Complex Modified KdV Equation Research Articles

Wave breaking, dispersive shock wave, and phase shift for the defocusing complex modified KdV equation.

We study the problem of wave breaking for a simple wave propagating to a quiescent medium in the framework of the defocusing complex modified KdV (cmKdV) equation. It is assumed that a cubic root singularity is formed at the wave-breaking point. The dispersive regularization of wave breaking leads to the generation of a dispersive shock wave (DSW). We describe the DSW as a modulated periodic wave in the framework of the Gurevich-Pitaevskii approach based on the Whitham modulation theory. The generalized hodograph method is used to solve the Whitham equations, and the boundaries of the DSW are found. Most importantly, we determine the correct phase shift for the DSW from the generalized phase relationships and the modified Gurevich-Pitaevskii matching conditions, so that a complete description of the DSW is obtained rather than just its envelope. All of our analytical predictions agree well with the numerical simulations.

Evolution of initial discontinuities in a particular case of two-step initial problem for the defocusing complex modified KdV equation

In this paper, the complete classification of solutions of defocusing complex modified KdV equation with a particular case of two-step initial condition is investigated by the finite-gap integration approach and Whitham modulation theory. The periodic wave solution and corresponding Whitham modulation equations for zero-phase, one-phase, two-phase are found. The self-similar wave structures in each case are composed of different building blocks: the plateau, vacuum, rarefaction wave, dispersive shock wave and two-genus region, which are studied in detail. The results of direct numerical simulations for the defocusing complex modified KdV equation agree well with those of Whitham modulation theory.

-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.

Rogue waves on the periodic background in the complex modified KdV equation with higher-order effects

We construct rogue wave solutions of the complex modified Korteweg–de Vries (cmKdV) equation with higher-order effects on the periodic background. The rogue waves are obtained by nonlinearization of spectral problem associated with the travelling periodic waves and using the one-fold Darboux transformation. We consider the dnoidal and cnoidal travelling periodic waves as the seed solutions. Since the dnoidal and cnoidal travelling periodic waves are modulationally unstable as regards long wave perturbations, the rogue waves generated on the periodic background.

The Whitham Modulation Solution of the Complex Modified KdV Equation

This paper primarily concerns the Whitham modulation equation of the complex modified Korteweg–de Vries (cmKdV) equation with a step-like initial value. By utilizing the Lax pair, we derive the N-genus Whitham equations via the averaging method. The Whitham equation can be integrated using the hodograph transformation. We investigate Krichever’s algebro-geometric scheme to propose the averaging method for the cmKdV integrable hierarchy and obtain the Whitham velocities of the integrable hierarchy and the hodograph transformation. The connection between the equations of the Euler–Poisson–Darboux type linear overdetermined system, which determines the solutions of the hodograph transformation, is constructed through Riemann integration, which demonstrates that the Whitham equation can be solved. Finally, a step-like initial value problem is solved and an exotic wave pattern is discovered. The results of direct numerical simulation agree well with the Whitham theory solution, which shows the validity of the theory.

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.

Rogue Waves on the Periodic Background in the Complex Modified Kdv Equation with Higher-Order Effects

Rogue Waves on the Periodic Background in the Complex Modified Kdv Equation with Higher-Order Effects

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.

The Complete Classification of Solutions to the Riemann Problem of the Defocusing Complex Modified KdV Equation

The Complete Classification of Solutions to the Riemann Problem of the Defocusing Complex Modified KdV Equation

Gradient-optimized physics-informed neural networks (GOPINNs): a deep learning method for solving the complex modified KdV equation

Recently, the physics-informed neural networks (PINNs) have received more and more attention because of their ability to solve nonlinear partial differential equations via only a small amount of data to quickly obtain data-driven solutions with high accuracy. However, despite their remarkable promise in the early stage, their unbalanced back-propagation gradient calculation leads to drastic oscillations in the gradient value during model training, which is prone to unstable prediction accuracy. Based on this, we develop a gradient optimization algorithm, which proposes a new neural network structure and balances the interaction between different terms in the loss function during model training by means of gradient statistics, so that the newly proposed network architecture is more robust to gradient fluctuations. In this paper, we take the complex modified KdV equation as an example and use the gradient-optimized PINNs (GOPINNs) deep learning method to obtain data-driven rational wave solution and soliton molecules solution. Numerical results show that the GOPINNs method effectively smooths the gradient fluctuations and reproduces the dynamic behavior of these data-driven solutions better than the original PINNs method. In summary, our work provides new insights for optimizing the learning performance of neural networks and improves the prediction accuracy by a factor of 10 to 30 when solving the complex modified KdV equation.

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.

On the solution of certain higher-order local and nonlocal nonlinear equations in optical fibers using Kudryashov's approach

Using Kudryashov's method we construct soliton solutions of a number of local partial differential equations relevant to optical fibers such as the Hirota equation and a fourth-order nonlinear PDE applicable for femtosecond pulses. We also explore the nature of transformations between such equations and their nonlocal counterparts and explicitly illustrate this connection for the complex modified KdV equation.

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.

Novel soliton molecules and breather-positon on zero background for the complex modified KdV equation

Based on Darboux transformation, we generate breather solutions sitting on zero background via module resonance for complex modified KdV equation. Then, some novel soliton molecules, breather molecules and breather-soliton molecules are obtained theoretically by module resonance and velocity resonance. Further, we study the higher-order breather-positons with zero background on the basis of degenerate Darboux transformation and module resonance for the first time. We discuss the dynamics of higher-order breather-positons in detail and propose related propositions. Finally, we find new elastic interactions between smooth positons and breather-positons by means of degenerate Darboux transformation and partial module resonance.

Dynamics of various waves in nonlinear Schrödinger equation with stimulated Raman scattering and quintic nonlinearity

In this work, we consider a nonlinear Schrodinger equation with stimulated Raman scattering and quintic nonlinearity, which works as a model for the propagation of ultrashort pulse in optical fiber and also corresponds to one-dimensional anisotropic Heisenberg ferromagnetic spin chain. Introducing a Galilean transformation, we transform the model into a fifth-order complex modified KdV equation, the integrability of which has been considered in the AKNS technique framework. The N-fold Darboux transformation for the model is derived in terms of the gauge transformation of the associated $$3 \times 3$$ matrix spectral problem. As applications, abundant intriguing types of nonlinear waves are obtained in zero and nonzero boundary conditions. The dynamical evolution of these nonlinear waves can be well controlled under stimulated Raman scattering and quintic nonlinearity management. Beside that, we also find some novel and interesting conversion phenomena, where a high-order term plays a pivotal role.

Soliton molecules and novel smooth positons for the complex modified KdV equation

In this research, based on Darboux transformation, a molecule consisting of two identical soliton waves is firstly obtained by velocity resonance for modified KdV equation. And we also get molecules containing a plurality of solitons. Further, we study the elastic interaction between soliton molecules and typical smooth higher-order positon via semi-degenerate Darboux transformation. Last but not least, we find a new type of smooth positons called rational positons. The dynamic properties of higher-order rational positon are discussed in detail and related propositions are given in this paper. The nature of rational positons is fundamentally different from that of typical smooth positons and breather positons. The method used in this paper to get interaction solutions and rational positons can be applied to other integrable equations as well.

Evolution of initial discontinuity for the defocusing complex modified KdV equation

The complete classification of solutions to the defocusing complex modified Korteweg-de Vries (cmKdV) equation with the step-like initial condition is given by Whitham theory. The process of studying the solution of cmKdV equation can be reduced to explore four quasi-linear equations, which predicts the evolution of dispersive shock wave. The results obtained here are quite different from the defocusing nonlinear Schrodinger equation: the bidirectionality of defocusing nonlinear Schrodinger equation determines that there are two basic rarefaction and shock structures while in the cmKdV case three basic rarefaction structures and four basic dispersive shock structures are constructed which lead to more complicated classification of step-like initial condition, and wave patterns even consisted of six different regions while each of wave patterns is consisted of five regions in the defocusing nonlinear Schrodinger equation. Direct numerical simulations of cmKdV equation are agreed well with the solutions corresponding to Whitham theory.

Dynamics of the smooth positons of the complex modified KdV equation

We provide the explicit formulae of the smooth positon solutions of the complex modified KdV (mKdV) equation using degenerate Darboux transformation with respect to the eigenvalues. The dynamics of the smooth positons of the complex mKdV are discussed in details using the method, i.e. decomposition of the modulus square. For this kind solution, we show explicitly the decomposition, bent trajectory and variable ‘phase shift’ after collision, which are remarkably different from the singular positons of the real-valued mKdV equation.

1-Soliton solutions of complex modified KdV equation with time-dependent coefficients

In this paper, we have obtained exact 1-soliton solutions of complex modified KdV equation with variable—coefficients using solitary wave ansatz. Restrictions on parameters of the soliton have been observed in course of the derivation of soliton solutions. Finally, a few numerical simulations of dark and bright solitons have been given.

On the Whitham Equations for the Defocusing Complex Modified KdV Equation

We study the Whitham equations for the defocusing complex modified KdV (mKdV) equation. These Whitham equations are quasi-linear hyperbolic equations and describe the averaged dynamics of the rapid oscillations which appear in the solution of the mKdV equation when the dispersive parameter is small. The oscillations are referred to as dispersive shocks. The Whitham equations for the mKdV equation are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the solutions of the Whitham equations when the initial values are given by a step-like function. We also compare the results with those of the defocusing nonlinear Schrodinger (NLS) equation. For the NLS equation, the Whitham equations are strictly hyperbolic and genuinely nonlinear. We show that the weak hyperbolicity of the mKdV–Whitham equations is responsible for some new structure in the dispersive shocks which has not been found in the NLS case.