The four-dimensional Martínez Alonso–Shabat equation: Reductions and nonlocal symmetries

Morozov, Oleg I.; Sergyeyev, Artur

doi:10.1016/j.geomphys.2014.05.025

Cited by 47 publications

(31 citation statements)

References 35 publications

(98 reference statements)

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“…Another interesting reduction admitted by arises when we put z = t . This produces the equation

u_{yt} = u_{t} u_{xy} - u_{y} u_{xt},

which is related with the so‐called ABC equation

A q_{x} q_{ty} + B q_{y} q_{tx} + C q_{t} q_{xy} = 0, A + B + C = 0,

by its another differential covering . In this case, the differential covering becomes

\begin{matrix} s_{y} = λ u_{y} s_{x} - λ s u_{xy}, \\ s_{t} = \frac{λ}{λ + 1} (u_{t} s_{x} - s u_{xt}) . \end{matrix}

The Lie point symmetry admitted by Equation is

X = a \partial_{x} + b \partial_{y} + e \partial_{t} + (a^{'} u + d) \partial_{u},

where a = a ( x ), b = b ( y ), e = e ( t ), and d = d ( x ) are arbitrary functions, while a nonlocal symmetry of equation

X = g_{1} \partial_{x} + h \partial_{y}

…”

Section: Resultsmentioning

confidence: 99%

“…Next, we consider nonlocal symmetry of Equation (1), that is, search for Lie point symmetry of system (2). Assuming the infinitesimal operator of symmetry for Equation (2) (1), we obtain a nonlocal symmetry of Equation (1) in the form…”

Section: Local and Nonlocal Symmetriesmentioning

confidence: 99%

“…In and , the authors considered the four‐dimensional Martínez Alonso–Shabat equation

u_{ty} = u_{z} u_{xy} - u_{y} u_{xz},

which was introduced in . Differential coverings, recursion operator, and nonlocal symmetries for equation were studied and used to prove its integrability .…”

Section: Introductionmentioning

confidence: 99%

“…In [1] and [2], the authors considered the four-dimensional Martínez Alonso-Shabat equation u ty D u z u xy u y u xz ,…”

Section: Introductionmentioning

confidence: 99%

“…which was introduced in [3]. Differential coverings, recursion operator, and nonlocal symmetries for equation (1) were studied and used to prove its integrability [1,2]. The aim of the paper is to study conservation laws of Equation (1) from the point of view of nonlinear self-adjointness with differential substitution.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The four‐dimensional Martínez Alonso–Shabat equation: nonlinear self‐adjointness and conservation laws

Zhang∗†

Guo

Huang

2016

Math Methods in App Sciences

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We show that the four‐dimensional Martínez Alonso–Shabat equation is nonlinearly self‐adjoint with differential substitution and the required differential substitution is just the admitted adjoint symmetry and vice versa. By means of computer algebra system, we obtain a number of local and nonlocal symmetries admitted by the equations under study. Then such symmetries are used to construct conservation laws of the equation under study and its reductions. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Another interesting reduction admitted by arises when we put z = t . This produces the equation

u_{yt} = u_{t} u_{xy} - u_{y} u_{xt},

which is related with the so‐called ABC equation

A q_{x} q_{ty} + B q_{y} q_{tx} + C q_{t} q_{xy} = 0, A + B + C = 0,

by its another differential covering . In this case, the differential covering becomes

\begin{matrix} s_{y} = λ u_{y} s_{x} - λ s u_{xy}, \\ s_{t} = \frac{λ}{λ + 1} (u_{t} s_{x} - s u_{xt}) . \end{matrix}

The Lie point symmetry admitted by Equation is

X = a \partial_{x} + b \partial_{y} + e \partial_{t} + (a^{'} u + d) \partial_{u},

where a = a ( x ), b = b ( y ), e = e ( t ), and d = d ( x ) are arbitrary functions, while a nonlocal symmetry of equation

X = g_{1} \partial_{x} + h \partial_{y}

…”

Section: Resultsmentioning

confidence: 99%

Section: Local and Nonlocal Symmetriesmentioning

confidence: 99%

“…In and , the authors considered the four‐dimensional Martínez Alonso–Shabat equation

u_{ty} = u_{z} u_{xy} - u_{y} u_{xz},

which was introduced in . Differential coverings, recursion operator, and nonlocal symmetries for equation were studied and used to prove its integrability .…”

Section: Introductionmentioning

confidence: 99%

“…In [1] and [2], the authors considered the four-dimensional Martínez Alonso-Shabat equation u ty D u z u xy u y u xz ,…”

Section: Introductionmentioning

confidence: 99%