We study higher order KdV equations from the GL(2,$\mathbb{R}$) $\cong$ SO(2,1) Lie group point of view. We find elliptic solutions of higher order KdV equations up to the ninth order. We argue that the main structure of the trigonometric/hyperbolic/elliptic $N$-soliton solutions for higher order KdV equations is the same as that of the original KdV equation. Pointing out that the difference is only the time dependence, we find $N$-soliton solutions of higher order KdV equations can be constructed from those of the original KdV equation by properly replacing the time-dependence. We discuss that there always exist elliptic solutions for all higher order KdV equations.
Regarding $N$-soliton solutions, the trigonometric type, the hyperbolic type, and the exponential type solutions are well studied. While for the elliptic type solution, we know only the one-soliton solution so far. Using the commutative Bäcklund transformation, we have succeeded in constructing the KdV static elliptic $N$-soliton solution, which means that we have constructed infinitely many solutions for the $\wp$-function type differential equation.
This paper presents two new Weierstrass elliptic function solutions of the projective Riccati equations and four conversion formulas for converting the Weierstrass elliptic functions to the hyperbolic and trigonometric functions. The Weierstrass elliptic function solutions to the projective Riccati equations and the conversion formulas are used to propose the called Weierstrass type projective Riccati equation expansion method. The Weierstrass elliptic function solutions, the solitary wave and the periodic wave solutions of the KdV equation are constructed by using the proposed method. The sol…
We provide an overview of elliptic algebro-geometric solutions of the KdV and AKNS hierarchies, with special emphasis on Floquet theoretic and spectral theoretic methods. Our treatment includes an effective characterization of all stationary elliptic KdV and AKNS solutions based on a theory developed by Hermite and Picard.
A general structure is developed from which a system of integrable partial difference equations is derived generalising the lattice KdV equation. The construction is based on an infinite matrix scheme with as key ingredient a (formal) elliptic Cauchy kernel. The consistency and integrability of the lattice system is discussed as well as special solutions and associated continuum equations.
In this work, the exact solutions for combined KdV-mKdV generalized equation as a linear superposition of Jacobi elliptic functions, $c_n(ξ,m)$, $d_n(ξ,m)$. When $m$ is set to one, the solution matches with well-known hyperbolic solutions of generalized combined KdV-mKdV equation. Similar solution is also derived for the case for time dependent co-efficient. In the latter case the velocity term become time dependent.
We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards. Namely, generic complex invariant manifolds are not Abelian varieties, and the billiard map is no more algebraic. A Poncelet-like theorem for such system is known. We give explicit sufficient conditions both for closed geodesics and periodic billiard orbits on Q and discuss their relation with the elliptic KdV solutions and elliptic Calogero system
A simple and general approach for calculating the elliptic finite-gap solutions of the Korteweg-de Vries (KdV) equation is proposed. Our approach is based on the use of the finite-gap equations and the general representation of these solutions in the form of rational functions of the elliptic Weierstrass function. The calculation of initial elliptic finite-gap solutions is reduced to the solution of the finite-band equations with respect to the parameters of the representation. The time evolution of these solutions is described via the dynamic equations of their poles, integrated with the help…
We give a detailed study of the traveling wave solutions of $(\ell=2)$ Kaup-Boussinesq type of coupled KdV equations. Depending upon the zeros of a fourth degree polynomial, we have cases where there exist no nontrivial real solutions, cases where asymptotically decaying to a constant solitary wave solutions, and cases where there are periodic solutions. All such possible solutions are given explicitly in the form of Jacobi elliptic functions. Graphs of some exact solutions in solitary wave and periodic shapes are exhibited. Extension of our study to the cases $\ell=3$ and $\ell=4$ are also me…
We establish a bilinear framework for elliptic soliton solutions which are composed by the Lamé-type plane wave factors. $τ$ functions in Hirota's form are derived and vertex operators that generate such $τ$ functions are presented. Bilinear identities are constructed and an algorithm to calculate residues and bilinear equations is formulated. These are investigated in detail for the KdV equation and sketched for the KP hierarchy. Degenerations by the periods of elliptic functions are investigated, giving rise to the bilinear framework associated with trigonometric/hyperbolic and rational func…
Poisson Pencils, Lie Symmetries and Hamiltonian Reductions of a Coupled Nonlinear Wave System:Tangent KdV Geometry and Elliptic Moduli
Department of Mathematics and Statistics, Universidad Nacional de Colombia, Sede Manizales（哥伦比亚国立大学马尼萨莱斯校区数学与统计系）；FIZMAKO Research Group, Manizales, Colombia（哥伦比亚马尼萨莱斯 FIZMAKO 研究组）
研究双场非线性色散系统，将其作为 KdV 的切覆盖和 Hamilton 流，证明 Magri 铅笔的切提升、识别对称代数，并通过行波约化得到四维系统，在椭圆轨迹上显式分类切行波。
Elliptic soliton solutions, i.e., a hierarchy of functions based on an elliptic seed solution, are constructed using an elliptic Cauchy kernel, for integrable lattice equations of Kadomtsev-Petviashvili (KP) type. This comprises the lattice KP, modified KP (mKP) and Schwarzian KP (SKP) equations as well as Hirota's bilinear KP equation, and their successive continuum limits. The reduction to the elliptic soliton solutions of KdV type lattice equations is also discussed.
Even though the KdV and modified KdV equations are nonlinear, we show that suitable linear combinations of known periodic solutions involving Jacobi elliptic functions yield a large class of additional solutions. This procedure works by virtue of some remarkable new identities satisfied by the elliptic functions.
We study to unify soliton systems, KdV/mKdV/sinh-Gordon, through SO(2,1) $\cong$ GL(2,$\mathbb R$) $\cong$ Möbius group point of view, which might be a keystone to exactly solve some special non-linear differential equations. If we construct the $N$-soliton solutions through the KdV type Bäcklund transformation, we can transform different KdV/mKdV/sinh-Gordon equations and the Bäcklund transformations of the standard form into the same common Hirota form and the same common Bäcklund transformation except the equation which has the time-derivative term. The difference is only the time-dependenc…
We calculate infinite set of initial profiles of higher integer KdV solitons, which are both exactly solvable for the Schrodinger equation and for the Gel'fand-Levitan-Marchenko equation in the inverse scattering transform method of KdV equation. The calculation of these higher integer soliton solutions is based on the recently developed multi-indexed extensions of the reflectionless soliton potential.
A nonlocal form of a two-layer fluid system is proposed by a simple symmetry reduction, then by applying multiple scale method to it a general nonlocal two place variable coefficient modified KdV (VCmKdV) equation with shifted space and delayed time reversal is derived. Various exact solutions of the VCmKdV equation, including elliptic periodic waves, solitary waves and interaction solutions between solitons and periodic waves are obtained and analyzed graphically. As an illustration, an approximate solution of the original nonlocal two-layer fluid system is also given.
Utilizing some conservation laws of (1+1)-dimensional integrable local evolution systems, it is conjectured that higher dimensional integrable equations may be regularly constructed by a deformation algorithm. The algorithm can be applied to Lax pairs and higher order flows. In other words, if the original lower dimensional model is Lax integrable (possesses Lax pairs) and symmetry integrable (possesses infinitely many higher order symmetries), then the deformed higher order systems are also Lax integrable and symmetry integrable. For concreteness, the deformation algorithm is applied to the u…
We analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are characterized by a Riemann--Hilbert problem which we show arises as the limit $N\to \infty$ of a gas of $N$-solitons. We show that this gas of solitons in the limit $N \to \infty$ is slowly approaching a cnoidal wave solution for $x \to - \infty$ (up to terms of order $\mathcal{O} (1/x)$), while approaching zero exponentially fast for $x\to+\infty$. We establish an asymptotic description of the gas of solitons…
We argue the integrability of the generalized KdV(GKdV) equation using the Painlevé test. For $d( \le 2)$ dimensional space, GKdV equation passes the Painlevé test but does not for $d \geq 3$ dimensional space. We also apply the Ablowitz-Ramani-Segur's conjecture to the GKdV equation in order to complement the Painlevé test.
Traveling wave solutions to Kawahara equation (KE), transmission line (TL), and Korteweg-de Vries (KdV) equation are found by using an elliptic function method which is more general than the $\mathrm{tanh}$-method. The method works by assuming that a polynomial ansatz satisfies a Weierstrass equation, and has two advantages: first, it reduces the number of terms in the ansatz by an order of two, and second, it uses Weierstrass functions which satisfy an elliptic equation for the dependent variable instead of the hyperbolic tangent functions which only satisfy the Riccati equation with constant…
We extend the definition of the pentagram map from 2D to higher dimensions and describe its integrability properties for both closed and twisted polygons by presenting its Lax form. The corresponding continuous limit of the pentagram map in dimension $d$ is shown to be the $(2,d+1)$-flow of the KdV hierarchy, generalizing the Boussinesq equation in 2D.
Periodic waves of the modified Korteweg-de Vries (mKdV) equation are identified in the context of a new variational problem with two constraints. The advantage of this variational problem is that its non-degenerate local minimizers are stable in the time evolution of the mKdV equation, whereas the saddle points are unstable. We explore the analytical representation of periodic waves given by Jacobi elliptic functions and compute numerically critical points of the constrained variational problem. A broken pitchfork bifurcation of three smooth solution families is found. Two families represent (…
We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) $u_t +\partial_{x_2}^n u_{x_1} - u_{x_1} u =0$ (here $n$ is any integer) reducing it to the ordinary differential equation (ODE). In a simplest case, $n=1$, the ODE is solvable in terms of elementary functions. Next choice, $n=2$, yields the cnoidal waves for the special case of Zakharov-Kuznetsov equation. The proposed method is based on the deformation of the characteristic of the equation $u_t-uu_{x_1}=0$ and might also be useful in study the higher dimensional PDEs with arbitr…
We calculate quantum mechanical scattering problems for multi-indexed extensions of soliton potential by Darboux transformations in terms of pseudo virtual wavefunctions. As an application, we calculate infinite set of higher integer KdV solitons by the inverse scattering transform method of KdV equation.
We seek multi-order exact solutions of a generalized shallow water wave equation along with those corresponding to a class of nonlinear systems described by the KdV, modified KdV, Boussinesq, Klein-Gordon and modified Benjamin-Bona-Mahony equation. We employ a modified version of a generalized Lame equation and subject it to a perturbative treatment identifying the solutions order by order in terms of Jacobi elliptic functions. Our multi-order exact solutions are new and hold the key feature that they are expressible in terms of an auxiliary function f in a generic way. For appropriate choices…
In the context of the emergence of new classes of superposed solutions such as dn2 (x,m) \pm \sqrt m cn(x,m) dn(x,m) for a variety of nonlinear equations we present some additional new ones for the KdV and QNLS equations and show that they also admit exact solutions of the type nd2 (x,m) \pm \sqrt sd(x,m) nd(x,m). We also determine different types of superposed solutions for a generalized shallow water wave equation.
We present the hierarchy and soliton solutions associated to a multi-component generalisation of the modified Korteweg-de Vries equation. A recursive formula for obtaining the Lax operators associated to the higher flows of the hierarchy is provided. Using the method of rational dressing and the symmetries of the Lax operators we obtain the one-soliton solution. We also derive the general rank one-breather solution and express it in terms of determinants. We present the simplest case of the one-breather solution, which is parametrised by two constant unit vectors normal to each other. Finally,…
Starting from the complete integrable lattice super-KdV equation, two super-mappings are obtained by performing a travelling-wave reduction. The first one is linear and the second is a four dimensional super-QRT mapping containing both Grassmann commuting and anti-commuting dependent variables. Adapting the classical "staircase" method to the Lax super-matrices of the lattice super-KdV equation, we compute the Lax super-matrices of the mapping and the two invariants; the first one is a pure nilpotent commuting quantity and the second one is given by an elliptic curve containing nilpotent commu…
For weak dispersion and weak dissipation cases, the (1+1)-dimensional KdV-Burgers equation is investigated in terms of approximate symmetry reduction approach. The formal coherence of similarity reduction solutions and similarity reduction equations of different orders enables series reduction solutions. For weak dissipation case, zero-order similarity solutions satisfy the Painlevé II, Painlevé I and Jacobi elliptic function equations. For weak dispersion case, zero-order similarity solutions are in the form of Kummer, Airy and hyperbolic tangent functions. Higher order similarity solutions c…
The elliptic Korteweg-de Vries (KdV) system is a multi-component generalization of the lattice potential KdV equation, whose soliton solutions are associated with an elliptic Cauchy kernel (i.e., a Cauchy kernel on the torus). In this paper we generalize the class of solutions by using a Sylvester type matrix equation and rederiving the system from the associated Cauchy matrix. Our starting point is the Sylvester equation in the form of $~\boldsymbol{k} \boldsymbol{M}+ \boldsymbol{M} \boldsymbol{k} = \boldsymbol{r} {\boldsymbol{c}}^{T}-g\boldsymbol{K}^{-1} \boldsymbol{r} {\boldsymbol{c}}^{T} \…
We present compacton-like solution of the modified KdV equation and compare its properties with those of the compactons and solitons. We further show that, the nonlinear Schr{ö}dinger equation with a source term and other higher order KdV-like equations also possess compact solutions of the similar form.
Recently proposed nonholonomic deformation of the KdV equation is solved through inverse scattering method by constructing AKNS-type Lax pair. Exact and explicit N-soliton solutions are found for the basic field and the deforming function showing an unusual accelerated (decelerated) motion. A two-fold integrable hierarchy is revealed, one with usual higher order dispersion and the other with novel higher nonholonomic deformations.
The discrete KdV (dKdV) equation, the pinnacle of discrete integrability, is often thought to possess the singularity confinement property because it confines on an elementary quadrilateral. Here we investigate the singularity structure of the dKdV equation through reductions of the equation, obtained for initial conditions on a staircase with height 1, and show that it is much more subtle than one might assume. We first study the singularities for the mappings obtained after reduction and contrast these with the singularities that arise in non-integrable generalizations of those mappings. We …
We study the Whitham equations for all the higher order KdV equations. The Whitham equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the solution of the Whitham equations when the initial values are given by a step function.
It is shown that the hodograph solutions of the dispersionless coupled KdV (dcKdV) hierarchies describe critical and degenerate critical points of a scalar function which obeys the Euler-Poisson-Darboux equation. Singular sectors of each dcKdV hierarchy are found to be described by solutions of higher genus dcKdV hierarchies. Concrete solutions exhibiting shock type singularities are presented.
In order to find higher dimensional integrable models, we study differential equations of hyperelliptic $\wp$ functions up to genus four. For genus two, differential equations of hyperelliptic $\wp$ functions can be written in the Hirota form. If the genus is more than one, we have KdV equation. If the genus is more than two, we have KdV and another KdV equations. If the genus becomes more than three, there appear differential equations which cannot be written in the Hirota form, which means that the Hirota form is not enough to characterize the integrable differential equations. We have shown…
We address the most general periodic travelling wave of the modified Korteweg-de Vries (mKdV) equation written as a rational function of Jacobian elliptic functions. By applying an algebraic method which relates the periodic travelling waves and the squared periodic eigenfunctions of the Lax operators, we characterize explicitly the location of eigenvalues in the periodic spectral problem away from the imaginary axis. We show that Darboux transformations with the periodic eigenfunctions remain in the class of the same periodic travelling waves of the mKdV equation. In a general setting, there …
We derive a general theorem relating the energy, momentum and velocity of any solitary wave solution of the generalized KdV equation which enables us to relate the amplitude, width, and momentum to the velocity of these solutions. We obtain the general condition for linear and Lyapunov stability. We then obtain a two parameter family of exact solutions to these equations which include elliptic and hyper-elliptic compacton solutions. For this general family we explicitly verify both the theorem and the stability criteria.
An analysis of extension of Hamiltonian operators from lower order to higher order of matrix paves a way for constructing Hamiltonian pairs which may result in hereditary operators. Based on a specific choice of Hamiltonian operators of lower order, new local bi-Hamiltonian coupled KdV systems are proposed. As a consequence of bi-Hamiltonian structure, they all possess infinitely many symmetries and infinitely many conserved densities.
In this paper, some notes of the homogeneous balance (HB) method are discussed by a kind of general fifth-order KdV (fKdV) equation. Frist, the auto-Bäcklund transformation and lax represents of the higher-order KdV equation(a specific forms of fKdV equation) are obtained by the HB method. Then, the connection of the Hirota transformation and the HB method is discussed by two specific fKdV equations: the Sawada-Kotera equation and the Lax's equation. At the same time, the solitary wave solution of the general fifth-order KdV equation is obtained.
We present a multi-parameter family of rational solutions to the complex Korteweg-de Vries(KdV) equations. This family of solutions includes particular cases with high-amplitude peaks at the centre, as well as a multitude of cases in which high-order rogue waves are partially split into lower-order fundamental components. We present an empirically-found symmetry which introduces a parameter controlling the splitting of the rogue wave components into multi-peak solutions, and allows for nonsingular solutions at higher order in certain cases.
Multicomponent KdV-systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having higher-order symmetries, recursion operators and hierarchies of conservation laws. In this paper the dispersionless limits of these Jordan KdV equations are studied, under the assumptions that the Jordan algebra has a unity element and a compatible non-degenerate inner product. Much of this structure may be encoded in a so-called Jordan manifold, akin to a Frobenius…
We examine the integrability of two models used for the interaction of long and short waves in dispersive media. One is more classical but arguably cannot be derived from the underlying water wave equations, while the other one was recently derived. We use the method of Zakharov and Schulman to attempt to construct conserved quantities for these systems at different orders in the magnitude of the solutions. The coupled KdV-NLS model is shown to be nonintegrable, due to the presence of fourth-order resonances. A coupled real KdV - complex KdV system is shown to suffer the same fate, except for …
The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include higher order effects. Although this equation has only one conservation law, exact periodic and solitonic solutions exist. Khare and Saxena \cite{KhSa,KhSa14,KhSa15} demonstrated the possibility of generating new exact solutions by combining known ones for several fundamental equations (e.g., Korteweg - de Vries, Nonlinear Schrödinger). Here we find that this construction can be repeated for higher order, non-integrable extensions of these equa…
The KdV-Sawada-Kotera equation has single-, two- and three-soliton solutions. However, it is not known yet whether it has N-soliton solutions for any N. Viewing it as a perturbed KdV equation, the asymptotic expansion of the solution is developed through third order within the framework of a Normal Form analysis. It is shown that the equation is asymptotically integrable through the order considered. Focusing on the soliton sector, it is shown that the higher-order corrections in the Normal Form expansion represent purely inelastic KdV-soliton-collision processes, and vanish identically in the…
A theory for constructing integrable couplings of soliton equations is developed by using various perturbations around solutions of perturbed soliton equations being analytic with respect to a small perturbation parameter. Multi-scale perturbations can be taken and thus higher dimensional integrable couplings can be presented. The theory is applied to the KdV soliton hierarchy. Infinitely many integrable couplings are constructed for each soliton equation in the KdV hierarchy, which contain integrable couplings possessing quadruple Hamiltonian formulations and two classes of hereditary recursi…
Residual Symmetry Reductions and Painlevé Solitons
School of Artificial Intelligence and Computer Science, Shaanxi Normal University（陕西师范大学人工智能与计算机科学学院）；School of Information and Engineering, Xian University（西安大学信息与工程学院）；School of Physical Science and Technology, Ningbo University（宁波大学物理科学与技术学院）；Institute of Fundamental Physics and Quantum Technology, Ningbo University（宁波大学基础物理与量子技术研究院）
In this paper we construct a higher order expansion of the manifold of quasi unidirectional waves in the Fermi-Pasta-Ulam (FPU) chain. We also approximate the dynamics on this manifold. As perturbation parameter we use $h^2=1/n^2$, where $n$ is the number of particles of the chain. It is well known that the dynamics of quasi unidirectional waves is described to first order by the Korteweg-de Vries (KdV) equation. Here we show that the dynamics to second order is governed by a combination of the first two nontrivial equations in the KdV hierarchy -- for any choice of parameters in the FPU poten…
A method for introducing the higher order terms in the potential expansion to study the continuous limits of the Toda hierarchy is proposed in this paper. The method ensures that the higher order terms are differential polynomials of the lower ones and can be continued to be performed indefinitly. By introducing the higher order terms, the fewer equations in the Toda hierarchy are needed in the so-called recombination method to recover the KdV hierarchy. It is shown that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda hierarchy tend towards the corresponding ones of the Kd…
