Residual symmetries of the modified Korteweg-de Vries equation and its localization

The residual symmetries of the famous modified Korteweg-de Vries (mKdV) equation are researched in this paper. The initial problem on the residual symmetry of the mKdV equation is researched. The residual symmetries for the mKdV equation are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries by means of enlarging the mKdV equations. One-parameter invariant subgroups and the invariant solutions for the extended system are listed. Eight types of similarity solutions and the reduction equations are demonstrated. It is noted that we researched the twofold residual symmetries by means of taking the mKdV equation as an example. Similarity solutions and the reduction equations are demonstrated for the extended mKdV equations related to the twofold residual symmetries.     

Long-Time Asymptotics for the Nonlocal MKdV Equation

In this paper, we study the Cauchy problem with decaying initial data for the nonlocal modified Korteweg-de Vries equation(nonlocal mKdV) qt(x, t)+qxxx(x, t)-6 q(x, t)q(-x,-t)qx(x, t) = 0, which can be viewed as a generalization of the local classical mKdV equation. We first formulate the Riemann-Hilbert problem associated with the Cauchy problem of the nonlocal mKdV equation. Then we apply the Deift-Zhou nonlinear steepest-descent method to analyze the long-time asymptotics for the solution of the nonlocal m KdV equation. In contrast with the classical mKdV equation,we find some new and different results on long-time asymptotics for the nonlocal mKdV equation and some additional assumptions about the scattering data are made in our main results.     

局地外源激发的孤波的增强和演变

利用扰动法由准地转涡度方程导出了强迫mKdV方程，讨论了强迫mKdV孤波的质量和能量的时间演变，并通过拟谱法求得了强迫mKdV方程的数值解。计算结果显示，局地外源强迫激发的mKdV孤波与失谐参数α和外源强度有密切关系。与强迫KdV方程相比，在强迫mKdV方程中，外源强迫可以激发出振幅更大的更不稳定的孤波。     

A Super mKdV Equation: Bosonization,Painlevé Property and Exact Solutions

The symmetry of the fermionic field is obtained by means of the Lax pair of the mKdV equation. A new super mKdV equation is constructed by virtue of the symmetry of the fermionic form. The super mKdV system is changed to a system of coupled bosonic equations with the bosonization approach. The bosonized SmKdV(BSmKdV)equation admits Painlevé property by the standard singularity analysis. The traveling wave solutions of the BSmKdV system are presented by the mapping and deformation method. We also provide other ideas to construct new super integrable systems.     

A new high-order spectral problem of the mKdV and its associated integrable decomposition

A new approach to formulizing a new high-order matrix spectral problem from a normal 2× 2 matrix modified Korteweg--de Vries (mKdV) spectral problem is presented. It is found that the isospectral evolution equation hierarchy of this new higher-order matrix spectral problem turns out to be the well-known mKdV equation hierarchy. By using the binary nonlinearization method, a new integrable decomposition of the mKdV equation is obtained in the sense of Liouville. The proof of the integrability shows that r-matrix structure is very interesting.     

非线性发展方程的Riemann theta 函数等几种新解

为了构造非线性发展方程的复合型无穷序列精确解, 获得了第二种椭圆方程的Riemann theta 函数等几种新解.在此基础上,利用第二种椭圆方程与Riccati方程的Bäcklund变换和解的非线性叠加公式, 借助符号计算系统 Mathematica, 以mKdV方程为应用实例, 构造了该方程的复合型无穷序列新精确解.这里包括Riemann theta 函数、Jacobi椭圆函数、双曲函数、 三角函数和有理函数,通过几种形式构成的复合型无穷序列新精确解. 关键词： 第二种椭圆方程 Riccati方程 非线性发展方程 Riemann theta 函数无穷序列解     

µ -symmetry and µ -conservation law for the extended mKdV equation

In this paper, we obtain µ -symmetry and µ -conservation law of the extended mKdV equation. The extended mKdV equation dose not admit a variational problem since it is of odd order. First we obtain µ -conservation law of the extended mKdV equation in potential form because it admits a variational problem, using it, we can obtain µ -conservation law of the extended mKdV equation.     

The time-dependent Ginzburg-Landau equation for the two-velocity difference model

A thermodynamic theory is formulated to describe the phase transition and critical phenomenon in traffic flow.Based on the two-velocity difference model,the time-dependent Ginzburg-Landau (TDGL) equation under certain condition is derived to describe the traffic flow near the critical point through the nonlinear analytical method.The corresponding two solutions,the uniform and the kink solutions,are given.The coexisting curve,spinodal line and critical point are obtained by the first and second derivatives of the thermodynamic potential.The modified Korteweg de Vries (mKdV) equation around the critical point is derived by using the reductive perturbation method and its kink-antikink solution is also obtained.The relation between the TDGL equation and the mKdV equation is shown.The simulation result is consistent with the nonlinear analytical result.     

Darboux Transformation and Multi-Solitons for Complex mKdV Equation

An explicR N-fold Darboux transformation with multi-parameters for coupled mKdV equation is constructed with the help of a gauge transformation of the Ablowitz-Kaup-Newell-Segur （AKNS） system spectral problem. By using the Darboux transformation and the reduction technique, some multi-soliton solutions for the complex mKdV equation are obtained.     

An integrable coupled Alice–Bob modified Korteweg de-Vries system: Lax pairs,Bäcklund transformations,residual symmetries and exact solutions

ABSTRACT

A coupled Alice–Bob modified Korteweg de-Vries (mKdV) system is established from the mKdV equation in this paper, which is nonlocal and suitable to model two-place entangled events. The Lax integrability of the coupled Alice–Bob mKdV system is proved by demonstrating three types of Lax pairs. By means of the truncated Painlevé expansion, auto-Bäcklund transformation of the coupled Alice–Bob mKdV system and Bäcklund transformation between the coupled Alice–Bob mKdV system and the Schwarzian mKdV equation are demonstrated. Nonlocal residual symmetries of the coupled Alice–Bob mKdV system are researched. To obtain localized Lie point symmetries of residual symmetries, the coupled Alice–Bob mKdV system is extended to a system consisting six equations. Calculation on the prolonged system shows that it is invariant under the scaling transformations, space-time translations, phase translations and Galilean translations. One-parameter group transformation and one-parameter subgroup invariant solutions are obtained. The consistent Riccati expansion (CRE) solvability of the coupled Alice–Bob mKdV system is proved and some interaction structures between soliton–cnoidal waves are obtained by CRE. Moreover, Jacobi periodic wave solutions, solitary wave solutions and singular solutions are obtained by elliptic function expansion and exponential function expansion.     

New Solitary Wave Solution of the Combined KdV and mKdV Equation

This paper presents two different methods forthe construction of new exact solitary wave solutions tothe combined KdV and mKdV equation. There exist 12 typesof soliton solutions which reduce to those of the mKdV equation. There exist three typesof solurion solutions which reduce to those of the KdVequation.     

Breather Lattice Solutions to Negative mKdV Equation

In this paper, dependent and independent variable transformations are introduced to solve the negative mKdV equation systematically by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different kinds of solutions can be obtained to the negative mKdV equation, including breather lattice solution and periodic wave solution.     

Rational Solutions with Non-zero Asymptotics of the Modified Korteweg-de Vries Equation

Using the relation between the mKdV equation and the KdV-mKdV equation, we derive non-singular rational solutions for the mKdV equation. The solutions are given in terms of Wronskians. Dynamics of some solutions is investigated by means of asymptotic analysis. Wave trajectories of high order rational solutions are asymptotically governed by cubic curves.     

A Discrete Lax-Integrable Coupled System Related to Coupled KdV and Coupled mKdV Equations

A modified Korteweg-de Vries （mKdV） lattice is found to be also a discrete Korteweg-de Vries （KdV） equation. A discrete coupled system is derived from the single lattice equation and its Lax pair is proposed. The coupled system is shown to be related to the coupled KdV and coupled mKdV systems which are widely used in physics.     

Adomian Decomposition Method for Nonlinear Differential-Difference Equation

Adomian decomposition method is applied to find the analytical and numerical solutions for the discretized mKdV equation. A numerical scheme is proposed to solve the long-time behavior of the discretized mKdV equation. The procedure presented here can be used to solve other differential-difference equations.     

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.     

Solitary Solution of Discrete mKdV Equation by Homotopy Analysis Method

In this paper, we apply homotopy analysis method to solve discrete mKdV equation and successfully obtain the bell-shaped solitary solution to mKdV equation. Comparison between our solution and the exact solution shows that homotopy analysis method is effective and validity in solving hybrid nonlinear problems, including solitary solution of difference-differential equation.     

Solving Non-Isospectral mKdV Equation and Sine-Gordon Equation Hierarchies with Self-Consistent Sources via Inverse Scattering Transform

N-soliton solutions of the hierarchy of non-isospectral mKdV equation with self-consistent sources and the hierarchy of non-isospectral sine-Gordon equation with self-consistent sources are obtained via the inverse scattering transform.     

Soliton molecules and the CRE method in the extended mKdV equation

The soliton molecules of the(1+1)-dimensional extended modified Korteweg–de Vries(mKdV)system are obtained by a new resonance condition, which is called velocity resonance. One soliton molecule and interaction between a soliton molecule and one-soliton are displayed by selecting suitable parameters. The soliton molecules including the bright and bright soliton, the dark and bright soliton, and the dark and dark soliton are exhibited in figures 1–3, respectively.Meanwhile, the nonlocal symmetry of the extended mKdV equation is derived by the truncated Painlevé method. The consistent Riccati expansion(CRE) method is applied to the extended mKdV equation. It demonstrates that the extended mKdV equation is a CRE solvable system. A nonauto-B?cklund theorem and interaction between one-soliton and cnoidal waves are generated by the CRE method.     

Integrable systems from inelastic curve flows in 2– and 3– dimensional Minkowski space

Integrable systems are derived from inelastic flows of timelike, spacelike, and null curves in 2– and 3– dimensional Minkowski space. The derivation uses a Lorentzian version of a geometrical moving frame method which is known to yield the modified Korteveg-de Vries (mKdV) equation and the nonlinear Schrödinger (NLS) equation in 2– and 3– dimensional Euclidean space, respectively. In 2–dimensional Minkowski space, timelike/spacelike inelastic curve flows are shown to yield the defocusing mKdV equation and its bi-Hamiltonian integrability structure, while inelastic null curve flows are shown to give rise to Burgers’ equation and its symmetry integrability structure. In 3–dimensional Minkowski space, the complex defocusing mKdV equation and the NLS equation along with their bi-Hamiltonian integrability structures are obtained from timelike inelastic curve flows, whereas spacelike inelastic curve flows yield an interesting variant of these two integrable equations in which complex numbers are replaced by hyperbolic (split-complex) numbers.