Backlund Transformation Research Articles

Bubbles interactions in fluidized granular medium for the van der Waals hydrodynamic regime

This paper investigates the fluidized granular materials (FGM) with the van der Waals normal form (VDWF) under the effects of friction and viscosity. The system of macroscopic balance is presented, including the mass, momentum, and energy equations of local densities. For two different types of collisions, elastic and inelastic collisions, analytical solutions of the nonlinear PDEs governing the granular model are investigated using the hydrodynamic equations for granular matter motion. The integrability of the proposed model is analyzed by applying the Painleve analysis. Moreover, the Backlund transformation (BT) is established using the Painleve truncation expansion. New traveling wave solutions of the VDWF within FGM are obtained by using the BT, tanh function, Jacobi elliptic function methods to study the phase separation phenomenon. As two pairs of rarefaction and shock waves emerge and travel away giving the appearance of bubbles, the resulting solutions of the proposed model show a behavior similar to those found in the molecular dynamic simulations. The dispersion relation and their properties to the model equation are investigated. Besides, stability analysis of the VDWF in its ODE form is demonstrated using the phase portrait classifications. Finally, using two- and three-dimensional graphics for seeking model solutions under the influence of friction and viscosity, qualitative agreements with previous related works are shown.

Bilinear Bäcklund transformation, kink and breather-wave solutions for a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation in fluid mechanics

Fluid mechanics studies are applied in the turbines, airplanes, ships, rivers, windmills, pipes, etc. Under investigation is a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation in fluid mechanics. Via the Hirota bilinear method, bilinear Backlund transformation is obtained, based on which the kink solutions are constructed. Breather-wave solutions are constructed via the extended homoclinic test approach. When the periods of the breather-wave solutions tend to infinity, the lump solutions are derived. It is found that the amplitudes and shapes of the breather waves keep unchanged during the propagation. Effects of the coefficients in the equation on the amplitudes and velocities for the breather waves are analyzed graphically.

A Note on a Three-Dimensional Bäcklund Transformation

The Backlund transformation (BT) for a three-dimensional nonlinear wave equation and its nonlinear superposition formula are studied in this note. We prove that the three dimensional Backlund transformation obtained by Leibbrandt, et al. can be decomposed into three two-dimensional BTs. Some results on the N-dimensional Liouville equation are also discussed in the article.

Non-differentiable solutions for non-linear local fractional heat conduction equation

Fractional calculus has many advantages. Under consideration of this paper is a (2+1)-dimensional non-linear local fractional heat conduction equation with arbitrary degree non-linearity. Backlund transformation of a reduced form of the local fractional heat conduction equation is constructed by Painleve analysis. Based on the Backlund transformation, some exact non-differentiable solutions of the local fractional heat conduction equation are obtained. To gain more insights of the obtained solutions, two solutions are constrained to a Cantor set and then two spatio-temporal fractal structures with profiles of these two solutions are shown. This paper further reveals by local fractional heat conduction equation that fractional calculus plays important role in dealing with non-differentiable problems.

Bilinear Bäcklund transformation, Lax pair and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics

In this paper, outcomes of the study on the Backlund transformation, Lax pair, and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics, fluid mechanics, and plasma physics are presented. Via the Hirota bilinear method, a bilinear Backlund transformation is obtained, based on which a Lax pair is constructed. Via the symbolic computation, mixed rogue–solitary and rogue–periodic wave solutions are derived. Interactions between the rogue waves and solitary waves, and interactions between the rogue waves and periodic waves, are studied. It is found that (1) the one rogue wave appears between the two solitary waves and then merges with the two solitary waves; (2) the interaction between the one rogue wave and one periodic wave is periodic; and (3) the periodic lump waves with the amplitudes invariant are depicted. Furthermore, effects of the noise perturbations on the obtained solutions will be investigated.

Bäcklund transformation, Pfaffian, Wronskian and Grammian solutions to the $$(3+1)$$-dimensional generalized Kadomtsev–Petviashvili equation

With the Hirota bilinear method and symbolic computation, we investigate the $$(3+1)$$ -dimensional generalized Kadomtsev–Petviashvili equation. Based on its bilinear form, the bilinear Backlund transformation is constructed, which consists of four equations and five free parameters. The Pfaffian, Wronskian and Grammian form solutions are derived by using the properties of determinant. As an example, the one-, two- and three-soliton solutions are constructed in the context of the Pfaffian, Wronskian and Grammian forms. Moreover, the triangle function solutions are given based on the Pfaffian form solution. A few particular solutions are plotted by choosing the appropriate parameters.

Explicit solutions and Darboux transformations of a generalized D-Kaup–Newell hierarchy

Darboux and Backlund transformations on integrable couplings are formulated and generalized. The generalization of the theory pertains to spectral problems where the spatial spectral matrix is a polynomial in $$\lambda $$ of any order. A specific application to a generalized D-Kaup–Newell integrable couplings system is worked out, along with an explicit formula for the associated Backlund transformation. Solutions are given for the 0, 1, 2, 3-order generalized D-Kaup–Newell integrable coupling system. Formulas for the general m-th-order integrable couplings system are seen. Graphs of explicit solutions to the fourth-order integrable couplings are presented for chosen parameters showing solitons. A brief discussion about open problems and physical implications of the paper concludes the paper.

Binary relations, Bäcklund transformations, and wave packet propagation

We propose a mathematical apparatus based on binary relations that expands the possibility of traditional analysis applied to problems in mathematical and theoretical physics. We illustrate the general constructions with examples with an algebraic description of Backlund transformations of nonlinear systems of partial differential equations and the dynamics of wave packet propagation.

Painlevé test, complete symmetry classifications and exact solutions to R–D types of equations

In this paper, the combination of Painlevé analysis and symmetry classification is performed on the reaction-diffusion (R–D) types of equations, the Painlevé properties (PPs) and Bäcklund transformations (BTs) are obtained under some conditions. Then, all of the point symmetries of the equations are given by Lie group classification method, moreover, the complete generalized symmetry classifications of the general R–D equation are provided using characteristics of predetermined order, and the integrability of the nonlinear equations are considered by the generalized symmetry classification method. Furthermore, the exact solutions to the equations generated from Painlevé expansions and symmetry reductions are investigated.

The short pulse equation: Bäcklund transformations and applications

AbstractA Bäcklund transformation (BT), which involves both independent and dependent variables, is established and studied for the short pulse (SP) equation. Based it, the nonlinear superposition formulae for 2‐, 3‐, and 4‐BT are presented. The general result for the composition of ‐BTs is achieved and given in terms of determinants. As applications, various solutions including loop solitons, breather solutions, and their interaction solutions are worked out.

Analytic Properties of Solutions to Equations in the Generalized Hierarchy of the Second Painlevé Equation

We consider the analytic properties of solutions to equations of arbitrary order in the generalized hierarchy of the second Painleve equation. The local properties of solutions, the Backlund transformations, and rational solutions and their representations via generalized Yablonskii–Vorob’ev polynomials are studied.

Bäcklund transformations and Riemann–Bäcklund method to a (3 + 1)-dimensional generalized breaking soliton equation

In this paper, the bilinear method is employed to investigate the N-soliton solutions of a (3 + 1)-dimensional generalized breaking soliton equation. Three sets of bilinear Backlund transformations are obtained by means of gauge transformation. The Riemann–Backlund method is further extended to the (3 + 1)-dimensional nonlinear integrable systems. The quasiperiodic wave solutions of the (3 + 1)-dimensional generalized breaking soliton equation are systematically analyzed. The asymptotic properties of the quasiperiodic solutions are discussed by using a limiting procedure. The one-periodic and two-periodic waves tend to the 1-soliton and 2-soliton under a small amplitude limit, respectively. The dynamical characteristics of the one- and two-periodic waves are summarized by selecting different parameters. Furthermore, we obtain some new types of the quasiperiodic wave solutions of the variable coefficient (3 + 1)-dimensional generalized breaking soliton equation. These solutions present the dynamical behaviors of C-type, anti-C-type and Z-type periodic waves moving on the background of the periodic waves of bell type.

Lump solutions and bilinear Bäcklund transformation for the $$(4+1)$$-dimensional Fokas equation

The $$(4+1)$$ -dimensional Fokas (4DFK) equation is explicitly written out by means of the linearized operator of the 4DFK equation after introducing an additional auxiliary variable. Based on the bilinear form, bilinear Backlund transformations consisting of three bilinear equations are provided. A new class of lump solutions is given and analyzed using the extremum theory of multivariate function, of which the peculiar dynamic property is demonstrated graphically.

A New $$(3+1)$$-dimensional Hirota Bilinear Equation: Its Bäcklund Transformation and Rational-type Solutions

The behavior of specific dispersive waves in a new $$(3+1)$$ -dimensional Hirota bilinear (3D-HB) equation is studied. A Backlund transformation and a Hirota bilinear form of the model are first extracted from the truncated Painleve expansion. Through a series of mathematical analyses, it is then revealed that the new 3D-HB equation possesses a series of rational-type solutions. The interaction of lump-type and 1-soliton solutions is studied and some interesting and useful results are presented.

Luigi Bianchi’s early investigations on pseudospherical surfaces

The paper provides a historical analysis of Luigi Bianchi’s early contributions to the transformation theory of pseudospherical surfaces. The origin of a geometrical construction known as complementary transformation is carefully studied by emphasizing the connection of Bianchi’s achievement with the “classical” methods of Gauss and Weingarten. Furthermore, the reception process of Bianchi’s theory is discussed in the context of Lie’s integration theory of PDE’s and of Backlund transformations.

Bäcklund Transformations for the Degasperis-Procesi Equation

We consider the Backlund transformation for the Degasperis-Procesi (DP) equation. Using the reciprocal transformation and the associated DP equation, we construct the Backlund transformation for the DP equation involving both dependent and independent variables. We also obtain the corresponding nonlinear superposition, which we use together with the Backlund transformation to derive some soliton solutions of the DP equation.

On Backlund transformation of Riccati equation method and its application to nonlinear partial differential equations and differential-difference equations

On Backlund transformation of Riccati equation method and its application to nonlinear partial differential equations and differential-difference equations

Applications of three methods for obtaining optical soliton solutions for the Lakshmanan–Porsezian–Daniel model with Kerr law nonlinearity

This paper examines new travelling wave solutions to the Lakshmanan–Porsezian–Daniel (LPD) model with Kerr nonlinearity using Backlund transformation method based on Riccati equation, Kudryashov method and a new auxiliary ordinary differential equation (ODE). The three methods are adequately utilised, and some new rational-type hyperbolic and trigonometric function solutions are derived in different shapes for the aforementioned model. We confirm that our methods are more efficient than the other methods and it might be used in many other such types of nonlinear equations arising in the basic fabric of communications network technology and nonlinear optics.

Painlevé analysis, group classification and exact solutions to the nonlinear wave equations

This paper is concerned with the general regular long-wave (RLW) types of equations. By the combination of Painleve analysis and Lie group classification method, the conditional Painleve property (PP) and Backlund transformations (BTs) of the nonlinear wave equations are provided under some conditions. Then, all of the point symmetries of the nonlinear RLW types of equations are obtained, the exact solutions to the equations are investigated. Particularly, some explicit solutions are provided by the special function and Φ-expansion method.

Multiple residual symmetries and soliton-cnoidal wave interaction solution of the $$(2+1)$$-dimensional negative-order modified Calogero–Bogoyavlenskii–Schiff equation

The residual symmetry for the \((2+1)\)-dimensional negative-order modified Calogero–Bogoyavlenskii–Schiff (nmCBS) equation is derived from the truncated Painleve expansion, and is extended to the multiple residual symmetries, which can be transformed to Lie point symmetries by introducing a suitable prolonged system. The nth Backlund transformation (BT) related to multiple residual symmetries is given in terms of determinant. More importantly, we obtain the explicit soliton-cnoidal wave interaction solution from a consistent differential equation.