How can a system in a macroscopically stable state explore energetically more favorable states, which are far away from the current equilibrium state? Based on continuum mechanical considerations we derive a Boussinesq-type equation ρü = ∂xσ(∂xu) + β∂ 2 x ü, x ∈ (0, 1), β > 0, which models the dynamics of martensitic phase transformations. Here ρ > 0 is the mass density, β∂ x ü is a regularizat...

Using a traveling wave reduction technique, we have shown that Maccari equation, (2?1)-dimensional nonlinear Schrödinger equation, medium equal width equation, (3?1)-dimensional modified KdV–Zakharov– Kuznetsev equation, (2?1)-dimensional long wave-short wave resonance interaction equation, perturbed nonlinear Schrödinger equation can be reduced to the same family of auxiliary elliptic-like equ...

This paper is concerned with interacting wave packet dynamics for long waves. The Kortweg-de Vries equation is the most well-known model for weakly nonlinear long waves. Although the dynamics of a single wave packet in this model is governed by the defocusing nonlinear Schrödinger equation, implying that a plane wave is modulationally stable, the dynamics of two interacting wave packets is gove...

We show that the Herglotz wave function with kernel the Tikhonov regularized solution of the far field equation becomes unbounded as the regularization parameter tends to zero iff the wavenumber k belongs to a discrete set of values. When the scatterer is such that the total field vanishes on the boundary, these values correspond to the square root of Dirichlet eigenvalues for −∆. When the scat...

Exact explicit rogue-wave solutions of intricate structures are presented for the long-wave-short-wave resonance equation. These vector parametric solutions feature coupled dark- and bright-field counterparts of the Peregrine soliton. Numerical simulations show the robustness of dark and bright rogue waves in spite of the onset of modulational instability. Dark fields originate from the complex...

We recover the amplitudes of the reflectivity function obtained by wave-equation migration by compensating for the amplitude distortions created by the imaging condition and by the incomplete reflector illumination. The amplitude effects produced by the imaging condition must be taken into account even for simple velocity models, and they are perfectly compensated by a diagonal scaling in the f...

The solution of partial differential equations has generally been one the most-vital mathematical tools for describing physical phenomena in different scientific disciplines. previous studies performed with classical derivative on this model cannot express propagating behavior at heavy infinite tails. In order to address problem, study addressed fractional regularized long-wave Burgers problem ...

In this work, we design, analyze and test a conservative discontinuous Galerkin method for solving the Degasperis–Procesi equation. This model is integrable and admits possibly discontinuous solutions, and therefore suitable for modeling both short wave breaking and long wave propagation phenomena. The proposed numerical method is high order accurate, and preserves two invariants, mass and ener...

The dynamics of spinorial wave functions in a causal fermion system is studied. A so-called dynamical equation derived. Its solutions form Hilbert space, whose scalar product represented by conserved surface layer integral. We prove under general assumptions that the initial value problem for admits unique global solution. Causal Green's operators are constructed and analyzed. Our findings illu...