CASORATIAN SOLUTIONS AND NEW SYMMETRIES OF THE DIFFERENTIAL-DIFFERENCE KADOMTSEV–PETVIASHVILI EQUATION

2009 ◽  
Vol 23 (17) ◽  
pp. 2107-2114 ◽  
Author(s):  
DA-JUN ZHANG ◽  
JIE JI ◽  
XIAN-LONG SUN
Keyword(s):  
Lie Algebra ◽  
Three Dimensional ◽  
Single Parameter ◽  
Petviashvili Equation ◽  
Invariant Group

This paper first discusses the condition in which Casoratian entries satisfy for the differential-difference Kadomtsev–Petviashvili equation. Then from the Casoratian condition we find a transformation under which the differential-difference Kadomtsev–Petviashvili equation is invariant. The transformation, consisting of a combination of Galilean and scalar transformations, provides a single-parameter invariant group for the equation. We further derive the related symmetry, and the symmetry together with other two symmetries form a closed three-dimensional Lie algebra.

scholarly journals GROUP PROPERTIES OF A ONE-DIMENSIONAL NONLINEAR POROELASTICITY EQUATION

2019 ◽  
Vol 4 (1) ◽  
pp. 149-155
Author(s):  
Kholmatzhon Imomnazarov ◽  
Ravshanbek Yusupov ◽  
Ilham Iskandarov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Group Classification ◽  
Second Order ◽  
Partial Differential ◽  
One Dimensional ◽  
Preliminary Group

This paper studies a class of partial differential equations of second order , with arbitrary functions and , with the help of the group classification. The main Lie algebra of infinitely infinitesimal symmetries is three-dimensional. We use the method of preliminary group classification for obtaining the classifications of these equations for a one-dimensional extension of the main Lie algebra.

Localized waves and interaction solutions to an extended (3+1)-dimensional Kadomtsev–Petviashvili equation

2020 ◽  
Vol 34 (06) ◽  
pp. 2050076 ◽  
Author(s):  
Han-Dong Guo ◽  
Tie-Cheng Xia ◽  
Wen-Xiu Ma
Keyword(s):  
Three Dimensional ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Dark Soliton ◽  
Complex Conjugate ◽  
Bilinear Method ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Parameter Constraints ◽  
Localized Waves

In this paper, an extended (3[Formula: see text]+[Formula: see text]1)-dimensional Kadomtsev–Petviashvili (KP) equation is studied via the Hirota bilinear derivative method. Soliton, breather, lump and rogue waves, which are four types of localized waves, are obtained. N-soliton solution is derived by employing bilinear method. Then, line or general breathers, two-order line or general breathers, interaction solutions between soliton and line or general breathers are constructed by complex conjugate approach. These breathers own different dynamic behaviors in different planes. Taking the long wave limit method on the multi-soliton solutions under special parameter constraints, lumps, two- and three-lump and interaction solutions between dark soliton and dark lump are constructed, respectively. Finally, dark rogue waves, dark two-order rogue waves and related interaction solutions between dark soliton and dark rogue waves or dark lump are also demonstrated. Moreover, dynamical characteristics of these localized waves and interaction solutions are further vividly demonstrated through lots of three-dimensional graphs.

scholarly journals Three-dimensional higher-order Schrödinger algebras and Lie algebra expansions

2020 ◽  
Vol 2020 (4) ◽  
Author(s):  
Oguzhan Kasikci ◽  
Nese Ozdemir ◽  
Mehmet Ozkan ◽  
Utku Zorba
Keyword(s):  
Lie Algebra ◽  
Three Dimensional ◽  
Higher Order

A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems

1993 ◽  
Vol 48 (4) ◽  
pp. 535-550 ◽  
Author(s):  
H. Kötz
Keyword(s):  
Differential Equations ◽  
Symmetry Group ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Classification Problem ◽  
Similarity Solutions ◽  
Point Symmetry ◽  
Point Symmetry Group ◽  
Relativistic Case ◽  
Lie Point Symmetry

"Optimal systems" of similarity solutions of a given system of nonlinear partial (integro-)differential equations which admits a finite-dimensional Lie point symmetry group Gare an effective systematic means to classify these group-invariant solutions since every other such solution can be derived from the members of the optimal systems. The classification problem for the similarity solutions leads to that of "constructing" optimal subalgebraic systems for the Lie algebra Gof the known symmetry group G. The methods for determining optimal systems of s-dimensional Lie subalgebras up to the dimension r of Gvary in case of 3 ≤ s ≤ r, depending on the solvability of G. If the r-dimensional Lie algebra Gof the infinitesimal symmetries is nonsolvable, in addition to the optimal subsystems of solvable subalgebras of Gone has to determine the optimal subsystems of semisimple subalgebras of Gin order to construct the full optimal systems of s-dimensional subalgebras of Gwith 3 ≤ s ≤ r. The techniques presented for this classification process are applied to the nonsolvable Lie algebra Gof the eight-dimensional Lie point symmetry group Gadmitted by the three-dimensional Vlasov-Maxwell equations for a multi-species plasma in the non-relativistic case.

scholarly journals GENERALIZED QUANTUM RELATIVISTIC KINEMATICS: A STABILITY POINT OF VIEW

2004 ◽  
Vol 13 (10) ◽  
pp. 2003-2034 ◽  
Author(s):  
C. CHRYSSOMALAKOS ◽  
E. OKON
Keyword(s):  
Lie Algebra ◽  
Quantum Effect ◽  
Three Dimensional ◽  
Heisenberg Algebra ◽  
Computer Code ◽  
Point Of View ◽  
Stable Form ◽  
Relativistic Kinematics ◽  
Warm Up ◽  
Stability Point

We apply Lie algebra deformation theory to the problem of identifying the stable form of the quantum relativistic kinematical algebra. As a warm up, given Galileo's conception of spacetime as input, some modest computer code we wrote zeroes in on the Poincaré-plus-Heisenberg algebra in about a minute. Further ahead, along the same path, lies a three-dimensional deformation space, with an instability double cone through its origin. We give physical as well as geometrical arguments supporting our view that moment, rather than position operators, should enter as generators in the Lie algebra. With this identification, the deformation parameters give rise to invariant length and mass scales. Moreover, standard quantum relativistic kinematics of massive, spinless particles corresponds to non-commuting moment operators, a purely quantum effect that bears no relation to spacetime non-commutativity, in sharp contrast to earlier interpretations.

scholarly journals Model Equations for Three-Dimensional Nonlinear Water Waves under Tangential Electric Field

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Bo Tao
Keyword(s):  
Electric Field ◽  
Water Waves ◽  
Three Dimensional ◽  
Liquid Sheet ◽  
Capillary Waves ◽  
Uniform Electric Field ◽  
Nonlinear Water Waves ◽  
Petviashvili Equation ◽  
Model Equations ◽  
Layer Problem

We are concerned with gravity-capillary waves propagating on the surface of a three-dimensional electrified liquid sheet under a uniform electric field parallel to the undisturbed free surface. For simplicity, we make an assumption that the permittivity of the fluid is much larger than that of the upper-layer gas; hence, this two-layer problem is reduced to be a one-layer problem. In this paper, we propose model equations in the shallow-water regime based on the analysis of the Dirichlet-Neumann operator. The modified Benney-Luke equation and Kadomtsev-Petviashvili equation will be derived, and the truly three-dimensional fully localized traveling waves, which are known as “lumps” in the literature, are numerically computed in the Benney-Luke equation.

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