Lax Pair Research Articles

Higher dimensional Degasperis–Procesi–Camassa–Holm systems

The (1+1)-dimensional models can be applied to derive corresponding higher dimensional models by a deformation algorithm with the help of their conservation laws, their Lax integrability can be proved in the same way. In this paper, we modify the original deformation algorithm such that the conserved density should not be derivative dependent. The modified deformation algorithm is mainly applied to the (1+1)-dimensional Degasperis-Procesi equation and Camassa–Holm equation to establish some brand-new (N+1)-dimensional coupled Degasperis–Procesi–Camassa–Holm systems and their (1+1)-dimensional reductions, whose Lax pairs have been given. On the other hand, we can find a new type of peakon solution of the (2+1)-dimensional deformed Degasperis-Procesi equation.

Spectral and soliton structures for the four-component Kaup–Newell type negative flow equation

Based on the spectral analysis of Lax pairs and the inverse scattering method, we construct a matrix Riemann–Hilbert problem of the four-component Kaup–Newell type negative flow equation associated with a 4 × 4 matrix spectral problem and investigate the evolution of scattering data. Then N-soliton formulas of the four-component Kaup–Newell type negative flow equation are obtained by solving the irregular Riemann–Hilbert problem in the reflectionless case. As an application, the one-soliton solutions and the two-soliton solutions of the four-component Kaup–Newell type negative flow equation are given explicitly. In addition, the interaction dynamics of the soliton solutions are analyzed and illustrated.

Analysis of the seventh-order Caputo fractional KdV equation: applications to the Sawada–Kotera–Ito and Lax equations

In this study, we investigate the seventh-order nonlinear Caputo time-fractional KdV equation. The suggested model’s solutions, which have a series form, are obtained using the hybrid ZZ-transform under the aforementioned fractional operator. The proposed approach combines the homotopy perturbation method (HPM) and the ZZ-transform. We consider two specific examples with suitable initial conditions and find the series solution to test their applicability. To demonstrate the utility of the presented technique, we explore its applications to the fractional Sawada–Kotera–Ito problem and the Lax equation. We observe the impact of a few fractional orders on the wave solution evolution for the problems under consideration. We provide the efficiency and reliability of the ZZHPM by calculating the absolute error between the series solution and the exact solution of both the Sawada–Kotera–Ito and Lax equations. The convergence and uniqueness of the solution are portrayed via fixed-point theory.

Equivalent transformation and integrability of the nonlinear Schrödinger equations with time-dependent coefficients

The nonlinear Schrödinger (NLS) types of equations play a key role in quantum mechanics, Quantum communication and physical applications. However, how to deal with explicit solutions and other properties of the NLS equations, especially for the variable-coefficient NLS (vc-NLS) types of equations is a difficult problem. In this paper, we construct the form-preserving equivalent transformations (ETs) to transform the vc-NLS systems into constant-coefficient NLS (cc-NLS) systems, and the form-preserving ETs are given explicitly. Then, based on the equivalent transformation method, we deal with the integrability of the NLS equations, and the Lax pairs (LPs) are provided as verification of the integrability.

Hungry Lotka–Volterra lattice under nonzero boundaries, block‐Hankel determinant solution, and biorthogonal polynomials

AbstractIn this paper, we first extend the hungry Lotka–Volterra lattice to a case of nonzero boundary conditions and present its corresponding exact solution expressed in terms of a block‐Hankel determinant. Then, we establish a connection between this hungry Lotka–Volterra lattice under nonzero boundary conditions and a set of biorthogonal polynomials. It turns out that the hungry Lotka–Volterra lattice under nonzero boundary conditions possesses a Lax pair expressed in terms of the biorthogonal polynomials. Moreover, we consider two special cases of the hungry Lotka–Volterra lattice. For the case , it reduces to the Lotka–Volterra lattice under nonzero boundary condition, which has been discussed in literature. We also present the result for in detail, which extends a known result to a case of nonzero boundary functions. All these results are obtained by virtue of Hirota's bilinear method and determinant techniques.

A new look into the concept of separable system of Classical Mechanics

The theory of integrable systems of Classical Mechanics has been deeply affected, in the second half of the last century, by the discovery of the integrability of the Korteweg–de Vries equation. New ideas appeared at that time (such as, for instance, the concepts of Lax and bihamiltonian representations, and the notions of Poisson manifold, Poisson pencil, and recursion operator) which have changed our way of seeing the geometry of integrable systems. They also changed our way of seeing the geometry of separable systems. The paper is an attempt to explain why and in what form the idea of recursion operator leads to reshape the definition of separable system, and to discover a new set of separability conditions called “Kowalevski separability conditions”.

AKNS type reduced integrable bi-Hamiltonian hierarchies with four potentials

The aim of this paper is to derive a kind of integrable hierarchies with four potentials, which possess bi-Hamiltonian structures, from reduced AKNS matrix spectral problems. The associated recursion operators are worked out explicitly. The Lax pair formulation and the trace identity are basic tools in the analysis. Two nonlinear examples in the resulting integrable hierarchies are integrable nonlinear Schrödinger type equations and integrable modified Korteweg–de Vries type equations.

Searching For (2+1)-dimensional nonlinear Boussinesq equation from (1+1)-dimensional nonlinear Boussinesq equation

A novel (2+1)-dimensional nonlinear Boussinesq equation is derived from a (1+1)-dimensional Boussinesq equation in nonlinear Schrödinger type based on a deformation algorithm. The integrability of the obtained (2+1)-dimensional Boussinesq equation is guaranteed by its Lax pair obtained directly from the Lax pair of the (1+1)-dimensional Boussinesq equation. Because of the effects of the deformation, the (2+1)-dimensional Boussinesq equation admits a special travelling wave solution with a shape that can be deformed to be asymmetric and/or multi-valued.

On the complete integrability of gradient systems on manifold of the lognormal family

The aim of this paper is to show that the lognormal statistical manifold admits a potential-gradient system which is Hamiltonian, completely integrable and has a Lax pair representation by a symmetric matrix. The explicit solutions of this gradient system are also determined by means of the associated linear system.

Higher-dimensional integrable deformations of the modified KdV equation

The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability. The well-known modified KdV equation is a prototypical example of an integrable evolution equation in one spatial dimension. Do there exist integrable analogs of the modified KdV equation in higher spatial dimensions? In what follows, we present a positive answer to this question. In particular, rewriting the (1+1)-dimensional integrable modified KdV equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations. Further, we illustrate this idea with examples from the modified KdV hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.

Solving the (2+1)-Dimensional Derivative Toda Equation

The main objectives of this work are to deduce the Lax pair for the Modified Toda lattice hierarchy and to find whether there exists a transformation between (2+1)-dimensional derivative Toda lattice equation and Modified Toda equation. By a new spectral problem, we obtain the Modified Toda lattice hierarchy, furthermore we give the Hirota’s form solution for the Modified Toda lattice equation, thus the Hirota’s solution for the (2+1)-dimensional derivative Toda equation can also be attained correspondingly.

Long-time asymptotics of solution to the coupled Hirota system with 4 × 4 Lax pair

We study the Cauchy problem for the integrable coupled Hirota system with a 4×4 Lax pair on the line with decaying initial data. By deriving a Riemann–Hilbert representation for the solution, we compute the precise leading-order terms for long-time asymptotics based on the nonlinear steepest descent arguments.

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.

Multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan-Porsezian-Daniel equations via Riemann-Hilbert approach

In this paper, we study multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan-Porsezian-Daniel equations, which describe the simultaneous propagation of nonlinear waves in the inhomogeneous optical fibers. We analyze the spectrum of the Lax pair to establish the Riemann-Hilbert problem. Using such Riemann-Hilbert problem, we calculate various multi-soliton solutions without reflection, including breather-like and mixed solitons. We illustrate the propagation and interaction dynamics of the solitons through appropriate parameter selection and asymptotic analysis. We find that the interaction between solitons is elastic, the amplitudes of solitons are only determined by the initial velocity and interaction, and the soliton with lower energy always yields a position shift when elastic interaction occurs. In addition, we observe that the existence time of the wave changes with energy and that multiple elastic interactions between solitons can be obtained when we choose appropriate variable coefficients. Then, we investigate the influences of group velocity dispersions and fourth-order dispersions on the interactions of solitons through parameter modulation mode and asymptotic analysis. Furthermore, we present several new types of nonlinear phenomena graphically, including elastic interactions between parabolic solitons and hump-type solitons, elastic interactions between cubic solitons and hump-type solitons, and periodic-changing propagations.

Local Yang–Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and their matrix Lax representations defined by the local Yang–Baxter equation. Sergeev (1998 Lett. Math. Phys. 45 113–9) presented classification results on three-dimensional tetrahedron maps obtained from the local Yang–Baxter equation for a certain class of matrix-functions in the situation when the equation possesses a unique solution which determines a tetrahedron map. In this paper, using correspondences arising from the local Yang–Baxter equation for some simple matrix-functions, we show that there are (non-unique) solutions to the local Yang–Baxter equation which define tetrahedron maps that do not belong to the Sergeev list; this paves the way for a new, wider classification of tetrahedron maps. We present invariants for the derived tetrahedron maps and prove Liouville integrability for some of them. Furthermore, using the approach of solving correspondences arising from the local Yang–Baxter equation, we obtain several new birational tetrahedron maps with Lax representations and invariants, including maps on arbitrary groups, a nine-dimensional map associated with a Darboux transformation for the derivative nonlinear Schrödinger equation, and a nine-dimensional generalisation of the three-dimensional Hirota map.

On classification of non-abelian Painlevé type systems

We find all non-abelian generalizations of P1−P6 Painlevé systems such that the corresponding autonomous system obtained by freezing the independent variable is integrable. All these systems have isomonodromic Lax representations.

Soliton solution, breather solution and rational wave solution for a generalized nonlinear Schrödinger equation with Darboux transformation

In this paper, the exact solutions of generalized nonlinear Schrödinger (GNLS) equation are obtained by using Darboux transformation(DT). We derive some expressions of the 1-solitons, 2-solitons and n-soliton solutions of the GNLS equation via constructing special Lax pairs. And we choose different seed solutions and solve the GNLS equation to obtain the soliton solutions, breather solutions and rational wave solutions. Based on these obtained solutions, we consider the elastic interactions and dynamics between two solitons.

Loop dynamics of a fully discrete short pulse equation

In this paper, a fully discrete short pulse (SP) equation is presented as an integrability condition of a linear system of difference equations (also known as discrete Lax pair). Additionally, two semi-discrete versions of the SP equation have also been obtained from fully discrete SP equation under continuum limits. Darboux transformation is employed to compute multi-soliton solutions of fully discrete and semi-discrete SP equations. Explicit expressions of first and second nontrivial soliton solutions are computed. We also derived explicit expression of breather solution for fully discrete SP equation. The dynamics of single loop soliton and interaction mechanism of loop–loop and loop–antiloop solutions have been explored and illustrated.

Prolongation structure of supersymmetric nonlinear equation and its Bäcklund transformation

The Heisenberg supermagnet model is a supersymmetric system and has a close relationship with the strong electron-correlated Hubbard model. In this paper, the supersymmetric prolongation structure is used to analyze the high order supersymmetric nonlinear equation. The Lax representation is constructed for the prolongation algebra of this equation. The Bäcklund transformation of the supersymmetric nonlinear Schrödinger equation is obtained by simplified calculation.

Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy

Solutions of the discrete Painlevé II hierarchy are shown to be in relation with a family of Toeplitz determinants describing certain quantities in multicritical random partitions models, for which the limiting behavior has been recently considered in the literature. Our proof is based on the Riemann-Hilbert approach for the orthogonal polynomials on the unit circle related to the Toeplitz determinants of interest. This technique allows us to construct a new Lax pair for the discrete Painlevé II hierarchy that is then mapped to the one introduced by Cresswell and Joshi.