Group theoretic aspects of conservation laws of nonlinear dispersive waves: KdV type equations and nonlinear Schrödinger equations

1977 ◽  
Vol 18 (2) ◽  
pp. 256 ◽  
Author(s):  
Sukeyuki Kumei
Keyword(s):  
Conservation Laws ◽  
Dispersive Waves ◽  
Kdv Type Equations

scholarly journals Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1077
Author(s):  
Yarema A. Prykarpatskyy
Keyword(s):  
Conservation Laws ◽  
Hamiltonian Structure ◽  
Necessary Condition ◽  
Type Representation ◽  
Infinite Hierarchy ◽  
Polynomial Coefficients ◽  
Kdv Type Equations ◽  
Special Case

Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s criterion inherits important parts of the gradient-holonomic scheme properties, especially the necessary condition of suitably ordered reduction expansions with certain types of polynomial coefficients. In addition, we also analyze a special case of a new infinite hierarchy of Riemann-type hydrodynamical systems using a gradient-holonomic approach that was suggested jointly with M. Pavlov and collaborators. An infinite hierarchy of conservation laws, bi-Hamiltonian structure and the corresponding Lax-type representation are constructed for these systems.

scholarly journals Singular vectors and conservation laws of quantum KdV type equations

1992 ◽  
Vol 278 (1-2) ◽  
pp. 79-84 ◽  
Author(s):  
P. Di Francesco ◽  
P. Mathieu
Keyword(s):  
Conservation Laws ◽  
Singular Vectors ◽  
Kdv Type Equations

scholarly journals Time fractional modified KdV-type equations: Lie symmetries, exact solutions and conservation laws

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 480-488
Author(s):  
Fangqin He ◽  
Lianzhong Li
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Lie Symmetries ◽  
Symmetry Reductions ◽  
Kdv Type Equations

Abstract In the paper, we research a time fractional modified KdV-type equations.We give the symmetry reductions and exact solutions of the equations, and we investigate the convergence of the solutions. In addition, the conservation laws of the equations are constructed.

Relaxation and diffusion enhanced dispersive waves

1994 ◽  
Vol 446 (1928) ◽  
pp. 555-563 ◽  
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Dispersive Waves ◽  
Hyperbolic Conservation ◽  
Physical Problems ◽  
And Diffusion ◽  
Nonlinear Hyperbolic Conservation Laws

The development of shocks in nonlinear hyperbolic conservation laws may be regularized through either diffusion or relaxation. However, we have observed surprisingly that for some physical problems, when both of the smoothing factors – diffusion and relaxation – coexist, under appropriate asymptotic assumptions, the dispersive waves are enhanced. This phenomenon is studied asymptotically in the sense of the Chapman–Enskog expansion and demonstrated numerically.

scholarly journals Hamiltonian Structure, Symmetries and Conservation Laws for a Generalized (2 1)-Dimensional Double Dispersion Equation+

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1031 ◽  
Author(s):  
Elena Recio ◽  
Tamara M. Garrido ◽  
Rafael de la Rosa ◽  
María S. Bruzón
Keyword(s):  
Conservation Laws ◽  
Dispersion Equation ◽  
Physical Meaning ◽  
Hamiltonian Structure ◽  
Nonlinear Function ◽  
Symmetry Transformation ◽  
Transformation Groups ◽  
Conserved Quantities ◽  
Dispersive Waves ◽  
Double Dispersion

This paper considers a generalized double dispersion equation depending on a nonlinear function f ( u ) and four arbitrary parameters. This equation describes nonlinear dispersive waves in 2 + 1 dimensions and admits a Lagrangian formulation when it is expressed in terms of a potential variable. In this case, the associated Hamiltonian structure is obtained. We classify all of the Lie symmetries (point and contact) and present the corresponding symmetry transformation groups. Finally, we derive the conservation laws from those symmetries that are variational, and we discuss the physical meaning of the corresponding conserved quantities.

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