An efficient algorithm of logarithmic transformation to Hirota bilinear form of KdV-type bilinear equation

Abstract In this paper, an efficient algorithm of logarithmic transformation to Hirota bilinear form of the KdV-type bilinear equation is established. In the algorithm, some properties of Hirota operator and logarithmic transformation are successfully applied, which helps to prove that the linear terms of the nonlinear partial differential equation play a crucial role in finding the Hirota bilinear form. Experimented with various integro-differential equations, our algorithm is proven to be more efficient than the algorithm referred by Zhou, Fu, and Li in getting the Hirota bilinear form, especially in achieving the coefficient of the logarithmic transformation.

[1]  Mingliang Wang,et al.  Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics , 1996 .

[2]  Willy Hereman,et al.  Symbolic Computation of , 2011 .

[3]  Ruan Hang-Yu,et al.  A Maple Package on Symbolic Computation of Hirota Bilinear Form for Nonlinear Equations , 2009 .

[4]  広田 良吾,et al.  The direct method in soliton theory , 2004 .

[5]  R. Hirota,et al.  N-Soliton Solutions of Model Equations for Shallow Water Waves , 1976 .

[6]  Jarmo Hietarinta,et al.  A Search for Bilinear Equations Passing Hirota''s Three-Soliton Condition , 1987 .

[7]  D. Zwillinger.  Handbook of differential equations , 1990 .

[8]  Jarmo Hietarinta,et al.  Introduction to the Hirota bilinear method , 1997, solv-int/9708006.

[9]  B. Kadomtsev,et al.  On the Stability of Solitary Waves in Weakly Dispersing Media , 1970 .

[10]  Jingsong He,et al.  SOLUTIONS OF THE (2 + 1)-DIMENSIONAL KP, SK AND KK EQUATIONS GENERATED BY GAUGE TRANSFORMATIONS FROM NONZERO SEEDS , 2008, Journal of Nonlinear Mathematical Physics.

[11]  M. Tabor,et al.  The Painlevé property for partial differential equations , 1983 .

[12]  S. Sawada,et al.  A Method for Finding N-Soliton Solutions of the KdV and KdV-Like Equation , 1974 .

[13]  A. Scott.  Encyclopedia of nonlinear science , 2006 .

[14]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.

[15]  Masaaki Ito,et al.  An Extension of Nonlinear Evolution Equations of the K-dV (mK-dV) Type to Higher Orders , 1980 .

[16]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[17]  Allen Parker,et al.  A bidirectional Kaup–Kupershmidt equation and directionally dependent solitons , 2002 .

[18]  Wen-Xiu Ma,et al.  EXACT ONE-PERIODIC AND TWO-PERIODIC WAVE SOLUTIONS TO HIROTA BILINEAR EQUATIONS IN (2+1) DIMENSIONS , 2008, 0812.4316.

[19]  Wen-Xiu Ma,et al.  Computers and Mathematics with Applications Linear Superposition Principle Applying to Hirota Bilinear Equations , 2022 .

[20]  Zhen-jiang Zhou,et al.  An implementation for the algorithm of Hirota bilinear form of PDE in the Maple system , 2006, Appl. Math. Comput..

[21]  Zhen-jiang Zhou,et al.  Maple packages for computing Hirota's bilinear equation and multisoliton solutions of nonlinear evolution equations , 2010, Appl. Math. Comput..

[22]  Liu Yu-lu,et al.  Doubly Periodic Propagating Wave for (2+1)-Dimensional Breaking Soliton Equation , 2008 .

[23]  John D. Gibbon,et al.  A new hierarchy of Korteweg–de Vries equations , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.