Darboux transformation, respectively.

Soliton molecules, rational positons and rogue waves for the extended complex modified KdV equation

In this paper, we consider the integrable extended complex modified Korteweg–de Vries equation. Based on Darboux transformation, we obtain soliton molecules, positon solutions, rational positon solutions and rogue waves for integrable extended complex modified Korteweg–de Vries equation. Further, under the standard decomposition, we divide the rogue waves into three patterns: fundamental pattern, triangular pattern and ring pattern. On the basis of fundamental pattern, we define the length and width of rogue waves and discuss the effect of different parameters on rogue waves.

Rogue waves on an elliptic function background in complex modified Korteweg–de Vries equation

With the assistance of one fold Darboux transformation formula, we derive rogue wave solutions of the complex modified Korteweg–de Vries equation on an elliptic function background. We employ an algebraic method to find the necessary squared eigenfunctions and eigenvalues. To begin we construct the elliptic function background. Then, on top of this background, we create a rogue wave. We demonstrate the outcome for three distinct elliptic modulus values. We find that when we increase the modulus value the amplitude of rogue waves on the dn-periodic background decreases whereas it increases in the case of cn-periodic background.

Linearly implicit energy-preserving Fourier pseudospectral schemes for the complex modified Korteweg-de Vries equation

We 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. doi:10.1017/S1446181120000218

Numerical simulation of the soliton solutions for a complex modified Korteweg–de Vries equation by a finite difference method

In this paper, a Crank–Nicolson-type finite difference method is proposed for computing the soliton solutions of a complex modified Korteweg–de Vries (MKdV) equation (which is equivalent to the Sasa–Satsuma equation) with the vanishing boundary condition. It is proved that such a numerical scheme has the second-order accuracy both in space and time, and conserves the mass in the discrete level. Meanwhile, the resulting scheme is shown to be unconditionally stable via the von Nuemann analysis. In addition, an iterative method and the Thomas algorithm are used together to enhance the computational efficiency. In numerical experiments, this method is used to simulate the single-soliton propagation and two-soliton collisions in the complex MKdV equation. The numerical accuracy, mass conservation and linear stability are tested to assess the scheme’s performance.