Integrable Deformations of the (2+1)-Dimensional Heisenberg Ferromagnetic Model

Based on the covariant prolongation structure technique,we construct the integrable higher-order deformations of the (2+1)-dimensional Heisenberg ferromagnet model and obtain their su(2)×R(λ) prolongation structures.By associating these deformed multidimensional Heisenberg ferromagnet models with the moving space curve in Euclidean space and using the Hasimoto function,we derive their geometrical equivalent counterparts,i.e.,higher-order (2+1)-dimensional nonlinear Schrödinger equations.     

Prolongation Approach to the Liouville Equation and Its Algebraic Structures

The Riccati type equation, the Biicklund transformation and the Lax pair of the Liouville equation are obtained by using the prolongation theory. It is shown that the prolongation structures generate the SL(2,R) Lie algebra.     

ON THE PROLONGATION STRUCTURE OF ERNST EQUATION

An SL(2R) ×R¹(l) prolongation structure of Ernst equation with a real parameter l and the corresponding Riccati equation as well as a pair of linear equations which are in principle equivalent to the inverse scattering problem due to Belinsky and Zakharov are obtained by solving the fundamental equation for the prolongation structure. A necessary condition which should be satisfied by the Bäcklund transformations is pfesented in terms of prolongation structure. And it is indicated that in the, case of Ernst equation the Harrison transformation, Neugebauer transformations and other available Bäcklund transformations as well as Belinsky-Zakharov's Riemann transformation, i.e., the homogeneous Hilbertproblem (HHP), would be covered by this condition.     

SL(2,R) PRINCIPAL PROLONGATION STRUCTURES OF SOLITON EQUATIONS AND THEIR CONSERVATION LAWS

We present the concept of principal prolongation structure (PPS) and a covariant criterion of the completeness of conserva-tion currents for the PPS of class of nonlinear evolution equations (NEES).The SL(2,R) × R'(l) PPS for AKNS systems is constructed, a new set of infinite number of polynomial conservation currents (PCCs) corresponding to the nonlinearity of SL (2,R) group manifold is given. These currents together with the usual PCCS of AKNS systems satisfy a covariant equation for the SL(2,R) × R'(l) PPS. This equation gives rise to a criterion of completeness of these currents. As an example,the sine-Gordon system is analysed.     

PROLONGATION STRUCTURE APPROACH TO THE GROUP-THEORETICAL TECHNIQUE FOR GENERATING STATIONARY AXISYMMETRIC SPACE-TIMES

The group-theoreti cal technique for generating stationary axisymmetric gravitational fields is approached by means of the prolongation structure theory for soliton systems. An sp(2)xc(t) structure is obtained via solving the fundamental equation for prolongation structures and the F-equation for Kinnersley-Chitre's generating function is naturally introduced as an inverse scattering equation. A homogeneous Hilbert problem(HRP) associated with the Geroch group K and a corresponding linear singular integral equation are derived based upon a general condition satisfied by the auto-Bäcklund transformations in the sense of prolongation structure theory.     

与一个非局域对称有关的Kaup-Kupershmidt方程新的精确解

利用Kaup-Kupershmidt（KK）方程的一个非局域对称，可在两种不同的方法上找到方程新的精确解．首先，用标准的展开近似得到KK方程有限的Lie-B?cklund变换和单孤子解．其次，把一些局域对称与这个非局域对称组合起来，给出其群不变解，进而可求得新的孤子解 关键词：     

Generating a New Higher-Dimensional Coupled Integrable Dispersionless System: Algebraic Structures,Bäcklund Transformation and Hidden Structural Symmetries

The prolongation structure methodologies of Wahlquist-Estabrook [H.D. Wahlquist and F.B. Estabrook, {J. Math. Phys.} 16 (1975) 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system. Based on the obtained prolongation structure, a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed. A Lie-Algebra representation of some hidden structural symmetries of the previous system, its Bäcklund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived. In the wake of the previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new (2+1)-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation, which applications are straightforward in nonlinear optics where additional propagating dimension deserves some attention.     

Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy

In this paper, a new nonlinear integrable coupling system of the soliton hierarchy is presented. From the Lax pairs, the coupled KdV equations are constructed successfully. Based on the prolongation method of Wahlquist and Estabrook, we study the prolongation structure of the nonlinear integrable couplings of the KdV equation.     

A note on the prolongation structure of the cubically nonlinear integrable Camassa-Holm type equation

In this Letter, we formulate an exterior differential system for the newly discovered cubically nonlinear integrable Camassa-Holm type equation. From the exterior differential system we establish the integrability of this equation. We then study Cartan prolongation structure of this equation. We also discuss the method of identifying conservation laws and Bäcklund transformation for this equation from the identified exterior differential system.     

PROLONGATION METHOD OF FINDING BACKLUND TRANSFORMATION AND LAX-PAIR OF ERNST EQUATION AND ERNST-MAXWELL EQUATIONS

In this papsr all the known Backlund trans- formations of the Ernst equation are de}tved by ua}ing prolongation: method. As tile nonlinear realizationsp and infinite-dimensional linear realization for the prolongation structure are used, the inverse scattering equation (Lax-pair) is also derived. And the connection between BT and Lax-pair becomes evident. A systematic procedure for deriving the non-Linear realizations of the algebras is sugested based on the prolongation structure. The two knowniBackl and transformations as well as the Lax-pair of the Ernst-Maxwell equations are obtained bu the same method.     

On the integrability of a system of coupled KdV equations

We consider a system of equations, a two component generalisation of the KdV equation, which Hirota and Satsuma recently conjectured to be integrable. Using the Wahlquist-Estabrook prolongation technique, we derive a scattering problem for this system.     

Lie Algebraic Structures and Integrability of Long-Short Wave Equation in （2＋1）Dimensions

The hidden symmetry and integrability of the long-short wave equation in (2 1) dimensions are considered using the prolongation approach. The internal algebraic structures and their linear spectra are derived in detail which show that the equation is integrable.     

Deep learning neural networks for the third-order nonlinear Schrödinger equation: bright solitons,breathers, and rogue waves

The dimensionless third-order nonlinear Schrödinger equation (alias the Hirota equation) is investigated via deep leaning neural networks. In this paper, we use the physics-informed neural networks (PINNs) deep learning method to explore the data-driven solutions (e.g. bright soliton, breather, and rogue waves) of the Hirota equation when the two types of the unperturbated and perturbated (a 2% noise) training data are considered. Moreover, we use the PINNs deep learning to study the data-driven discovery of parameters appearing in the Hirota equation with the aid of bright solitons.     

CONNECTION THEORY OF FIBRE BUNDLE AND PROLONGATION STRUCTURES OF NONLINEAR EVOLUTION EQUATIONS

It is shown that the proper geometrical framework for the nonlinear evolution equations (NEEs) and the soliton equations Should be the fibre bundle theory, the principal bundle and its associated bundle and their connection theory. Based upon the requirement of covariance of the geometrical quantities, a covariant generic geometry theory for the prolongation strutures of the NEEs is proposed and the fundamental equations for the prolongation structures are presented. From the fundamental equations it immediately follows that the comections corresponding to these NEEs always flat but with torsion and the covariant formulae satisfied by the conservation quantities associated with these NEEs are obtained. The prolongation structure of the MKdV equation, as an example, is concretely worked out by means of the covariant theory of the prolongation structure presented in this paper.     

New Soliton Solutions with Compact Support for a Family of Two-Parameter Regularized Long-Wave Boussinesq Equations

Searching for special solitary wave solutions with compact support is of important significance in soliton theory. In this paper, to understand the role of nonlinear dispersion in pattern formation, a family of the regularized longwave Boussincsq equations with fully nonlinear dispersion (simply called R(m, n) equations), utt + a( un )xx + b(um )xxtt = 0(a, b const.), is studied. New solitary wave solutions with compact support of R(m, n) equations are found. In addition we find another compacton solutions of the two special cases, R(2, 2) equation and R(3, 3) equation. It is found that the nonlinear dispersion term in a nonlinear evolution equation is not a necessary condition of that it possesses compacton solutions.     

Prolongation Structure of Semi-discrete Nonlinear Evolution Equations

Based on noncommutative differential calculus, we present a theory of prolongation structure for semidiscrete non/inear evolution equations. As an illustrative example, a semi-discrete model of the non/inear SchrSdinger equation is discussed in terms of this theory and the corresponding Lax pairs are also given.     

Integrability of fermionic extensions of the Burgers equation

Using the prolongation structure technique, we investigate the fermionic extensions of the Burgers equation suggested by Hlavaty. Besides those which pass the Painlevé test, some new integrable cases are found. Through the representations of the Lie superalgebras, the Lax pairs for these integrable cases are constructed.     

Darboux Transformations,Higher-Order Rational Solitons and Rogue Wave Solutions for a(2+1)-Dimensional Nonlinear Schrdinger Equation

By Taylor expansion of Darboux matrix, a new generalized Darboux transformations(DTs) for a(2 + 1)-dimensional nonlinear Schrdinger(NLS) equation is derived, which can be reduced to two(1 + 1)-dimensional equation:a modified KdV equation and an NLS equation. With the help of symbolic computation, some higher-order rational solutions and rogue wave(RW) solutions are constructed by its(1, N-1)-fold DTs according to determinants. From the dynamic behavior of these rogue waves discussed under some selected parameters, we find that the RWs and solitons are demonstrated some interesting structures including the triangle, pentagon, heptagon profiles, etc. Furthermore, we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter. These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems.     

A(2+1)-dimensional nonlinear model for Rossby waves in stratified fluids and its solitary solution

In this paper,we investigate a(2+1)-dimensional nonlinear equation model for Rossby waves in stratified fluids.We derive a forced Zakharov–Kuznetsov(ZK)–Burgers equation from the quasigeostrophic potential vorticity equation with dissipation and topography under the generalized beta effect,and by utilizing temporal and spatial multiple scale transform and the perturbation expansion method.Through the analysis of this model,it is found that the generalized beta effect and basic topography can induce nonlinear waves,and slowly varying topography is an external impact factor for Rossby waves.Additionally,the conservation laws for the mass and energy of solitary waves are analyzed.Eventually,the solitary wave solutions of the forced ZK–Burgers equation are obtained by the simplest equation method as well as the new modified ansatz method.Based on the solitary wave solutions obtained,we discuss the effects of dissipation and slowly varying topography on Rossby solitary waves.     

Adaptive multiscale finite-volume method for nonlinear multiphase transport in heterogeneous formations

In the previous multiscale finite-volume (MSFV) method, an efficient and accurate multiscale approach was proposed to solve the elliptic flow equation. The reconstructed fine-scale velocity field was then used to solve the nonlinear hyperbolic transport equation for the fine-scale saturations using an overlapping Schwarz scheme. A coarse-scale system for the transport equations was not derived because of the hyperbolic character of the governing equations and intricate nonlinear interactions between the saturation field and the underlying heterogeneous permeability distribution. In this paper, we describe a sequential implicit multiscale finite-volume framework for coupled flow and transport with general prolongation and restriction operations for both pressure and saturation, in which three adaptive prolongation operators for the saturation are used. In regions with rapid pressure and saturation changes, the original approach, with full reconstruction of the velocity field and overlapping Schwarz, is used to compute the saturations. In regions where the temporal changes in velocity or saturation can be represented by asymptotic linear approximations, two additional approximate prolongation operators are proposed. The efficiency and accuracy are evaluated for two-phase incompressible flow in two- and three-dimensional domains. The new adaptive algorithm is tested using various models with homogeneous and heterogeneous permeabilities. It is demonstrated that the multiscale results with the adaptive transport calculation are in excellent agreement with the fine-scale solutions. Furthermore, the adaptive multiscale scheme of flow and transport is much more computationally efficient compared with the previous MSFV method and conventional fine-scale reservoir simulation methods.