Abstract. Using the reality condition of the solutions, one constructs the Mach-type soliton of the Novikov-Veselov equation by the minor-summation formula of the Pfaffian. We study the evolution of the Mach-type soliton and find that the amplitude of the Mach stem wave is less than two times of the one of the incident wave. It is shown that the length of the Mach stem wave is linear with time. One discusses the relations with V -shape initial value wave for different critical values of Miles parameter.

Soliton solutions are studied for paraxial wave propagation with intensity-dependent dispersion.Although the corresponding Lagrangian density has a singularity, analytical solutions, derived by the pseudo-potential method and the corresponding phase diagram, exhibit one-and two-humped solitons with almost perfect agreement to numerical solutions. The results obtained in this work reveal a hitherto unexplored area of soliton physics associated with nonlinear corrections to wave dispersion. 1 arXiv:2001.01631v1 [nlin.PS] 6 Jan 2020 Chromatic dispersion is the dependence of the phase velocity of a wave on its frequency [1] or, equivalently, frequency dependence of the refractive index. Nonlinear corrections to the chromatic dispersion as a function of the wave intensity arise for various waves, such as shallow water waves [2, 3], acoustic waves in micro-inhomogeneous media [4], or ultrafast coherent pulses in GaAs/AlGaAs quantum well waveguide structures [5]. In the context of photon-atom interactions, nonlinear dispersion effects may come about from the saturation of the atomic-level population [6], electromagnetically-induced transparency (EIT) in a chain-Λ configuration [7], or nonlocal nonlinearity mediated by dipole-dipole interactions [8].

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We investigate the Miura map between the dispersionless KP and dispersionless modified KP hierarchies. We show that the Miura map is canonical with respect to their bi-Hamiltonian structures. Moreover, inspired by the works of Takasaki and Takebe, the twistor construction of solution structure for the dispersionless modified KP hierarchy is given.

Inspired by the works of Y. Ohta and J. Yang, one constructs the lumps solutions in the Kadomtsev-Petviashvili-(I) equation using the Grammian determinants. It is shown that the locations of peaks will depend on the real roots of Wronskian of the orthogonal polynomials for the asymptotic behaviors in some particular cases. Also, one can prove that all the locations of peaks are on a vertical line when time approaches -∞, and then they will be on a horizontal line when time approaches ∞, i.e., there is a rotation π 2 after interaction.