## Mini-symposium

Analysis of Nonlinear Waves

#### in the frame of the International Conference “Mathematics Days in Sofia”

**T**

**his mini-symposium** is focused on a variety of problems concerning solutions of a large class of partial differential equations of mathematical physics. These equations support coherent structures like solitary or periodic traveling waves, as well as wave trains and multi-pulses. Such solutions are very important objects when modeling physical processes and their stability is essential in practical applications. Stable states of the system are key because they attract all nearby configurations, while the loss of stability or being able to control it is of practical importance as well. Speakers will showcase the use of dynamical systems, spectral and variational methods, as well as the techniques of Fourier analysis, to resolve an array of open problems in the theory of existence, long-time asymptotic and stability for nonlinear wave solutions.

#### Organizing Committee

Sevdzhan Hakkaev (Institute of Mathematics and Informatics, Bulgaria and Trakya University, Turkiye)

Milena Stanislavova (University of Alabama at Birmingham, USA)

Atanas Stefanov (University of Alabama at Birmingham, USA)

### Participants

#### Invited Speaker

**Dmitry Pelinovsky**, McMaster University, Canada

#### Contributors

**Abba Ramadan**, University of Alabama at Birmingham, USA**Ademir Pastor**, University of Campinas, Brazil**Atanas Stefanov**, University of Alabama at Birmingham, USA**Dionyssios Mantzavinos**, University of Kansas, USA**Dmitry Pelinovsky**, McMaster University, Canada**Fabio Natali**, Universidade Estadual de Maringa, Brazil**George Venkov**, Technical University of Sofia, Bulgaria**Iryna Petrenko**, Florida International University, USA**John Albert**, University of Oklahoma, USA**Milen Ivanov**, Institute of Mathematics and Informatics, Bulgaria

**Milena Stanislavova**, University of Alabama at Birmingham, USA**Nilay Duruk**, Sabanci University, Turkiye**Sevdzhan Hakkaev**, Institute of Mathematics and Informatics, Bulgaria and Trakya University, Turkiye**Stephane Lafortune**, College of Charleston, USA**Svetlana Roudenko**, Florida International University, USA**Vahagn Manukyan**, Miami University, USA**Vladimir Georgiev**, University of Pisa, Italy**Yuri Latushkin**, University of Missouri, USA**Xing Cheng**, Hohai University, China

### Program and Abstracts

In this talk, we will consider the NLS system of the third-harmonic generation. Our interest is in solitary wave solutions and their stability properties. This enhanced the recent work of Oliveira and Pastor. In this work, we systematically build and study solitary waves for this important model. We construct the waves in the largest possible parameter space, and we provide a complete classification of their spectral stability. Finally, we showed instability by a blow-up, for dimension 3, and for a more restrictive set of parameters, we use virial identities methods to derive the strong instability, in the spirit of Ohta’s approach.

This is joint work with Prof. Atanas Stefanov.

We consider a diffusive Rosenzweig-MacArthur predator-prey model in the situation when the prey diffuses at the rate much smaller than that of the predator. The existence of fronts in the system is proved using the Geometric Singular Perturbation Theory. We show that, in a limit, the underlying system is reduced to a scalar Fisher-KPP equation and the fronts in the full system are small perturbations of the fronts in that Fisher-KPP equation. We then investigate whether the stability of the fronts is also governed by the scalar Fisher-KPP equation. The techniques of the analysis include a construction of unstable augmented bundles and their treatment as multi-scale topological structures.

We consider the nonlinear Schrödinger equation with a nonzero boundary condition of Robin type and establish its Hadamard well-posedness in one as well as in two spatial dimensions (half-line and half-plane, respectively). The Neumann problem is also discussed as a special case. The results are proved by using the solution formulae for the linear Schrödinger equation obtained via the unified transform of Fokas. These formulae allow us to establish suitable linear estimates which are then combined with a contraction mapping argument to yield well-posedness for the nonlinear problems.

In this talk, we determine the transversal instability of periodic traveling wave solutions of the generalized Zakharov-Kuznetsov equation in two space dimensions. Using an adaptation in the periodic context of a well-known criterion used for solitary waves, it is possible to prove that all positive and one-dimensional L´periodic waves are spectrally (transversally) unstable. In addition, when periodic sign-changing waves exist, we also obtain the same property when the associated projection operator defined in the zero-mean Sobolev space has only one negative eigenvalue.

We first review solutions to a standard nonlinear Schrödinger (NLS) equation with various nonlinearities and recall several properties of solitary waves. Then we look at the higher order dispersion NLS and discuss rather peculiar properties of solitary waves in that case.

Joint work with Svetlana Roudenko.

Solutions of reaction-diffusion systems exhibit a wide variety of patterns like spirals, stripes and Turing patterns. In particular, the Belousov-Zhabotinsky (BZ) reaction produces spiral patterns, which sometimes have a defect: there is a line defect, emitted from the center of the spiral, and alongside that defect the pattern jumps by half a period. In order to study this phenomenon, we view the defect as a so-called contact defect, studied by Bjorn Sandstede and Arnd Scheel: time-periodic functions, which converge to a spatially homogeneous oscillation as the space variable diverges to infinity. The context of spiral waves requires a large bounded domain, so it is natural to inquire about the definition and existence of truncated contact defects. We prove that said truncated defects exist and are unique, and we explain how this result can be seen as a homoclinic bifurcation. In addition, we prove that our solutions are spectrally stable with periodic boundary conditions and spectrally unstable with Neumann boundary conditions.

The deep connection between dispersive effect and regularity of the solutions corresponding to Cauchy problems arising in wave theory brings many problems in qualitative analysis. In this talk, we will see

ut + a(u)u = b(u)

quasi-linear form of evolution equations for certain choices of a(u) and b(u). The literature on dispersive nonlinear equations will be introduced. Camassa-Holm equation, which is one of the most studied examples, and its generalizations will be analyzed in terms of regularity. Recent results on well-posedness regarding generalized Camassa-Holm equation will be introduced.

The Camassa-Holm equation with linear dispersion was originally derived as an asymptotic equation in shallow water wave theory. Among its many interesting mathematical properties, which include complete integrability, perhaps the most striking is the fact that in the case where linear dispersion is absent it admits weak multi-soliton solutions – peakons’ – with a peaked shape corresponding to a discontinuous first derivative. There is a one-parameter family of generalized Camassa-Holm equations, most of which are not integrable, but which all admit peakon solutions. Numerical studies reported by Holm and Staley indicate changes in the stability of these and other solutions as the parameter b varies through the family. In this talk, we describe analytical results on some of these bifurcation phenomena, showing that in a suitable parameter range there are stationary solutions – leftons’ – which are orbitally stable. We establish information about the point spectrum of the peakon solutions and notably find that for suitably smooth perturbations there exists point spectrum in the right half plane rendering the peakons unstable for b¡1. We also show that the peakons are linearly unstable on L 2 for all values of b. Finally, we establish the orbital stability of the smooth solitary waves for b > 1. We also obtain stability results about the peakon solutions of the Novikov equation.

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We study a class of two-dimensional non-linear Schroеdinger equations with point-like singular perturbation and Hartree or local type non-linearity. Our analysis has the following key points: we establish existence, symmetry, and regularity of ground states, and we demonstrate the well-posedness of the associated Cauchy problem in the singular perturbed energy space. Stability/instability properties for the local nonlinearities are discussed too.

The talk is based on collaborations with Masahiro Ikeda, Noriyoshi Fukaya, Alessandro Michelangeli and Raffaele Scandone.

The Evans function is a well known tool for locating spectra of non-self-adjoint differential operators in one spatial dimension widely used to detect stability and instability of traveling and standing waves. A major unsolved problem in many spatial dimensions is to construct its analogue, that is, a function whose zeros and poles would correspond to the eigenvalues of the respective PDE operators. In this work joint with G. Cox and A. Sukhtayev we construct a multidimensional analogue of the Evans function as the modified Fredholm determinant of a ratio of Dirichlet-to-Robin operators on the boundary. This gives a tool for studying the eigenvalue counting functions of second-order elliptic operators that need not be self-adjoint.

We discuss relations between the spectra of the non-self-adjoint elliptic multidimensional differential operators and nonlinear Robin-to-Robin, Robin-to-Dirichlet, and Dirichlet-to-Robin operator pencils. In the self-adjoint case we relate our construction to the Maslov index, another well-known tool in the spectral theory of differential operators. This gives new insight into connections between the Evans function and the Maslov index, allowing us to obtain crucial monotonicity results for the Maslov index.

In this talk, we show the existence of global weak solution of the three-dimensional focusing energy-critical nonlinear Schrödinger equation. To do this, we need to show the global well-posedness of energy-critical Ginzburg-Landau equation in the energy space when the energy is less than the energy of the stationary solution. The global well-posedness of Ginzburg-Landau equation can be reduced to the exclusion of a critical element by the concentration-compactness argument. The critical element is ruled out by dissipation of the Ginzburg-Landau equation, including local smoothness, backwards uniqueness and unique continuation. After establishing the global well-posedness of the Ginzburg-Landau equation, the existence of global weak solution of the three dimensional focusing energy-critical nonlinear Schrödinger equation then follows from the global well-posedness of the energy-critical Ginzburg-Landau equation via a limitation argument.