It is shown that the backlund transformation generates an important class of nonlinear evolution equations exhibiting nsoliton solutions. The implicit midpoint rule is used to advance the solution in time. The same stencil is used everywhere on the lattice. This equation was developed as a model for the unidirectional. The proposed method in this work is based on a hirota bilinear differential equation. The goal of this paper is to provide an intuition for some of these results. A petrovgalerkin method and product approximation technique are used to solve numerically the hirotasatsuma coupled kortewegde vries equation, using cubic splines as test functions and a linear spline as trial functions.
Pdf discrete hirota reductions associated with the. The he can be written in the following operator form with two free parameters. It is known as schrodingerhirota equation she, which is moderately different from ordinary nonlinear schrodingers equation that involves the solitons investigation for optical fibers propagation. Hirota quadratic equations for the extended toda hierarchy milanov, todor e. We present explicit forms for the two lowerorder solutions. Mutual conversions between the two become possible when the evolution equation contains a sufficient number of free parameters which can be used to control their solutions. Hirota satsuma equation appeared in the theory of shallow water waves, first discussed by hirota, ryogo. Differential equations department of mathematics, hkust. We considered the relation between two famous integrable equations. The values of these special eigenvalues depend on two free parameters that are present in. In the basic case n3, when the system reduces to a single equation, it was discovered, up to a change of independent variables, by hirota 1981, who called it the discrete analogue of the twodimensional toda lattice, as a culmination of his studies on the bilinear form of nonlinear integrable equations. Pdf modelling of wave motion and propagation characteristics of waves plays an important role in coastal, ocean and maritime.
Lump solutions to nonlinear partial differential equations. This occurs for the firstorder, as well as higher orders, of breather solutions. The evolution operator is explicitly constructed in the quantum variant of the model and the integrability of the corresponding classical finitedimensional system is established. And then by exact analysis of the energy equation, it is shown that the global weak attractor is actually the global strong attractor in h per k.
Stages of the hirota method example of the kdv equation in order to identify the four stages of the hirota method we will pursue an. These equations are very good candidates to be integrable. In this paper, we investigate the nonlinear wave solutions for a dimensional equation which can be reduced to the potential kdv equation. The introduction of this approach provided a direct method for nding nsoliton solutions to nonlinear evolutionary equations and, by way of an example, hirota applied this method to the kortewegdevries kdv equation hir71, i. The global weak attractor for this system in h per k is constructed. If we analogize results of extended trial equation method with results of generalized kudryashov method, then we can conclude that the solutions of hirota equation and hirotamaccari system that we obtained by using extended trial equation. Dispersive optical solitons with schrodingerhirota equation. The hamiltonian formalism is developed for the sinegordon model on the spacetime lightlike lattice, first introduced by hirota. The hirota difference equation hde and the darboux system that describes conjugate curvilinear systems of coordinates in r 3. The form of the solutions to the equation is constructed and the solutions are improved through analysis and symbolic computations with maple.
One of the most famous method to construct multisoliton solutions is the hirota direct method. The integrability of the new models is established by providing their explicit forms of lax pairs or zero curvature conditions. We demonstrated that specific properties of solutions of the hde with respect to independent variables enabled introduction of an infinite set of discrete symmetries. Part 2 hirotas bilinear method for lattice equations. Solutions to hirotasatsuma kdv system 1819 4 rational solutions to shkdv system 1 we seek solutions to system 5 in the form f fx,t 3 i,j0 a ijt ixj, g gx,t 3 i,j0 b ijt ixj. Global attractor for hirota equation global attractor for hirota equation zhang, ruifeng. Mar 03, 2008 the long time behavior of solution for hirota equation with zero order dissipation is studied. The dependent variable u ux,t is a realvalued function of the two real variables xand t. Aug 23, 2019 we construct several new integrable systems corresponding to nonlocal versions of the hirota equation, which is a particular example of higher order nonlinear schrodinger equations.
Wronskian determinant solutions of hirota equations known in the literature. Dark soliton solutions of the coupled hirota equation in. Hbde is defined as hirota bilinear difference equation rarely. It is integrable in the sense that it arises as the compatibility condition of a linear system lax pair. The dynamics of dispersive optical solitons, modeled by schrodingerhirota equation, are studied in this paper. For example, the free motion in space is of course integrable but it is not a good starting point for a description. Currysoliton solutions of integrable systems and hirota s method 3 partial differential equations pdes and the methods for generating soliton solutions have led to many deep ideas in mathematics and physics. A backlund transformation for the boussinesq equation is given in the bilinear from. Hirota bilinear equations with linear subspaces of solutions. A study of optical wave propagation in the nonautonomous. We present generalized soliton solutions in which some arbitrarily differentiable functions are involved by using a simplified hirotas method. Hirota bilinear difference equation listed as hbde. Exact nenvelopesoliton solutions of the hirota equation.
We study the integrability of a family of birational maps obtained as reductions of the discrete hirota equation, which are related to travelling wave solutions of the lattice kdv equation. For many years the hirota direct method and painlev. Hirota equation as an example of an integrable symplectic map. Nonlinear evolution equations generated from the backlund. The application of homotopy analysis method to solve a generalized hirotasatsuma coupled kdv equation. In this paper, the variational iteraton method is used for solving the generalized. Similarities between elements of quantum and classical theories of integrable systems are discussed. On using the theorems, we can construct a new soliton equation through two soliton equations with similar properties. We show that each model gives multiple soliton solutions, where the structures of the obtained solutions differ.
On the soliton solutions of a family of tzitzeica equations babalic, corina n. We know that another famous test for integrability is painlev. Stages of the hirota method example of the kdv equation. The linear superposition principle of exponential travelling waves is analysed for equations of hirota bilinear type, with an aim to construct a specific subclass of n soliton solutions formed by linear combination of exponential travelling waves. New generalized soliton solutions for a dimensional. The hirotamiwa equation is studied from the view point of derived category. Dec 11, 2009 in the basic case n3, when the system reduces to a single equation, it was discovered, up to a change of independent variables, by hirota 1981, who called it the discrete analogue of the twodimensional toda lattice, as a culmination of his studies on the bilinear form of nonlinear integrable equations. July, 2005 the search for integrability of nonlinear partial di. Aug 06, 2001 we consider the coupled hirota equation which describes the pulse propagation in a coupled fiber with higherorder dispersion and selfsteepening. A complete determination of quadratic functions positive in space and time is given. Bright dark optical solitons for schrodingerhirota.
Discrete hirotas equation in quantum integrable models. Bright, dark and singular optical soliton solutions to this model are obtained in. A complete determination of quadratic functions positive. China bthe graduate school of china academy of engineering physics, beijing 88, p. We apply the reduction technique to the lax pair of the kadomtsevpetviashvili equation and demonstrate the integrability property of the new equation, because we obtain the corresponding lax pair.
Desargues maps and the hirotamiwa equation proceedings. Typical integrable equations, such as the kdv equation, the boussinesq equation and the kp equation, possess multisol iton solutions, generated from combinations of multiple exponential waves on the basis of their hirota bilinear forms 1. In addition to being quadratic in the dependent variables, an equation in the hirota bilinear form must also satisfy a condition with respect to the derivatives. If we analogize results of extended trial equation method with results of generalized kudryashov method, then we can conclude that the solutions of hirota equation and hirotamaccari system that we obtained by using extended trial equation method in this paper are more general solutions. Rogue waves and rational solutions of the hirota equation. Rogue waves and rational solutions of the hirota equation core. The evolution operator is explicitly constructed in the quantum variant of the model and the integrability of the corresponding classical finite. Meanwhile, exact nenvelopesoliton solutions of the hirota equation are derived through the trace method.
A few threedimensional plots and contour plots of three. Using the painleve analysis, we obtain the parametric conditions for the existence of bright and dark solitons. Department of physics, university of turku, turku, finland abstract in this lecture we will. A numerical solution for hirotasatsuma coupled kdv equation. The simplified hirotas method for studying three extended. On linear superposition principle applying to hirota. Hbde hirota bilinear difference equation acronymfinder. The hirota bilinear method is applied to construct exact analytical one solitary wave solutions of some class of nonlinear di erential equations. In describing wave propagation in the ocean and optical. The hirotamiwa equation also known as the discrete kp equation, or the octahedron recurrence, is a bilinear partial difference equation in three independent variables. Department of mathematics, umalqurah university, makkah, saudi arabia received 31 december 2007, accepted 12 october 2008 abstract. Longtime asymptotics for the hirota equation on the halfline.
Hirotasatsuma equation appeared in the theory of shallow water waves, first discussed by hirota, ryogo. The values of these special eigenvalues depend on two free parameters that are present in the hirota equation. The laxtype equation, the sawadakoteratype equation and the cdgtype equation are derived from the extended kdv equation. Hirota bilinear difference equation how is hirota bilinear. The basis of success is the hirota bilinear formulation and the primary object is the class of positive multivariate quadratic functions. Abbasbandy, the application of homotopy analysis method to solve a generalized hirotasatsuma coupled kdv equation, physics letters a. Numerical simulation of the generalized hirotasatsuma. How is hirota bilinear difference equation abbreviated. Shiesser traveling wave analysis of partial differential p5 equations academy press. In order to apply hirota s method it is necessary that the equation is quadratic and that the derivatives only appear in combinations that can be expressed using hirota s.
Pdf all exact travelling wave solutions of hirota equation and. On reductions of the hirotamiwa equation article pdf available in symmetry integrability and geometry methods and applications may 2017 with 83. Free differential equations books download ebooks online. Global attractor for hirota equation, applied mathematicsa. The multisoliton solutions to the kdv equation by hirota method. The multisoliton solutions to the kdv equation by hirota. Numerical simulation of the generalized hirotasatsuma coupled kdv equations by variational iteration method laila m. Hirotasatsuma equation has multiple soliton solutions and traveling wave solutions. All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. Some key ideas in quantum theory, now standard in the quantum inverse scattering method, are identified with typical constructions in classical soliton. The basic starting point is a hirota bilinear form of the hirotasatsumaito equation. This paper investigates the nonautonomous schrodingerhirota equation with. May 02, 2017 on reductions of the hirotamiwa equation article pdf available in symmetry integrability and geometry methods and applications may 2017 with 83 reads how we measure reads.
The hirota direct method was rst published in a paper by hirota in 1971. In this work we study three extended higherorder kdvtype equations. Soliton solutions of integrable systems and hirotas method justin m. Finally, figure of the solution is made for specific examples. Longtime asymptotics for the hirota equation on the halfline boling guoa, nan liub. Hirotas bilinear method and soliton solutions jarmo hietarinta. Desargues maps and the hirotamiwa equation proceedings of. We consider the hirota equation on the quarter plane with the initial. If you want to learn differential equations, have a look at differential equations for engineers if your interests are matrices and elementary linear algebra, try matrix algebra for engineers if you want to learn vector calculus also known as multivariable calculus, or calculus three, you can sign up for vector calculus for engineers. One of the simplest equations of this type is the socalled hirota equation he 4,5. Pdf the application of homotopy analysis method to solve. Hbde stands for hirota bilinear difference equation.
First one is the system of multidimensional nonlinear wave equation with the reaction part in form of the third order polynomial determined by three distinct constant vectors. Moving breathers and breathertosoliton conversions for the. Hirota bilinear equations with linear subspaces of solutions wenxiu maa,b. An analytic expression for the condition of the transformation is given and several examples of transformations are presented. For the identified cases, we also construct the common lax pair and the soliton solutions are constructed for the dark soliton case. In writing this book he had endeavoured to supply some elementary material suitable for the needs of students who are studying the subject for the first time, and also some more advanced work which may be useful to men who are interested more in physical mathematics than in the developments of differential geometry and the theory of functions. Moving breathers and breathertosoliton conversions for the hirota. Computing exact solutions to hirotasatsuma kdv system. We consider the coupled hirota equation which describes the pulse propagation in a coupled fiber with higherorder dispersion and selfsteepening. Moving breathers and breathertosoliton conversions for. The hirota equation is a modified nonlinear schrodinger equation nlse that takes. It results into dissimilar type of nonlinear evolution equation, describing the eminent characteristics of optical soliton transmission. All exact travelling wave solutions of hirota equation and. Global attractor for hirota equation, applied mathematics.
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