Symmetries, conservation laws, and generalized travelling waves for a forced Ostrovsky equation

Ostrovsky’s equation with timeand spacedependent forcing is studied. This equation is model for long waves in a rotating fluid with a non-constant depth (topography). A classification of Lie point symmetries and low-order conservation laws is presented. Generalized travelling wave solutions are obtained through symmetry reduction. These solutions exhibit a wave profile that is stationary in a moving reference frame whose speed can be constant, accelerating, or decelerating.

[1]  Jianming He,et al.  Terminal Damping of a Solitary Wave Due to Radiation in Rotational Systems , 1998 .

[2]  Chaudry Masood Khalique,et al.  Symmetries, solutions and conservation laws of a class of nonlinear dispersive wave equations , 2016, Commun. Nonlinear Sci. Numer. Simul..

[3]  A. I. Leonov.  THE EFFECT OF THE EARTH'S ROTATION ON THE PROPAGATION OF WEAK NONLINEAR SURFACE AND INTERNAL LONG OCEANIC WAVES * , 1981 .

[4]  S. Anco.  Generalization of Noether’s Theorem in Modern Form to Non-variational Partial Differential Equations , 2016, 1605.08734.

[5]  Y. Stepanyants,et al.  Nonlinear Internal Waves in Rotating Ocean , 2015 .

[6]  Yue Liu,et al.  On the stability of solitary waves of a generalized Ostrovsky equation , 2006, Analysis and Mathematical Physics.

[7]  Maria Luz Gandarias,et al.  Symmetry analysis and exact solutions of some Ostrovsky equations , 2011 .

[8]  Roger Grimshaw,et al.  The Reduced Ostrovsky Equation: Integrability and Breaking , 2012 .

[9]  G. Bluman,et al.  Direct construction method for conservation laws of partial differential equations Part II: General treatment , 2001, European Journal of Applied Mathematics.

[10]  Yury Stepanyants,et al.  On stationary solutions of the reduced Ostrovsky equation: periodic waves, compactons and compound solitons , 2006 .

[11]  A. Sjöberg,et al.  On double reductions from symmetries and conservation laws , 2009 .

[12]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[13]  Stephen C. Anco,et al.  Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications , 2001, European Journal of Applied Mathematics.

[14]  Ashfaque H. Bokhari,et al.  Generalization of the double reduction theory , 2009, 0909.4564.

[15]  Anjan Biswas,et al.  Solitons and conservation laws of coupled Ostrovsky equation for internal waves , 2015, Appl. Math. Comput..

[16]  M. Bruzón,et al.  Traveling wave solutions for a generalized Ostrovsky equation , 2018 .

[17]  Maria Luz Gandarias,et al.  Some conservation laws for a forced KdV equation , 2012 .

[18]  Solutions of the modified Ostrovskii equation with cubic non-linearity , 2000 .

[19]  Wen-Xiu Ma,et al.  Conservation laws by symmetries and adjoint symmetries , 2017, 1707.03496.

[20]  G. Bluman,et al.  Direct Construction of Conservation Laws from Field Equations , 1997 .

[21]  Roger Grimshaw,et al.  The Effect of Topography on the Dynamics of Interacting Solitary Waves in the Context of Atmospheric Blocking , 1999 .

[22]  Rossen I. Ivanov,et al.  Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovsky equation , 2006, math-ph/0606063.

[23]  O. A. Gilman,et al.  Approximate Analytical and Numerical Solutions of the Stationary Ostrovsky Equation , 1995 .

[24]  G. Bluman,et al.  Applications of Symmetry Methods to Partial Differential Equations , 2009 .

[25]  R. Grimshaw,et al.  Long‐time Solutions of the Ostrovsky Equation , 2008 .

[26]  Thomas Wolf,et al.  A comparison of four approaches to the calculation of conservation laws , 2002, European Journal of Applied Mathematics.

[27]  A. Sjöberg,et al.  Double reduction of PDEs from the association of symmetries with conservation laws with applications , 2007, Appl. Math. Comput..

[28]  G. Bluman,et al.  Symmetry and Integration Methods for Differential Equations , 2002 .