By Vasily Y. Belashov, Sergey V. Vladimirov
This e-book is dedicated to at least one of the main attention-grabbing and swiftly constructing parts of contemporary nonlinear physics and arithmetic - the theoretical, analytical and complex numerical learn of the constitution and dynamics of one-dimensional in addition to - and 3-dimensional solitons and nonlinear waves defined through Korteweg-de Vries (KdV), Kadomtsev-Petviashvili (KP), nonlinear Schrodinger (NLS) and spinoff NLS (DNLS) sessions of equations. detailed awareness is paid to generalizations (relevant to numerous advanced actual media) of those equations, accounting for higher-order dispersion corrections, impact of dissipation, instabilities, and stochastic fluctuations of the wave fields. The ebook addresses researchers operating within the concept and numerical simulations of dispersive advanced media in such fields as hydrodynamics, plasma physics, and aerodynamics. it's going to even be priceless as a reference paintings for graduate scholars in physics and arithmetic.
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Additional resources for Solitary Waves in Dispersive Complex Media: Theory, Simulation, Applications
In the second part, we investigate an algebraic equation of the fourth order appearing when analyzing possible extrema of the Hamiltonian of the GKP equation in Sect. 1. 1. 1 Korteweg–de Vries Equation and KdV-Class Equations In this section, we derive the KdV equation from the full set of hydrodynamic equations. , we do not consider any particular type of a medium (Sect. 1), and work further with the general equations. We demonstrate that the KdV model is universal in the sense that it describes the nonlinear wave dynamics in any medium where the wave dispersion is of a certain class, and consider the scale transformations and the similarity principle for the KdV equation (Sect.
61) is also important (see Sect. 1). They are in fact pseudo-eigenfunctions because they are not quadratically integrated in general. The functions r(k), a−1 (k), and ψj (x, k) satisfy the relations 36 1. KdV-Class Solitons ⎧ ⎨ r(−k) = r∗ (k), ∗ a−1 (−k) = a−1 (k), ⎩ ψj (x, k) = ψj∗ (x, −k). Note that the functions ψj and ψj∗ are linearly independent. It is obvious that (see Sect. 1) ϕj (x, k) = ψj (x, k) + rj (k)ψj∗ (x, k). ˆ j can have consists of a ﬁnite The discrete spectrum that the operator H number of the point-like eigenvalues represented by λjn = −κ2jn , κjn > 0.
20), we obtain the 2-soliton solution where the phase delay appearing as a result of the solitons’ interaction is deﬁned by the factor A12 . To conclude this brief remark on the construction of the soliton solutions, we note that the choice of values of the constants x0i is quite arbitrary. Thus, using the above direct method, we can also construct other classes of solutions. 21) for N = 1, we obtain a singular solution u(t, x) = − k1 k12 x − k12 t . sinh−2 2 2 In the long-wavelength limit k1 → 0, we obtain the solution u = −2/x2  which is the ﬁrst representative of the class of the rational solutions of the KdV equation.