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Plamen Koshlukov - One of the best experts on this subject based on the ideXlab platform.
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weak polynomial identities for a vector space with a symmetric Bilinear Form
arXiv: Rings and Algebras, 2019Co-Authors: Vesselin Drensky, Plamen KoshlukovAbstract:Let $V_k$ be a $k$-dimensional vector space with a non-degenerate symmetric Bilinear Form over a field $K$ of characteristic 0 and let $C_k$ be the Clifford algebra on $V_k$. We study the weak polynomial identities of the pair $(C_k,V_k)$. We establish that all they follow from $[x_1^2,x_2]=0$ when $k=\infty$ and from $[x_1^2,x_2]=0$ and $S_{k+1}(x_1,\ldots,x_{k+1})=0$ when $k<\infty$. We also prove that the weak identity $[x_1^2,x_2]=0$ satisfies the Specht property. As a consequence we obtain a new proof of the theorem of Razmyslov that the weak Lie polynomial identities of the pair $(M_2(K),sl_2(K))$ follow from $[x_1^2,x_2]=0$.
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Embeddings for the Jordan algebra of a Bilinear Form
Advances in Mathematics, 2018Co-Authors: Claudemir Fidelis, Diogo Diniz, Plamen KoshlukovAbstract:Abstract Let K be a field of characteristic zero and let J be a Jordan algebra with a Formal trace. We prove that the algebra J can be embedded into a Jordan algebra of a non-degenerate symmetric Bilinear Form over some associative and commutative K-algebra C if and only if J satisfies all trace identities of the Jordan algebra of a non-degenerate symmetric Bilinear Form over the field K. This is an extension of results of Procesi and Berele concerning the analogous problem for the associative matrix algebras and the matrix algebras with involution. As a consequence of these results we also prove that the ideal of all trace identities of the Jordan algebra of a non-degenerate symmetric Bilinear Form over K satisfies the Specht property.
Bo Tian - One of the best experts on this subject based on the ideXlab platform.
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Bilinear Form and n shock wave solutions for a 2 1 dimensional breaking soliton equation in certain fluids with the bell polynomials and auxiliary function
Studies in Applied Mathematics, 2013Co-Authors: Yan Jiang, Bo Tian, Pan WangAbstract:In this paper, we will investigate a (2+1)-dimensional breaking soliton (BS) equation for the (2+1)-dimensional collision of a Riemann wave with a long wave in certain fluids. Using the Bell polynomials and an auxiliary function, we derive a new Bilinear Form for the (2+1)-dimensional BS equation, which is different from those in the previous literatures. One-, two- and N-shock-wave solutions are obtained with the Hirota method and symbolic computation. One shock wave is found to be able to stably propagate. Two shock waves are observed to have the parallel collision, oblique collision, and stable propagation of the V-type structure. In addition, we present the collision between one shock wave and V-type structure, and the collision between two V-type structures.
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Bilinear Form and soliton interactions for the modified kadomtsev petviashvili equation in fluid dynamics and plasma physics
Nonlinear Dynamics, 2013Co-Authors: Yan Jiang, Bo Tian, Pan WangAbstract:In this paper, we investigate the modified Kadomtsev–Petviashvili (mKP) equation for the nonlinear waves in fluid dynamics and plasma physics. By virtue of the rational transFormation and auxiliary function, new Bilinear Form for the mKP equation is constructed, which is different from those in previous literatures. Based on the Bilinear Form, one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. Propagation and interactions of shock and solitary waves are investigated analytically and graphically. Parametric conditions for the existence of the shock, elevation solitary, and depression solitary waves are given. From the two-soliton solutions, we find that the (i) parallel elastic interactions can exist between the (a) shock and solitary waves, and (b) two elevation/depression solitary waves; (ii) oblique elastic interactions can exist between the (a) shock and solitary waves, and (b) two solitary waves; (iii) oblique inelastic interactions can exist between the (a) two shock waves, (b) two elevation/depression solitary waves, and (c) shock and solitary waves.
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Bilinear Form and soliton solutions for the coupled nonlinear klein gordon equations
International Journal of Modern Physics B, 2012Co-Authors: Xianghua Meng, Bo TianAbstract:With the coupling of a scalar field, a generalization of the nonlinear Klein–Gordon equation which arises in the relativistic quantum mechanics and field theory, i.e., the coupled nonlinear Klein–Gordon equations, is investigated via the Hirota method. With the truncated Painleve expansion at the constant level term with two singular manifolds, the coupled nonlinear Klein–Gordon equations are transFormed to a Bilinear Form. Starting from the Bilinear Form, with symbolic computation, we obtain the N-soliton solutions for the coupled nonlinear Klein–Gordon equations.
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backlund transFormation in Bilinear Form for a higher order nonlinear schrodinger equation
Nonlinear Analysis-theory Methods & Applications, 2008Co-Authors: Bo TianAbstract:Backlund transFormation in Bilinear Form is presented for a higher-order nonlinear Schrodinger equation, which describes the propagation of ultrashort light pulses in optical fibers. With symbolic computation and starting from the Backlund transFormation, the analytical soliton solution is obtained from a trivial solution and the inverse scattering transForm scheme is also derived. Furthermore, the N-soliton solution in double Wronskian Form is given, and the value of the arbitrary constant appearing in the Backlund transFormation is determined for a transFormation between the (N−1) and N-soliton solutions. The results obtained from the Backlund transFormation might be valuable in optical communications.
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variable coefficient higher order nonlinear schrodinger model in optical fibers variable coefficient Bilinear Form backlund transFormation brightons and symbolic computation
Physics Letters A, 2007Co-Authors: Bo TianAbstract:Symbolically investigated in this Letter is a variable-coefficient higher-order nonlinear Schrodinger (vcHNLS) model for ultrafast signal-routing, fiber laser systems and optical communication systems with distributed dispersion and nonlinearity management. Of physical and optical interests, with Bilinear method extend, the vcHNLS model is transFormed into a variable-coefficient Bilinear Form, and then an auto-Backlund transFormation is constructed. Constraints on coefficient functions are analyzed. Potentially observable with future optical-fiber experiments, variable-coefficient brightons are illustrated. Relevant properties and features are discussed as well. Backlund transFormation and other results of this Letter will be of certain value to the studies on inhomogeneous fiber media, core of dispersion-managed brightons, fiber amplifiers, laser systems and optical communication links with distributed dispersion and nonlinearity management.
Yitian Gao - One of the best experts on this subject based on the ideXlab platform.
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Bilinear Form soliton breather lump and hybrid solutions for a varvec 2 1 2 1 dimensional sawada kotera equation
Nonlinear Dynamics, 2020Co-Authors: Yitian Gao, Tingting Jia, Cuicui Ding, Yujie FengAbstract:In this paper, we investigate a ( $$2+1$$ )-dimensional Sawada–Kotera (SK) equation for the atmosphere, rivers, lakes, oceans, as well as the conFormal field and two-dimensional quantum gravity gauge field. Bilinear Form and N-soliton solutions, which are different from those in the existing literatures, are derived, where N is a positive integer. The higher-order breather, lump and hybrid solutions for the ( $$2+1$$ )-dimensional SK equation are also constructed based on the N-soliton solutions. Three kinds of the first-order breathers are obtained, and the higher-order breathers are constructed. The higher-order lump solutions are also derived via the long-wave limit method. Hybrid solutions composed of the solitons, breathers and lumps are worked out, and interaction between the waves is discussed graphically. Finally, similar solutions for a generalized Form of the ( $$2+1$$ )-dimensional SK equation are given.
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Bilinear Form solitons breathers and lumps of a 3 1 dimensional generalized konopelchenko dubrovsky kaup kupershmidt equation in ocean dynamics fluid mechanics and plasma physics
European Physical Journal Plus, 2020Co-Authors: Yujie Feng, Yitian Gao, Tingting JiaAbstract:A $$(3+1)$$-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation in ocean dynamics, fluid mechanics and plasma physics is investigated in this paper. Bilinear Form, soliton and breather solutions are derived via the Hirota method. Lump solutions are also obtained. Amplitudes of the solitons are proportional to the coefficient $$h_1$$, while inversely proportional to the coefficient $$h_2$$. Velocities of the solitons are proportional to the coefficients $$h_1$$, $$h_3$$, $$h_4$$, $$h_5$$ and $$h_9$$. Elastic and inelastic interactions between the solitons are graphically illustrated. Based on the two-soliton solutions, breathers and periodic line waves are presented. We find that the lumps propagate along the straight lines affected by $$h_4$$ and $$h_9$$. Both the amplitudes of the hump and valleys of the lump are proportional to $$h_4$$, while inversely proportional to $$h_2$$. It is also revealed that the amplitude of the hump of the lump is eight times as large as the amplitudes of the valleys of the lump. Graphical investigation indicates that the lump which consists of one hump and two valleys is localized in all directions and propagates stably.
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Bilinear Form and solutions of a 3 1 dimensional generalized nonlinear evolution equation for the shallow water waves
Applicable Analysis, 2019Co-Authors: Yujie Feng, Yitian Gao, Tingting JiaAbstract:A (3+1)-dimensional generalized nonlinear evolution equation for the shallow-water waves is investigated. Bilinear Form is derived and semi-rational solutions are constructed via the Kadomtsev–Petv...
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Bilinear Form and two backlund transFormations for the 3 1 dimensional jimbo miwa equation
Abstract and Applied Analysis, 2015Co-Authors: Yitian GaoAbstract:With Bell polynomials and symbolic computation, this paper investigates the (3+1)-dimensional Jimbo-Miwa equation, which is one of the equations in the Kadomtsev-Petviashvili hierarchy of integrable systems. We derive a Bilinear Form and construct a Bilinear Backlund transFormation (BT) for the (3+1)-dimensional Jimbo-Miwa equation, by virtue of which the soliton solutions are obtained. Bell-polynomial-typed BT is also constructed and cast into the Bilinear BT.
Yujie Feng - One of the best experts on this subject based on the ideXlab platform.
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Bilinear Form soliton breather lump and hybrid solutions for a varvec 2 1 2 1 dimensional sawada kotera equation
Nonlinear Dynamics, 2020Co-Authors: Yitian Gao, Tingting Jia, Cuicui Ding, Yujie FengAbstract:In this paper, we investigate a ( $$2+1$$ )-dimensional Sawada–Kotera (SK) equation for the atmosphere, rivers, lakes, oceans, as well as the conFormal field and two-dimensional quantum gravity gauge field. Bilinear Form and N-soliton solutions, which are different from those in the existing literatures, are derived, where N is a positive integer. The higher-order breather, lump and hybrid solutions for the ( $$2+1$$ )-dimensional SK equation are also constructed based on the N-soliton solutions. Three kinds of the first-order breathers are obtained, and the higher-order breathers are constructed. The higher-order lump solutions are also derived via the long-wave limit method. Hybrid solutions composed of the solitons, breathers and lumps are worked out, and interaction between the waves is discussed graphically. Finally, similar solutions for a generalized Form of the ( $$2+1$$ )-dimensional SK equation are given.
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Bilinear Form solitons breathers and lumps of a 3 1 dimensional generalized konopelchenko dubrovsky kaup kupershmidt equation in ocean dynamics fluid mechanics and plasma physics
European Physical Journal Plus, 2020Co-Authors: Yujie Feng, Yitian Gao, Tingting JiaAbstract:A $$(3+1)$$-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation in ocean dynamics, fluid mechanics and plasma physics is investigated in this paper. Bilinear Form, soliton and breather solutions are derived via the Hirota method. Lump solutions are also obtained. Amplitudes of the solitons are proportional to the coefficient $$h_1$$, while inversely proportional to the coefficient $$h_2$$. Velocities of the solitons are proportional to the coefficients $$h_1$$, $$h_3$$, $$h_4$$, $$h_5$$ and $$h_9$$. Elastic and inelastic interactions between the solitons are graphically illustrated. Based on the two-soliton solutions, breathers and periodic line waves are presented. We find that the lumps propagate along the straight lines affected by $$h_4$$ and $$h_9$$. Both the amplitudes of the hump and valleys of the lump are proportional to $$h_4$$, while inversely proportional to $$h_2$$. It is also revealed that the amplitude of the hump of the lump is eight times as large as the amplitudes of the valleys of the lump. Graphical investigation indicates that the lump which consists of one hump and two valleys is localized in all directions and propagates stably.
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Bilinear Form and solutions of a 3 1 dimensional generalized nonlinear evolution equation for the shallow water waves
Applicable Analysis, 2019Co-Authors: Yujie Feng, Yitian Gao, Tingting JiaAbstract:A (3+1)-dimensional generalized nonlinear evolution equation for the shallow-water waves is investigated. Bilinear Form is derived and semi-rational solutions are constructed via the Kadomtsev–Petv...
Julien Bichon - One of the best experts on this subject based on the ideXlab platform.
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the representation category of the quantum group of a non degenerate Bilinear Form
Communications in Algebra, 2003Co-Authors: Julien BichonAbstract:Abstract We show that the representation category of the quantum group of a non-degenerate Bilinear Form is monoidally equivalent to the representation category of the quantum group SL q (2) for a well chosen non-zero parameter q. The key ingredient for the proof of this result is the direct and explicit construction of an appropriate Hopf bigalois extension. Then we get, when the base field is of characteristic zero, a full description of cosemisimple Hopf algebras whose representation semi-ring is isomorphic to the one of SL(2).
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the representation category of the quantum group of a non degenerate Bilinear Form
arXiv: Quantum Algebra, 2001Co-Authors: Julien BichonAbstract:We show that the representation category of the quantum group of a non-degenerate Bilinear Form is monoidally equivalent to the representation category of the quantum group SL_q(2), for a well chosen non-zero parameter q. The main ingredient for the proof of this result is the direct and explicit construction of an appropriate Hopf bigalois extension. Then we get, in characteristic zero, a full description of cosemisimple Hopf algebras whose representation semi-ring is isomorphic to the one of SL(2).