Modeling chemotaxis with nonstandard production/degradation mechanisms from Doebner–Goldin theory: Existence of solitary waves

Highlights

• Schrödinger and Keller–Segel models are linked by the polar form of the wavefunction.

• Introduction of innovative production–degradation mechanisms of Hamilton–Jacobi type.

• Derivation of a nonstandard Keller–Segel system which admits solitary wave solutions.

Abstract

In this paper we investigate various forms of the chemical production–degradation mechanism in a chemotactic system of Keller–Segel type under which the existence of solitary wave solutions is guaranteed. Specifically, the existence of sech-type or compactly supported solitary wave solutions as well as exponential traveling wave profiles for the cell concentration is shown under certain conditions on the physical parameters.

Introduction

Chemotaxis is the process through which a population of motile cells $n(t,x)$ travels in the direction of the concentration of some chemical agent $S(t,x)$ released to the environment by themselves. From the perspective of partial differential equations, this mechanism is typically modeled by the well-known Keller–Segel system [1] (the interested reader may want to consult the review paper [2], as well as the references therein, for a comprehensive scrutiny of the subject and the mathematical progress made in last years). In a recent paper [3], it has been specified the exact Schrödinger model (cf. Eqs. (2)–(3)) from which the Keller–Segel equations of chemotaxis are deduced in the fluid regime accompanying the modulus-argument decomposition of the wavefunction (see also [4], where a derivation of the quantum hydrodynamic system associated with the most general class of nonlinear Schrödinger equations accounting for Fokker–Planck type diffusion of the probability density was carried out). This fact somehow links the qualitative properties of the solutions to both systems. In particular, the blow-up of solutions to the Keller–Segel model of chemotaxis is connected to the collapse of wavefunctions obeying a dissipative Schrödinger equation with potential nonlinearity. Indeed, the eventual blow-up of solutions depends in both cases on the size of the initial population for dimensions $N=2$ and higher (see for instance [5], [6], [7], [8], [9]).

The main objective of this paper consists of giving new insight on a crucial feature which is not typically available from the standard theory of Keller–Segel models (unless the diffusion mechanism underlying the chemotactic process is nonlinearly modified by an appropriate flux limiter, as for instance in [10], [11], [12], [13], [14], [15], or the chemotactic sensitivity is logarithmic and the consumption mechanism is of the type $f(n,S)=-n{S}^m$, as raised for the first time in the second reference of [1] and more recently in the review paper  [16]), namely the existence of solitary waves. To that aim we just focus on four different forms that the signal production–degradation balance function may take (or eventually a combination of them), inherited from well-known quantum dissipative mechanisms.

We start with the following nonlinear Doebner–Goldin–Schrödinger equation [17], [18], [19], [20]:

$$i{\partial }_t\psi +\frac{1}{2}{\Delta }_x\psi +\sigma n\psi -i\frac{\tau }{2}log\left(\frac{\psi }{\overline{\psi }}\right)\psi =\frac{i{D}_1}{2}\left(\frac{{\Delta }_{x}n}{n}\right)\psi +D^{\prime }{c}_1\left(\frac{{\nabla }_x\cdot J}{n}\right)\psi +D^{\prime }{c}_3\frac{{\left|J\right|}^2}{n^2}\psi +D^{\prime }{c}_4\left(\frac{J\cdot {\nabla }_{x}n}{n^2}\right)\psi +D^{\prime }\left(c_2\frac{{\Delta }_{x}n}{n}+c_5\frac{{\left|{\nabla }_{x}n\right|}^2}{n^2}\right)\psi ,$$

where $n(t,x)={\left|\psi (t,x)\right|}^2$ denotes the probability density associated with the complex wavefunction $\psi$, $J(t,x)=\mathrm{Im}\left(\overline{\psi }(t,x){\nabla }_x\psi (t,x)\right)$ is the electric current, and $\sigma ,\tau ,D_1$ are nonnegative parameters. Here, we denoted $\mathrm{Im}\left(F\right)$ the imaginary part of the complex function $F$. Choosing

$$D^{\prime }{c}_4=-D^{\prime }{c}_1=D_2>0,D^{\prime }{c}_3=c-\frac{1}{2},D^{\prime }{c}_5=-b-\frac{1}{8},D^{\prime }{c}_2=-a+\frac{1}{4},$$

with $a,b,c\in \mathbb{R}$, leads to

$$i{\partial }_t\psi +\frac{1}{2}{\Delta }_x\psi +\sigma n\psi -i\frac{\tau }{2}log\left(\frac{\psi }{\overline{\psi }}\right)\psi ={\mathcal{L}}_{abc}\left[\psi \right],$$

where

$${\mathcal{L}}_{abc}\left[\psi \right]=\left(\left(-a+i\frac{D_1}{2}\right)\frac{{\Delta }_{x}n}{n}-b\frac{{\left|{\nabla }_{x}n\right|}^2}{n^2}-D_2{\nabla }_x\cdot \left(\frac{J}{n}\right)\right.$$$$\left.+\left(c-\frac{1}{2}\right)\frac{{\left|J\right|}^2}{n^2}-Q\right)\psi .$$

Here, $Q$ denotes the quantum Bohm potential representing current arising as a result of density gradient effects, defined as

$$Q=-\frac{{\Delta }_x\sqrt{n}}{2\sqrt{n}}=-\frac{1}{4}\frac{{\Delta }_{x}n}{n}+\frac{1}{8}\frac{{\left|{\nabla }_{x}n\right|}^2}{n^2}.$$

The Doebner–Goldin kernel ${\mathcal{L}}_{abc}$ can be also written as follows:

$${\mathcal{L}}_{abc}\left[\psi \right]=\frac{i{D}_1}{2}\left(\frac{{\Delta }_{x}n}{n}\right)\psi +\left(\left(\frac{1}{4}-a\right)\frac{{\Delta }_{x}n}{n}-\left(\frac{1}{8}+b\right)\frac{{\left|{\nabla }_{x}n\right|}^2}{n^2}\right)\psi$$$$-\left(D_2{\nabla }_x\cdot \left(\frac{J}{n}\right)-\left(c-\frac{1}{2}\right)\frac{{\left|J\right|}^2}{n^2}\right)\psi =\left(\frac{i{D}_1}{2}\left(\frac{{\Delta }_{x}n}{n}\right)+F_{\frac{1}{4}-a,\frac{1}{8}-a-b}(log\left(n\right))-F_{D_2,D_2+\frac{1}{2}-c}\left(S\right)\right)\psi ,$$

with

$$F_{\alpha ,\beta }\left(u\right)=\alpha {\Delta }_{x}u+\beta {\left|{\nabla }_{x}u\right|}^2.$$

Here, we used the fact that $\frac{J}{n}={\nabla }_{x}S$ for a given wavefunction $\psi =\sqrt{n}{e}^{iS}$ (see  [21] for details). Thus, the first term in the right-hand side of Eq. (4) can be interpreted as a Fokker–Planck diffusive mechanism for the probability density ${\left|\psi \right|}^2$, while the two $F_{\alpha ,\beta }$-contributions adopt the form of viscous Hamilton–Jacobi operators acting on $log\left(n\right)$ and $S$, respectively. These fluxes are known to model the evolution of growing interfaces, for instance the growth of various types of aggregates, fronts or tumors, where the Laplacian term describes relaxational diffusion while the gradient square term typically accounts for lateral growth (see [22], [23]). These choices put Eq. (1) in a close perspective to that of the Keller–Segel equations of chemotaxis. As shown in [3], the standard parabolic–parabolic Keller–Segel model

$$\left\{\begin{array}{c}{\partial }_{t}n=D_1{\Delta }_{x}n-{\nabla }_x\cdot \left(n{\nabla }_{x}S\right)\hfill \\ {\partial }_{t}S=D_2{\Delta }_{x}S+n-\tau S\hfill \end{array}\right.,$$

is obtained from Eqs. (2)–(3) after setting $a=b=c=0$ and $\sigma =1$.

Indeed, after inserting the polar decomposition $\psi =\sqrt{n}{e}^{iS}$ in Eqs. (2)–(3) and then separating the real and imaginary parts, the following hydrodynamic system is obtained:

$$\left\{\begin{array}{c}{\partial }_{t}n=D_1{\Delta }_{x}n-{\nabla }_x\cdot \left(n{\nabla }_{x}S\right)\hfill \\ {\partial }_{t}S=D_2{\Delta }_{x}S-\tau S+f_{\sigma abc}(n,{\nabla }_{x}n,{\Delta }_{x}n,{\nabla }_{x}S)\hfill \end{array}\right.,$$

with

$$f_{\sigma abc}(n,{\nabla }_{x}n,{\Delta }_{x}n,{\nabla }_{x}S)=\sigma n+a\frac{{\Delta }_{x}n}{n}+b\frac{{\left|{\nabla }_{x}n\right|}^2}{n^2}+c{\left|{\nabla }_{x}S\right|}^2.$$

Our main purpose is to show the unavoidability of (at least one of) the derivative terms appearing in (7) (up to appropriate gauge transformations) on the right-hand side of the evolution equation for the chemoattractant provided that solitary wave cell profiles are expected to be found. To that purpose, in what follows we just deal with the case $\sigma =\tau =0$ and concentrate our attention on the range of validity and the eventual relations among $a,b,c$.

The paper is organized as follows: In Section 2 we construct solitary waves solving Eqs. (2)–(3) for which the argument (corresponding to the chemical concentration in (6)) is of logarithmic type and the amplitude (corresponding to the cell concentration in (6)) is of sech-type. In Section 3 we show that solitary wave solutions (with linear chemical concentration and again sech-type cell concentration) can also be found when there is no diffusion of the cell population ($D_1=0$). In this case, neither $a\frac{{\Delta }_{x}n}{n}$ nor $b\frac{{\left|{\nabla }_{x}n\right|}^2}{n^2}$ can be dismissed from the system (6). In Section 4 we introduce a change of variables which illustrates how a crossed gradient–gradient term of the type ${\nabla }_{x}log\left(n\right)\cdot {\nabla }_{x}S$ actually behaves like a combination $a\frac{{\Delta }_{x}n}{n}+b\frac{{\left|{\nabla }_{x}n\right|}^2}{n^2}$, thus leading to solitary wave solutions of the type specified before. Finally, in Section 5 we show that the Hamilton–Jacobi term ${\left|{\nabla }_{x}S\right|}^2$ also gives rise to one-dimensional solitary wave solutions in the absence of cell diffusion.