Pfaffian Technique Research Articles

Bilinear form, auto-Bäcklund transformations, Pfaffian, soliton, and breather solutions for a (3 + 1)-dimensional extended shallow water wave equation

Study of the water waves remains central to fluid physics, ocean dynamics, and engineering. In this paper, a (3 + 1)-dimensional extended shallow water wave equation is investigated via symbolic computation. Bilinear form and two kinds of the bilinear auto-Bäcklund transformations with some solutions are given via the Hirota method. The Nth-order Pfaffian solutions are worked out by means of the Pfaffian technique, where N is a positive integer. N-soliton solutions are exported through the Nth-order Pfaffian solutions. By virtue of the asymptotic analysis, elastic and inelastic interactions between the two solitons on some periodic backgrounds are discussed. Interaction among the three solitons is illustrated graphically. The higher-order breather solutions are investigated via the complex parameter relation. Elastic and inelastic interactions between the two breathers on the periodic backgrounds are depicted graphically. Hybrid solutions consisting of the solitons and breathers are obtained. Interaction between the one soliton and one breather on a periodic background is presented.

Pfaffian solutions and nonlinear waves of a (3 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt system in fluid mechanics

Fluid mechanics is concerned with the behavior of liquids and gases at rest or in motion, where the nonlinear waves and their interactions are important. Hereby, we study a (3 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt system in fluid mechanics. We determine a bilinear form of that system via the Hirota method. Nth-order Pfaffian solutions are obtained via the Pfaffian technique and our bilinear form, where N is a positive integer. Based on the Nth-order Pfaffian solutions, we derive the N-soliton, higher-order breather, and hybrid solutions. Using those solutions, we present the (1) elastic interaction between the two solitary waves with a short stem, (2) elastic interaction between the two solitary waves with a long stem, (3) fission between the two solitary waves, (4) fusion between the two solitary waves, (5) one breather wave, (6) elastic interaction between the two breather waves, (7) fission between the two breather waves, (8) fusion among the one breather wave and two solitary waves, and (9) elastic interaction between the one breather wave and one solitary wave.

Pfaffian, breather, and hybrid solutions for a (2 + 1)-dimensional generalized nonlinear system in fluid mechanics and plasma physics

Fluid mechanics is seen as the study on the underlying mechanisms of liquids, gases and plasmas, and the forces on them. In this paper, we investigate a (2 + 1)-dimensional generalized nonlinear system in fluid mechanics and plasma physics. By virtue of the Pfaffian technique, the Nth-order Pfaffian solutions are derived and proved, where N is a positive integer. Based on the Nth-order Pfaffian solutions, the first- and second-order breather solutions are obtained. In addition, Y-type and X-type breather solutions are constructed. Furthermore, we investigate the influence of the coefficients in the system on those breathers as follows: The locations and periods of those breathers are related to δ1, δ2, δ3, δ4, and δ5, where δc's (c=1,2,3,4,5) are the constant coefficients in the system. Moreover, hybrid solutions composed of the breathers and solitons are derived. Interactions between the Y/X-type breather and Y-type soliton are illustrated graphically, respectively. Then, we show the influence of the coefficients in the system on the interactions between the Y/X-type breather and Y-type soliton.

Hybrid waves for a (2 + 1)-dimensional extended shallow water wave equation

Shallow water waves are studied for the applications in hydraulic engineering and environmental engineering. In this paper, a (2 + 1)-dimensional extended shallow water wave equation is investigated. Hybrid solutions consisting of H-soliton, M-breather, and J-lump solutions have been constructed via the modified Pfaffian technique, where H, M, and J are the positive integers. One-breather solutions with a real function ϕ(y) are derived, where y is the scaled space variable; we notice that ϕ(y) influences the shapes of the background planes, and the one-breather solutions are localized along the curve (k1R+k2R)x+(k1Rk1I2+k1R2+k2Rk2I2+k2R2)ϕ(y)+w1t+Ω1R+Ω2R=0, while periodic along the curve (k1I+k2I)x−(k1Ik1I2+k1R2+k2Ik2I2+k2R2)ϕ(y)+w2t+Ω1I+Ω2I=0, where k1R, k1I, k2R, k2I, w1, w2, Ω1R, Ω1I, Ω2R, and Ω2I are the real constants. Discussions on the hybrid waves consisting of one breather and one soliton indicate that the one breather is not affected by one soliton after interaction. One-lump solutions with ϕ(y) are obtained with the condition k1R2<k1I2; we notice that the one lump consists of two low valleys and one high peak, and the amplitude and velocity keep invariant during its propagation. Hybrid waves consisting of the one lump and one soliton imply that the shape of the one soliton becomes periodic when ϕ(y) is changed from a linear function to a periodic function. Constant coefficient α can affect the propagation direction and velocity of the one breather and one lump, respectively.

Hybrid solutions for the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics

In fluid mechanics, the higher-dimensional and higher-order equations are constructed to describe the propagations of nonlinear waves. In this paper, we investigate the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics. The Nth-order Pfaffian solutions are constructed and proved via the modified Pfaffian technique. The higher-order soliton, first- and second-order breather solutions are constructed based on the Nth-order Pfaffian solutions. We graphically demonstrate that the amplitudes and velocities of the solitons are affected by some variable coefficients. Hybrid solutions composed of breathers, lumps and solitons are illustrated graphically. It can be found that when certain parameters are chosen, the breathers, lumps and solitons included in the hybrid solutions possess the same properties as those of the breather and lump solutions.

Topological sectors, dimer correlations, and monomers from the transfer-matrix solution of the dimer model

We solve the classical square-lattice dimer model with periodic boundaries and in the presence of a field t that couples to the (vector) flux, by diagonalizing a modified version of Lieb's transfer matrix. After deriving the torus partition function in the thermodynamic limit, we show how the configuration space divides into topological sectors corresponding to distinct values of the flux. Additionally, we demonstrate in general that expectation values are t independent at leading order, and obtain explicit expressions for dimer occupation numbers, dimer-dimer correlation functions, and the monomer distribution function. The last of these is expressed as a Toeplitz determinant, whose asymptotic behavior for large monomer separation is tractable using the Fisher-Hartwig conjecture. Our results reproduce those previously obtained using Pfaffian techniques.

Higher-order hybrid waves for the (2 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid via the modified Pfaffian technique

Fluids, as a phase of matter including liquids, gases and plasmas, are seen to be common in nature, the study of which helps the design in the related industries. In this paper, we optimize the Pfaffian technique and investigated the Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid. Higher-order hybrid solutions consisting of the L lumps, M breathers and N solitons are constructed with L, M and N being positive integers. Relative extrema of the breather and lump are presented, respectively. Breather is found to be localized along the curve $$a_1 x+b_1 \varphi (y)+\omega _1 t+\xi _1=0$$ and periodic along the curve $$\alpha _1 x+\beta _1 \varphi (y)+\gamma _1 t+\theta _1=0$$ . Under the lump existence condition, higher-order rogue wave solutions do not exist. Hybrid solutions composed of breathers, lumps and solitons are illustrated graphically. It can be found that when certain parameters are chosen, the breather, lump and soliton included in the hybrid solutions possess the same properties as those of the breather and lump solutions.

Solitons and breather waves for the generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt system in fluid mechanics, ocean dynamics and plasma physics

Abstract Under investigation in this paper is the (2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt system, which can be used to describe certain situations in fluid mechanics, ocean dynamics and plasma physics. The Nth-order Pfaffian and Wronskian solutions are derived via the Pfaffian and Wronskian techniques, respectively, where N is a positive integer. Asymptotic analysis implies that the interaction between the two solitons is elastic with certain conditions. Furthermore, we obtain the breather waves according to the extended homoclinic test technique. Propagation of the breather waves indicates that the breather waves can evolve periodically along a straight line with a certain angle with the x and y axes, and their wave lengthes, amplitudes and velocities remain unchanged during the propagation.

Solitons and periodic waves for the (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics

Fluid mechanics has the applications in a wide range of disciplines, such as oceanography, astrophysics, meteorology, and biomedical engineering. Under investigation in this paper is the ($$2+1$$)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics. Via the Pfaffian technique and certain constraint on the real constant $$\alpha $$, the Nth-order Pfaffian solutions are derived. One- and two-soliton solutions are obtained via the Nth-order Pfaffian solutions. Based on the Hirota–Riemann method, one- and two-periodic wave solutions are constructed. With the help of the analytic and graphic analysis, we notice that: (1) of the one soliton, amplitude is irrelevant to $$\gamma $$, a real constant coefficient in the equation, velocity along the x direction is independent of $$\gamma $$, while velocity along the y direction is proportional to $$\gamma $$; (2) one soliton keeps its amplitude and velocity invariant during the propagation and total amplitude of the two solitons in the interaction region is lower than that of any soliton; (3) one-periodic wave can be viewed as a superposition of the overlapping solitary waves, placed one period apart; (4) periodic behaviors for the two-periodic wave exist along the x and y directions, respectively; (5) under certain limiting conditions, one-periodic wave solutions approach to the one-soliton solutions and two-periodic wave solutions approach to the two-soliton solutions.

Solitons for the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid via the Pfaffian technique

Fluids are common in nature, the study of which helps the design of the related industries. Under investigation in this letter is the (2[Formula: see text]+[Formula: see text]1)-dimensional Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid. Pfaffian solutions have been obtained based on the Pfaffian technique with the assistance of the real auxiliary function [Formula: see text]. N-soliton solutions with [Formula: see text] are constructed, where y is the scaled space coordinate and [Formula: see text] is the steady stream function in the irrotational incompressible flow. Background shapes of the solutions are affected by [Formula: see text], but the structures of the solutions are affected by the derivative of the log terms in the solutions. Neighborhoods at the origins of the solitons are different in consequence of the different values of the real auxiliary parameter a. One- and two-soliton solutions are illustrated, which are the superpositions of the two kink solitons and different forms of [Formula: see text]. Interactions of the two solitons are presented, from which we see that the velocities, amplitudes and shapes of the two solitons remain unchanged before and after each interaction.

Multi-breather wave solutions for a generalized (3＋1)-dimensional Yu–Toda–Sasa–Fukuyama equation in a two-layer liquid

Two-layer fluid models are proposed to describe certain nonlinear phenomena in fluid mechanics, thermodynamics and medical sciences. For a generalized(3＋1)-dimensional Yu–Toda–Sasa–Fukuyama equation for the interfacial waves in a two-layer liquid, in which h0 is the constant coefficient of the development term, h1 is the constant coefficient of the dispersion term, h2 and h3 are the constant coefficients of the nonlinear terms, and h4 is the constant coefficient of the linear term. Based on the Pfaffian technique and certain constraint on h3, we obtain the multi-breather solutions in terms of the Gramian. For the one-breather waves, amplitudes are proportional to h1h2, velocity components along the x and z directions for the characteristic lines are proportional to h1h0, and velocity component along the y direction for the characteristic line is proportional to h1h43h02, where x, y and z are the spatial coordinates and t is the temporal coordinate. Interaction between the two-breather waves implies that the two-breather waves can evolve periodically along two straight lines on the x-y and y-z planes, period for the two-breather waves with t is proportional to h0h1, period for the two-breather waves along the y direction is proportional to h43h1, while the two-breather waves are not periodic along the x and z directions. Amplitudes, velocities and widths of the two-breather waves keep unchanged after the interaction between the two-breather waves on the x-y and y-z planes, which means that the interaction is elastic. On the x-z plane, the two-breather waves appear as two parallel solitons at certain angles with the x and z axes.

Bright–dark soliton solutions of the multi-component AB system

In this paper, we investigate a multi-component AB system that appears in geophysical fluid dynamics. We construct bright–dark soliton solutions of the system using the Hirota’s bilinear method. We obtain asymptotic behaviors of two-soliton solution for the two-component AB system and prove that the interactions between two bright and two dark solitons are elastic. Under various parameter conditions, the oblique interactions and bound states of solitons are analyzed in detail. By use of the Pfaffian technique, we present N-bright and N−dark soliton solutions of the two- and multi-component AB system. The results obtained in this paper can be helpful for the study of vector multi-dark solitons in many physical systems such as nonlinear optics and fluid dynamics.

Wronskian, Pfaffian and periodic wave solutions for a $$(2 + 1)$$ ( 2 + 1 ) -dimensional extended shallow water wave equation

Under investigation in this paper is a \((2 + 1)\)-dimensional extended shallow water wave equation. Bilinear form is obtained via the generalized dependent variable transformation. The Nth-order analytic solutions are, respectively, obtained via the Wronskian and Pfaffian techniques. Soliton solutions are constructed through the Nth-order solutions. Discussions on the propagation of the solitons indicate that the soliton solutions with \(\varphi (y)\) are more general than those without \(\varphi (y)\), and \(\varphi (y)\) could affect the features of the soliton solutions, where \(\varphi (y)\) is a real function related to the aforementioned transformation. One-periodic wave solutions are obtained via the Hirota–Riemann method. Relation between the one-periodic wave solutions and one-soliton solutions is studied, which indicates that the one-periodic wave solutions can approach to the one-soliton solutions under certain condition.

Bilinear Bäcklund transformation, Lax pair and multi-soliton solution for a vector Ramani equation

In this paper, a vector Ramani equation is proposed by using the bilinear approach. With the help of the bilinear exchange formulae, bilinear Bäcklund transformation and the corresponding Lax pair for the vector Ramani equation are derived. Besides, multi-soliton solution expressed by pfaffian is given and proved by pfaffian techniques.

Pfaffian-Type Soliton Solution to a Multi-Component Coupled Ito Equation

Multi-soliton solution to a multi-component coupled Ito system is investigated based on the Hirota bilinear method. By virtue of the perturbation method, we firstly derive one- and two-soliton solutions for the coupled Ito system possessing four components. Then the multi-soliton solution for the multi-component coupled Ito system is summarized into a general form expressed by pfaffians. Finally, this general pfaffian-type soliton solution is proved by pfaffian techniques.

Multi-component generalizations of the Hirota–Satsuma coupled KdV equation

In this paper, we consider multi-component generalizations of the Hirota–Satsuma coupled Korteweg–de Vries (KdV) equation. By introducing a Lax pair, we present a matrix generalization of the Hirota–Satsuma coupled KdV equation, which is shown to be reduced to a vector Hirota–Satsuma coupled KdV equation. By using Hirota’s bilinear method, we find a few soliton solutions to the vector Hirota–Satsuma coupled KdV equation in a symmetric case. Finally, in this symmetric case, we give a multi-soliton solution expressed by pfaffians and prove it by pfaffian techniques.

Rényi entanglement entropies in quantum dimer models: from criticality to topologicalorder

Thanks to Pfaffian techniques, we study the Rényi entanglement entropies and the entanglementspectrum of large subsystems for two-dimensional Rokhsar–Kivelson wavefunctionsconstructed from a dimer model on the triangular lattice. By including a fugacityt on some suitable bonds, one interpolates between the triangular lattice (t = 1) and thesquare lattice (t = 0). The wavefunction is known to be a massive topological liquid for t > 0 whereas it is a gapless critical state att = 0. We mainly consider two geometries for the subsystem: that of a semi-infinitecylinder and the disc-like set-up proposed by Kitaev and Preskill (2006 Phys.Rev. Lett. 96 110404). In the cylinder case, the entropies contain an extensiveterm—proportional to the length of the boundary—and a universal subleading constantsn(t). Fitting these cylinder data (up to a perimeter ofL = 32 sites) providessn with a very highnumerical accuracy (10 − 9 at t = 1 and 10 − 6 at t = 0.5). In the topological liquid phase we find sn(t > 0) = − ln2,independent of the fugacity t and the Rényi parameter n. At t = 0 we recover a previously known result, for n < 1 and sn(t = 0) = − ln(2)/(n − 1) for n > 1. In the disc-like geometry—designed to get rid of the boundary contributions—we find an entropysnKP(t0) = − ln2 in the whole massivephase whatever n > 0, in agreement with the result of Flammia et al (2009 Phys. Rev. Lett. 103 261601). Someresults for the gapless limit are discussed.

Universality in phase boundary slopes for spin glasses on self-dual lattices

We study the effects of disorder on the slope of the disorder–temperature phase boundary near the Onsagerpoint (Tc = 2.269...) in spin-glass models. So far, studies have focused on marginal or irrelevant cases ofdisorder. Using duality arguments, as well as exact Pfaffian techniques, we reproduce theseanalytical estimates. In addition, we obtain different estimates for spin-glass models onhierarchical lattices where the effects of disorder are relevant. We show that the phaseboundary slope near the Onsager point can be used to probe for the relevance of disordereffects.

Application of the Pfaffian technique to the KR and mNVN equations

In this paper we discuss the differentiation of Pfaffians which is useful in Pfaffian technique. We apply the Pfaffian technique to two integrable systems; the Konopelchenko–Rogers (KR) equations and the modified Novikov–Veselov–Nizik (mNVN) equations and show that these equations in the bilinear form reduce to a Pfaffian identity. We also derive a new Lax pair for the mNVN equations which is gauge equivalent to a pair of operators.

Determinant and Pfaffian solutions of the strong coupling limit of integrable discrete NLS systems

The strong coupling limits of the integrable semi-discrete and fully discrete nonlinear Schrödinger systems are studied by using the Hirota bilinear method. The determinant solutions (in both infinite and finite lattice cases) for the strong coupling limits of semi-discrete and fully discrete nonlinear Schrödinger systems are obtained using a determinant technique. The vector generalizations of the strong coupling limits of semi-discrete and fully discrete nonlinear Schrödinger systems are also presented. The Pfaffian solutions for vector systems are obtained using the Pfaffian technique.