Complex Modified Korteweg De Vries Research Articles

Exact Group Invariant Solutions and Conservation Laws of the Complex Modified Korteweg–de Vries Equation

We study the scalar complex modified Korteweg-de Vries (cmKdV) equation by analyzing a system of partial differential equations (PDEs) from the Lie symmetry point of view. These systems of PDEs are obtained by decomposing the underlying cmKdV equation into real and imaginary components. We derive the Lie point symmetry generators of the system of PDEs and classify them to get the optimal system of one-dimensional subalgebras of the Lie symmetry algebra of the system of PDEs. These subalgebras are then used to construct a number of symmetry reductions and exact group invariant solutions to the system of PDEs. Finally, using the Lie symmetry approach, a couple of new conservation laws are constructed. Subsequently, respective conserved quantities from their respective conserved densities are computed.

Nth-order rogue wave solutions of the complex modified Korteweg–de Vries equation

In this paper, a generalized Darboux transformation for the complex modified Korteweg–de Vries (mKdV) equation is constructed by the Darboux matrix method. As applications, Nth-order wave solutions of the complex mKdV equation have been obtained, part of which exhibit features of rogue waves, such as the first- to third-order rogue waves that are discussed and illustrated through some figures.

Solution of the Complex Modified Korteweg–de Vries equation by the projected differential transform method

In this paper, we first transform the Complex Modified Korteweg–de Vries (CMKdV) equation into a real-valued system of partial differential equations. Then the projected differential transformation method is used to compute an approximate solution of the CMKdV equation. Three illustrative examples are given to show the effectiveness of the method and comparing with the differential transformation method and the variational iteration method.

Multisymplectic box schemes for the complex modified Korteweg–de Vries equation

In this paper, two multisymplectic integrators, an eight-point Preissman box scheme and a narrow box scheme, are considered for numerical integration of the complex modified Korteweg–de Vries equation. Energy and momentum preservation of both schemes and their dispersive properties are investigated. The performance of both methods is demonstrated through numerical tests on several solitary wave solutions.

Stability of periodic traveling waves for complex modified Korteweg–de Vries equation

We study the existence and stability of periodic traveling-wave solutions for complex modified Korteweg–de Vries equation. We also discuss the problem of uniform continuity of the data-solution mapping.

Numerical solution of complex modified Korteweg–de Vries equation by mesh-free collocation method

A mesh-free method based on radial basis functions (RBFs) is proposed for the solution of complex modified Korteweg–de Vries (CMKdV) equation. The mesh-free algorithm is based on scattered data interpolation along with approximating functions known as radial basis functions (RBFs). A set of scattered nodes provided by initial data is used for solution of the problem. Computational experiments examine the accuracy of the method in terms of error norms L2, L∞ and the three invariants I1, I2 and I3. The results obtained by the method are in good agreement with exact solutions and the earlier work.

Solitary wave simulations of Complex Modified Korteweg–de Vries Equation using differential quadrature method

Complex Modified Korteweg–deVries Equation is solved numerically using differential quadrature method based on cosine expansion. Three test problems, motion of single solitary wave, interaction of solitary waves and wave generation, are simulated. The accuracy of the method is measured via the discrete root mean square error norm L 2 , maximum error norm L ∞ for the motion of single solitary wave since it has an analytical solution. A rate of convergency analysis for motion of single solitary wave containing both real and imaginary parts is also given. Lowest three conserved quantities are computed for all test problems. A comparison with some earlier works is given.

Integrable systems related to Su–Schrieffer–Heeger lattices

We present a novel approach to nonlinear lattices with fermions based on the exact solutions for the equations of motion, where fermions and bosons are treated as Grassmannian anticommuting and commuting classical fields. It is shown that the Su–Schrieffer–Heeger equations of motion can be solved exactly in continuum limit using Hirota bilinear formalism and bright and dark solitons appear for phononic and fermionic fields. The bosonic soliton has a tail due to the presence of fermions and the fermionic density has a strongly localised shape. Moreover, applying the multiple scales method, the emergent Grassmannian amplitude nonlinear equation is the complex modified Korteweg de Vries and at the resonance the fermionic version of Yajima–Oikawa system. As an interesting offshoot this system turns out to be a new completely integrable one having N-dark soliton solution.

Estimates of low Sobolev norms of solutions for some nonlinear evolution equations

In this work we obtain results on the estimates of low Sobolev norms for solutions of some nonlinear evolution equations, in particular we apply our method for the complex modified Korteweg–de Vries type equation and Benjamin–Ono equation.

Solitary wave solutions of the CMKdV equation by using the quintic B-spline collocation method

In this paper, the method based on the collocation method with quintic B-spline finite elements is set up to simulate the solitary wave solution of the complex modified Korteweg–de Vries (CMKdV) equation. The Crank–Nicolson central differencing scheme has been used for the time integration and quintic B-spline functions have been used for the space integration. Propagation of the solitary wave and the interaction of two solitary waves are studied.

Extramedullary plasmacytoma of maxillary sinus in 10-year-old girl: A rare presentation

Recently, Anjan Kundu, Abhik Mukherjee and Tapan Naskar (KMN) have introduced a (2+1)-dimensional equation as a new extension of the well-known nonlinear Schrödinger (NLS) equation, which is called KMN equation in this paper. We provide a triplet Lax pair of the KMN equation. Basing on triplet Lax pair, we present the first-order rogue wave (RW) solution of the KMN equation by the one-fold Darboux transformation from a non-zero “seed” solution. This RW solution also satisfies the complex modified Korteweg–de Vries (mKdV) and the NLS equation by two different transformations of variables. Moreover, we discuss the localization of rogue wave of KMN equation, observing that the area of rogue wave is a constant which is just related to the amplitude of seed solution.

Numerical solution of complex modified Korteweg–de Vries equation by Petrov–Galerkin method

In this paper, a Petrov–Galerkin method is used to derive a numerical scheme for the complex modified Korteweg–de Vries equation (CMKdV), where we have used cubic B-splines as test functions and linear B-splines as trial functions. An implicit midpoint rule is used to discretize in time. A block non-linear pentadiagonal system is obtained. We solve this system by Newton’s method. The resulting scheme has a fourth order accuracy in space direction and second order in time direction. It is unconditionally stable by von Neumann method. The exact soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the scheme. The interaction of two solitary waves for different parameters is discussed.

Numerical solution of complex modified Korteweg-de Vries equation by collocation method

The collocation method using quintic B-spline is derived for solving the complex modified Korteweg-de Vries (CMKdV). The method is based on Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The von Neumann stability is used to prove that the scheme is unconditionally stable. Newton’s method is used to solve the nonlinear block pentadiagonal system obtained. Numerical tests for single, two, and three solitons are used to assess the performance of the proposed scheme.

Soliton dynamics in the complex modified KdV equation with a periodic potential

We study the existence and stability of stationary and moving solitary waves in a periodically modulated system governed by an extended cmKdV (complex modified Korteweg–de Vries) equation. The proposed equation describes, in particular, the co-propagation of two electromagnetic waves with different amplitudes and orthogonal linear polarizations in a liquid crystal waveguide, the stronger (nonlinear) wave actually carrying the soliton, while the other (a nearly linear one) creates an effective periodic potential. A variational analysis predicts solitons pinned at minima and maxima of the periodic potential, and the Vakhitov–Kolokolov criterion predicts that some of them may be stable. Numerical simulations confirm the existence of stable stationary solitary waves trapped at the minima of the potential, and show that persistently moving solitons exist too. The dynamics of pairs of interacting solitons is also studied. In the absence of the potential, the interaction is drastically different from the behavior known in the NLS (nonlinear Schrödinger) equation, as the force of the interaction between the cmKdV solitons is proportional to the sine, rather than cosine, of the phase difference between the solitons. In the presence of the potential, two solitons placed in one potential well form a persistently oscillating bound state.

Sasa-Satsuma (complex modified Korteweg–de Vries II) and the complex sine-Gordon II equation revisited: Recursion operators, nonlocal symmetries, and more

We present a new symplectic structure and a hereditary recursion operator for the Sasa-Satsuma equation which is widely used in nonlinear optics. Using an integrodifferential substitution relating this equation to a third-order symmetry flow of the complex sine-Gordon II equation enabled us to find a hereditary recursion operator and higher Hamiltonian structures for the latter equation. We also show that both the Sasa-Satsuma equation and the third-order symmetry flow for the complex sine-Gordon II equation are bi-Hamiltonian systems, and we construct several hierarchies of local and nonlocal symmetries for these systems.

Standard and embedded solitons in nematic optical fibers.

A model for a non-Kerr cylindrical nematic fiber is presented. We use the multiple scales method to show the possibility of constructing different kinds of wave packets of transverse magnetic modes propagating through the fiber. This procedure allows us to generate different hierarchies of nonlinear partial differential equations which describe the propagation of optical pulses along the fiber. We go beyond the usual weakly nonlinear limit of a Kerr medium and derive a complex modified Korteweg-de Vries equation (CM KdV) which governs the dynamics for the amplitude of the wave packet. In this derivation the dispersion, self-focussing, and diffraction in the nematic fiber are taken into account. It is shown that this CM KdV equation has two-parameter families of bright and dark complex solitons. We show analytically that under certain conditions, the bright solitons are actually double-embedded solitons. We explain why these solitons do not radiate at all, even though their wave numbers are contained in the linear spectrum of the system. We study (numerically and analytically) the stability of these solitons. Our results show that these embedded solitons are stable solutions, which is an interesting property since in most systems the embedded solitons are weakly unstable solutions. Finally, we close the paper by making comments on the advantages as well as the limitations of our approach, and on further generalizations of the model and method presented.

A split-step Fourier method for the complex modified Korteweg-de Vries equation

In this study, the complex modified Korteweg-de Vries (CMKdV) equation is solved numerically by three different split-step Fourier schemes. The main difference among the three schemes is in the order of the splitting approximation used to factorize the exponential operator. The space variable is discretized by means of a Fourier method for both linear and nonlinear subproblems. A fourth-order Runge-Kutta scheme is used for the time integration of the nonlinear subproblem. Classical problems concerning the motion of a single solitary wave with a constant polarization angle are used to compare the schemes in terms of the accuracy and the computational cost. Furthermore, the interaction of two solitary waves with orthogonal polarizations is investigated and particular attention is paid to the conserved quantities as an indicator of the accuracy. Numerical tests show that the split-step Fourier method provides highly accurate solutions for the CMKdV equation.

Perturbation theory for nearly integrable multicomponent nonlinear PDEs

The Riemann–Hilbert problem associated with the integrable PDE is used as a nonlinear transformation of the nearly integrable PDE to the spectral space. The temporal evolution of the spectral data is derived with account for arbitrary perturbations and is given in the form of exact equations, which generate the sequence of approximate ordinary differential equations in successive orders with respect to the perturbation. For vector nearly integrable PDEs, embracing the vector nonlinear Schrödinger and complex modified Korteweg–de Vries equations, the main result is formulated in a theorem. For a single vector soliton the evolution equations for the soliton parameters and first-order radiation are given in explicit form.

Stability properties of solitary waves in a complex modified KdV system

Solitary wave solutions are studied for a system of coupled complex modified Korteweg-de Vries (KdV) equations with perturbations that break the Hamiltonian symmetry, and parameter regimes corresponding to linear instability and stability of the solitary waves are found. In addition, perturbation formulas are found for eigenvalues bifurcating from zero. The relative strengths of self- and cross-phase modulation determine the linear stability of the waves; in the non-Hamiltonian case, the location of bifurcating eigenvalues suggests a possible regime of bistability, where both “vector” (two-channel) and “scalar” (one-channel) solitary waves may persist. The proof requires the use of recently developed perturbation formulas based on the Evans function, as well as other spectral-theoretic tools.

Nonlinear Transverse Waves in a Generalized Elastic Solidand the Complex Modified Korteweg–deVries Equation

By using an asymptotic expansion technique based on the Lagrangiandensity formulation, it is demonstrated that the nonlinear interactionbetween two transverse waves propagating in a generalized elasticsolid is described asymptotically by the complex modifiedKorteweg–deVries (CMKdV) equation. Also, the CMKdV equation isshown to be Hamiltonian and its solitary wave solutions areinvestigated by means of a modified Hirota method.