In this paper, we investigate the anomalous dispersive relations, inverse scattering transform with a Riemann-Hilbert (RH) problem, and fractional multi-solitons of the integrable combined fractional higher-order mKdV (fhmKdV) hierarchy, including the fractional mKdV (fmKdV), fractional fifth-order mKdV (f5mKdV), fractional combined third-fifth-order mKdV (f35mKdV) equations, etc., which can be featured via completeness of squared scalar eigenfunctions of the ZS spectral problem. We construct a matrix RH problem to present three types of fractional N-solitons illustrating anomalous dispersions of the combined fhmKdV hierarchy for the reflectionless case. As some examples, we analyze the wave velocity of the fractional one-soliton such that we find that the fhmKdV equation predicts a power law relationship between the wave velocity and amplitude, and demonstrates the anomalous dispersion. Furthermore, we illustrate other interesting anomalous dispersive wave phenomena containing the elastic interactions of fractional bright and dark solitons, W-shaped soliton and dark soliton, as well as breather and dark soliton. These obtained fractional multi-solitons will be useful to understand the related nonlinear super-dispersive wave propagations in fractional nonlinear media.
The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing isometry and isoenergy conditions on a moduli space of plane loops. The conditions are compared to the constraints that define Euler's elastica. Moreover, the conditions are shown to be constraints on the curvature and other invariants of the loops which appear as coefficients of the generating function for the Faber polynomials.
The paper presents two results. First it is shown how the discrete potential modified KdV equation and its Lax pairs in matrix form arise from the Hirota-Miwa equation by a 2-periodic reduction. Then Darboux transformations and binary Darboux transformations are derived for the discrete potential modified KdV equation and it is shown how these may be used to construct exact solutions.
A one-to-one correspondence between triple Jordan systems and integrable multi-component models of the modified Korteveg--de Vries type is established.
In this note we present explicitly the construction of the mKdV hierarchy and show that it decomposes into positive and negative graded sub-hierarchies. We extend the construction of the Backlund transformation for the sinh-Gordon model to all other positive and negative odd graded equations of motion generated by the same affine algebraic structure. Some simple examples of solutions are explicitly verified to satisfy, in a universal manner, the Backlund transformations for the first few odd (positive and negative) sub-hierarchies.
In this paper, we prove that the nonlocal complex modified Korteweg-de Vries (mKdV) equation introduced by Ablowitz and Musslimani [Nonlinearity, 29, 915-946 (2016)] is gauge equivalent to a spin-like model. From the gauge equivalence, one can see that there exists significant difference between the nonlocal complex mKdV equation and the classical complex mKdV equation. Through constructing the Darboux transformation(DT) for nonlocal complex mKdV equation, a variety of exact solutions including dark soliton, W-type soliton, M-type soliton and periodic solutions are derived.
In this paper we consider the ${\cal N}=1$ supersymmetric mKdV hierarchy composed of positive odd flows embedded within an affine $\hat sl(2,1)$ algebra. Its Bäcklund transformations are constructed in terms of a gauge transformation preserving the zero curvature representation. The recursion operator relating consecutive flows is derived and shown to relate their Bäcklund transformations.
From an algebraic construction of the mKdV hierarchy we observe that the space component of the Lax operator play a role of an universal algebraic object. This fact induces the universality of a gauge transformation that relates two field configurations of a given member of the hierarchy. Such gauge transformation generates the Backlund transformation (BT). In this paper we propose a systematic construction of Backlund Transformation for the entire mKdV hierarchy form the known Type-II BT of the sinh-Gordon theory. We explicitly construct the BT of the first few integrable models associated to…
The integrability of the ${\cal N}=1$ supersymmetric modified Korteweg de-Vries (smKdV) hierarchy in the presence of defects is investigated through the construction of its super Bäcklund transformation. The construction of such transformation is performed by using essentially two methods: the Bäcklund-defect matrix approach and the superfield approach. Firstly, we employ the defect matrix associated to the hierarchy which turns out to be the same for the supersymmetric sinh-Gordon (sshG) model. The method is general for all flows and as an example we derive explicitly the Bäcklund equations i…
We consider the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the affine Lie algebra $A^2_2$. We describe how the critical points of this population define rational solutions of the equations of the mKdV hierarchy associated with $A^2_2$.
It is well known that the nonlinear Schrödinger (NLS) equation is a very important integrable equation. Ablowitz and Musslimani introduced and investigated an integrable nonlocal NLS equation through inverse scattering transform. Very recently, we proposed an integrable nonlocal modified Korteweg-de Vries equation (mKdV) which can also be found in a paper of Ablowitz and Musslimani. We have constructed the Darboux transformation and soliton solutions for the nonlocal mKdV equation. In this paper, we will investigate further the nonlocal mKdV equation. We will give its exact solutions including…
Superpositions of hierarchies of integrable equations are also integrable. The superposed equations, such as the Hirota equations in the AKNS hierarchy, cannot be considered as new integrable equations. Furthermore if one applies the Hirota bilinear method to these equations one obtains the same $N$-soliton solutions of the generating equation which differ only by the dispersion relations. Similar discussions can be made for the locally and nonlocally reduced equations as well. We give, as an example, AKNS system of equations in $(1+1)$-dimensions.
We consider the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra $A^{(2)}_{2n}$. The population is naturally partitioned into an infinite collection of complex cells $\mathbb{C}^m$, where $m$ are some positive integers. For each cell we define an injective rational map $\mathbb{C}^m \to M(A^{(2)}_{2n})$ of the cell to the space $M(A^{(2)}_{2n})$ of Miura opers of type $A^{(2)}_{2n}$. We show that the image of the map is invariant with respect to all mKd…
We consider the population of critical points, generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra $C_n^{(1)}$. The population is naturally partitioned into an infinite collection of complex cells $\mathbb C^m$, where $m$ are positive integers. For each cell we define an injective rational map $\mathbb C^m \to \mathcal M(C_n^{(1)})$ of the cell to the space $\mathcal M(C_n^{(1)})$ of Miura opers of type $C_n^{(1)}$. We show that the image of the map is invariant with respect to all mKd…
Bi-Hamiltonian hierarchies of soliton equations are derived from geometric non-stretching (inelastic) curve flows in the Hermitian symmetric spaces $SU(n+1)/U(n)$ and $SO(2n)/U(n)$. The derivation uses Hasimoto variables defined by a moving parallel frame along the curves. As main results, new integrable multi-component versions of the Sine-Gordon (SG) equation and the modified Korteveg-de Vries (mKdV) equation, as well as a novel nonlocal multi-component version of the nonlinear Schrödinger (NLS) equation are obtained, along with their bi-Hamiltonian structures and recursion operators. These …
Under investigation in this paper is the nonisospectral and variable coefficients modified Kortweg-de Vries (vc-mKdV) equation, which manifests in diverse areas of physics such as fluid dynamics, ion acoustic solitons and plasma mechanics. With the degrees of restriction reduced, a simplified constraint is introduced, under which the vc-mKdV equation is an integrable system and the spectral flow is time-varying. The Darboux transformation for such equation is constructed, which gives rise to the generation of variable kinds of solutions including the double-breather coherent structure, periodi…
In this paper, we consider Cauchy problem for the modified Korteweg-de Vries hierarchy on the real line with decaying initial data. Using the Riemann--Hilbert formulation and nonlinear steepest descent method, we derive a uniform asymptotic expansion to all orders in powers of $t^{-1/(2n+1)}$ with smooth coefficients of the variable $(-1)^{n+1}x((2n+1) t)^{-1/(2n+1)}$ in the self-similarity region for the solution of $n$-th member of the hierarchy. It turns out that the leading asymptotics is described by a family of special solutions of the Painlevé II hierarchy, which generalize the classica…
Bäcklund Transformations of MKdV and Painlevé Equations
对于$N\ge 3$，在标准MKdV方程的解空间上的$SL(N) MKdV hierarchy, generalizing the two $ Z_2$作用中，第一非平凡方程的解空间上有$S_N$和$D_N$作用.这些作用在标度缩减中幸存下来，并产生了某些常微分方程（系统）的变换群，包括第二、第四和第五潘勒韦方程。
We present a new variational characterization of breather solutions of any equation of the \emph{focusing} Gardner hierarchy. This hierarchy is characterized by a nonnegative index $n$, and $2n+1$ represents the order of the corresponding PDE member. In this paper, we first show the existence of such breathers, and that they are solutions of the (2n+1)th-order Gardner equation. Then we prove a \emph{variational universality property}, in the sense that all these breather solutions satisfy the \emph{same} fourth order stationary elliptic ODE, regardless the order of the hierarchy member. This f…
Relations between triple Jordan systems and integrable multi-component models of the modified Korteveg--de Vries type are established. The most general model is related to a pair consisting of a triple Jordan system and a skew-symmetric bilinear operation. If this operation is a Lie bracket, then we arrive at the Lie-Jordan algebras
We study existence of helical solitons in the vector modified Korteweg-de Vries (mKdV) equations, one of which is integrable, whereas another one is non-integrable. The latter one describes nonlinear waves in various physical systems, including plasma and chains of particles connected by elastic springs. By using the dynamical system methods such as the blow-up near singular points and the construction of invariant manifolds, we construct helical solitons by the efficient shooting method. The helical solitons arise as the result of co-dimension one bifurcation and exist along a curve in the ve…
Inspired by the forms of delay-Painleve equations, we consider some new differential-discrete systems of KdV, mKdV and Sine-Gordon - type related by simple one way Miura transformations to classical ones. Using Hirota bilinear formalism we construct their new integrable discretizations, some of them having higher order. In particular, by this procedure, we show that the integrable discretization of intermediate sine-Gordon equation is exactly lattice mKdV and also we find a bilinear form of the recently proposed lattice Tzitzeica equation. Also the travelling wave reduction of these new lattic…
In this paper, we systematically study the integrability and data-driven solutions of the nonlocal mKdV equation. The infinite conservation laws of the nonlocal mKdV equation and the corresponding infinite conservation quantities are given through Riccti equation. The data driven solutions of the zero boundary for the nonlocal mKdV equation are studied by using the multi-layer physical information neural network algorithm, which including kink soliton, complex soliton, bright-bright soliton and the interaction between soliton and kink-type. For the data-driven solutions with non-zero boundary,…
The unified transform method introduced by Fokas can be used to analyze initial-boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann-Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of $L^2$-RH problems, we consider the construction…
The construction of Miura and Bäcklund transformations for $A_n$ mKdV and KdV hierarchies are presented in terms of gauge transformations acting upon the zero curvature representation. As in the well known $sl(2)$ case, we derive and relate the equations of motion for the two hierarchies. Moreover, the Miura-gauge transformation is not unique, instead, it is shown to be connected to a set of generators labeled by the exponents of $A_n$ The construction of generalized gauge-Bäcklund transformation for the $A_n$-KdV hierarchy is obtained as a composition of Miura and Bäcklund-gauge transformatio…
We identify the self-similarity limit of the second flow of $sl(N)$ mKdV hierarchy with the periodic dressing chain thus establishing % a connection to $A^{(1)}_{N-1}$ invariant Painlevé equations. The $A^{(1)}_{N-1}$ Bäcklund symmetries of dressing equations and Painlevé equations are obtained in the self-similarity limit of gauge transformations of the mKdV hierarchy realized as zero-curvature equations on the loop algebra $\widehat{sl}(N)$ endowed with a principal gradation.
