Order Darboux Research Articles

PINN deep learning method for the Chen–Lee–Liu equation: Rogue wave on the periodic background

We consider the exact rogue periodic wave (rogue wave on the periodic background) and periodic wave solutions for the Chen–Lee–Liu equation via the odd-th order Darboux transformation. Then, the multi-layer physics-informed neural networks (PINNs) deep learning method is applied to research the data-driven rogue periodic wave, breather wave, soliton wave and periodic wave solutions of well-known Chen–Lee–Liu equation. Especially, the data-driven rogue periodic wave is learned for the first time to solve the partial differential equation. In addition, using image simulation, the relevant dynamical behaviors and error analysis for there solutions are presented. The numerical results indicate that the rogue periodic wave, breather wave, soliton wave and periodic wave solutions for Chen–Lee–Liu equation can be generated well by PINNs deep learning method.

Darboux transformation of the second-type nonlocal derivative nonlinear Schrödinger equation

The second-type nonlocal derivative nonlinear Schrödinger (NDNLSII) equation is studied in this paper. By constructing its [Formula: see text]-order Darboux transformations (DT) from the first-order DT, Vandermonde-type determinant solutions of the NDNLSII equation are obtained from zero seed solutions, which would be singular unless the square of eigenvalues are purely imaginary.

Finding higher order Darboux polynomials for a family of rational first order ordinary differential equations

For the Darbouxian methods we are studying here, in order to solve first order rational ordinary differential equations (1ODEs), the most costly (computationally) step is the finding of the needed Darboux polynomials. This can be so grave that it can render the whole approach unpractical. Hereby we introduce a simple heuristics to speed up this process for a class of 1ODEs. New version program summaryProgram title: LsolverCatalogue identifier: ADZF_v3_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZF_v3_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1831No. of bytes in distributed program, including test data, etc.: 20862Distribution format: tar.gzProgramming language: Maple, release 17.Computer: Any computer running Maple.Operating system: Windows XP, Windows 7.RAM: 20 MbClassification: 4.3.Catalogue identifier of previous version: ADZF_v2_0Journal reference of previous version: Comput. Phys. Comm. 185(2014)1496Does the new version supersede the previous version?: YesNature of problem: We deal with the analytical solving of 1ODEs via a Darboux method.Solution method: The method of solution is based on the standard Prelle–Singer (Darboux) method [1], with extensions of our own [2] which solves 1ODEs with Liouvillian solutions. The first part of the Method depends on the finding of the Darboux polynomials for the ODE. The integrability analysis mentioned above is performed at this stage.Reasons for new version: A great portion of our work has been concentrated on analyzing and solving systems of first and second order differential equations (1ODEs and 2ODEs, respectively) from a numerical point of view [3], using Lie methods [4, 5] and Darboux type approaches [2, 6]. The common factor among the algorithms in the Darbouxian approach has always been that the most costly step is the determination of the associated Darboux polynomials. So, we are hereby trying to improve on this aspect.Summary of revisions: The revision is the introduction of an option to the command Darboux that allows for faster finding of the necessary Darboux polynomials and co-factors, in some cases. In the last version of this package, we have introduced a possibility of using a defined degree for both the Darboux polynomial and co-factor. Previously, one could only change the degree of the Darboux polynomial and the program used the maximum value possible for the degree of the co-factor. It causes the program to be stuck (sometimes) in a very large problem and the package had problems both in time and computer memory management. Here we introduce a similar idea. We allow the program to look for Darboux polynomials with a fixed degree in one variable. We also fix the degree of the polynomial as a whole. This extra liberty (in restricting the problem to be dealt with) allows for the success of the package in solving some 1ODEs that were missed even with the improvement of the last version. In other words, by fixing the degree for the specific variable we divided the search into smaller steps, dividing the problem into smaller ones. This prevents the algorithm entering too large a calculation. In order to properly see the syntax of the usage of the command, please, see the example Maple file. Let us just display the general format of the command Darboux.>Darboux(ode,DPEXT=[[dx,dt],[x,y]])This would produce an output of the form:[[Darboux polynomials],[co-factors]]In the above calling of the command Darboux, the [dx,dt] terms refer to the degrees of x and the total degree of the Darboux polynomial, respectively.It is important to notice that, for some cases, if we do not instruct the Darboux command which degree to use for the variable x (for example) and the total degree of the Darboux polynomial, thus going step-by-step on the algorithm, (via the extra term DPEXT=[[dx,dt],[x,y]]), the command may not be able to solve the 1ODE. Furthermore, the Maple in-built dsolve command cannot solve some of the same class of 1ODEs. So the new improvement, albeit simple to implement, is of great practical value.Restrictions: The finding of the Darboux polynomials and the correspondent co-factors is the most time-demanding task of the method. So, one could say that, for the general case, if the degree for the Darboux polynomials is “4”, for instance, it would make the case hard to deal with. This situation is mitigated (for some 1ODEs) by the improvements hereby introduced.Unusual features: The faster way to find high (>3) degree Darboux polynomials and co-factors for some cases.Running time: This depends strongly on the 1ODE being analyzed. As mentioned above, a degree for the Darboux polynomials “4” or above could (in general) make dealing with the case unpractical. For the degree “3”, the routines may take more than a minute to deal with the task. Generally, the program takes, at most, for the other cases, a few seconds. Now, even for degrees >4 it can sometime only take a couple of seconds.

Higher order Darboux transformations in two dimensions, linearizable auxiliary equations, and invariant potentials

We show that the two-dimensional Schrödinger equation admits Darboux transformations only for a particular class of potentials. It is demonstrated that for such potentials, the associated auxiliary equation can be linearized to Schrödinger form. Furthermore, since the aforehand mentioned class of potentials turns out to remain invariant under Darboux transformations, construction of multiple, higher order Darboux transformations (chains) becomes possible. We present second-order chains in explicit form and give an algorithmic scheme for the construction of higher order chains.

Darboux transformations for the time-dependent nonhomogeneous Burgers equation in (1+1) dimensions

We extend the formalism of nth order Darboux transformations to the time-dependent nonhomogeneous Burgers equation (NBE) in (1+1) dimensions. Similar to the Schrödinger case, our Darboux transformation retains the form of the NBE, while changing the nonhomogeneous term. The transformed solution of the NBE and the corresponding transformed nonhomogeneity are given in closed form. Furthermore, properties of the transformation are discussed and an application is given.

P T symmetric models with nonlinear pseudosupersymmetry

By applying the higher order Darboux algorithm to an exactly solvable non-Hermitian PT symmetric potential, we obtain a hierarchy of new exactly solvable non-Hermitian PT symmetric potentials with real spectra. It is shown that the symmetry underlying the potentials so generated and the original one is nonlinear pseudosupersymmetry. We also show that this formalism can be used to generate a larger class of new solvable potentials when applied to non-Hermitian systems.

Second-order Darboux displacements

The potentials for a one dimensional Schroedinger equation that are displaced along the x axis under second order Darboux transformations, called 2-SUSY invariant, are characterized in terms of a differential-difference equation. The solutions of the Schroedinger equation with such potentials are given analytically for any value of the energy. The method is illustrated by a two-soliton potential. It is proven that a particular case of the periodic Lame-Ince potential is 2-SUSY invariant. Both Bloch solutions of the corresponding Schroedinger equation equation are found for any value of the energy. A simple analytic expression for a family of two-gap potentials is derived.

Systems with higher-order shape invariance: spectral and algebraic properties

We study systems of two intertwining relations of first or second order for the same (up to a constant shift) partner Schrödinger operators. It is shown that the corresponding Hamiltonians possess a higher order shape invariance which is equivalent to the ladder equation. We analyze with particular attention irreducible second order Darboux transformations which together with the first order act as building blocks. For the third order shape-invariance irreducible Darboux transformations entail only one sequence of equidistant levels while for the reducible case the structure consists of up to three infinite sequences of equidistant levels and, in some cases, singlets or doublets of isolated levels.

Darboux transformations for Lamé potentials

Irreducible second order Darboux (SUSY) transformations are applied to Lam periodic Hamiltonians in order to find physically acceptable partner potentials with the same band structure. The main differences with the usual SUSY approach of potentials allowing for a discrete spectrum are pointed out.

Rank 2 commuting ordinary differential operators and Darboux conjugates of KdV

By considering the factorizations (flags) and associated (simultaneous) second order Darboux transformations of the square and cube of an arbitrary second order Schrödinger operator, we generate commuting ordinary differential operators of orders four and six with a singular elliptic spectrum. This procedure generates true rank 2 commutative algebras. Under the KdV flow, each such factorization (flag) leads to an integrable equation for which the corresponding Darboux transformation generates a Lax-type operator as one of a commuting pair of orders four and six with singular elliptic spectrum. Hence, these integrable equations are Darboux conjugates of KdV.