Infinite series symmetry reduction solutions to the modified KdV--Burgers equation |
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Authors: | Yao Ruo-Xi Jiao Xiao-Yu and Lou Sen-Yue |
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Institution: | Department of Physics, Shanghai Jiao Tong University, Shanghai 200062, China; School of Computer Science, Shaanxi Normal University, Xi’an 710062, China;.; Department of Physics, Ningbo University, Ningbo 315211, China |
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Abstract: | From the point of view of approximate symmetry, the modified
Korteweg--de Vries--Burgers (mKdV--Burgers) equation with weak
dissipation is investigated. The symmetry of a system of the
corresponding partial differential equations which approximate the
perturbed mKdV--Burgers equation is constructed and the
corresponding general approximate symmetry reduction is derived;
thereby infinite series solutions and general formulae can be
obtained. The obtained result shows that the zero-order similarity
solution to the mKdV--Burgers equation satisfies the Painlevé II
equation. Also, at the level of travelling wave reduction, the
general solution formulae are given for any travelling wave solution
of an unperturbed mKdV equation. As an illustrative example, when
the zero-order tanh profile solution is chosen as an initial
approximate solution, physically approximate similarity solutions
are obtained recursively under the appropriate choice of parameters
occurring during computation. |
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Keywords: | modified Korteweg--de
Vries--Burgers (mKdV--Burgers) equation approximate symmetry
reduction series reduction solution |
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